Some topics related to differential forms on R^{n}

Stephen Semmes Rice University

Preface

These informal notes are largely concerned with “calculus on manifolds”, where “manifold” will normally mean something like a reasonably nice submanifold of R^{n} for some positive integer n, perhaps with some sort of reasonable nice boundary. The reader is expected to have some familiarity with basic analysis, at least on Euclidean spaces, as well as linear algebra. Of course, the reader should also be familiar with multivariable calculus, and my colleague Frank Jones’ book [93] is a very helpful resource.

A basic question to be addressed here is

What is a differential form?

Normally we might be concerned with differential forms that depend on a point, and we may also be interested in continuity or differentiability properties of the form, as a function of the point. In order to answer the previous question, a more basic question to be addressed is

What is a differential form at a point?

This involves multilinear functions, which will be discussed starting in Chapter 1.

More precisely, differential forms involve alternating multilinear functions, and we shall look at symmetric multilinear functions some too. This is related to permutations on finite sets, and the notion of even and odd permutations in particular. A bit of group theory is relevant here, but the reader is not necessarily expected to be familiar with that already.

We shall be interested in multiplying differential forms as well. This is related to some other aspects of abstract algebra, and again the reader is not necessarily expected to be familiar with that already. Certain products of multilinear forms will be discussed in Chapter 2, corresponding to multiplying differential forms at a point in particular. Differential forms are formally defined in Chapter 3, along with related continuity and differentiability properties.

In Chapter 2, we shall also consider linear mappings between vector spaces, and their relationship with multilinear functions on these vector spaces. This will be used in Chapter 4 to pull differential forms back using differentiable mappings. This works for some other types of tensor fields too, as we shall see.

There is a particular type of derivative of a differential form known as the exterior derivative, which will be discussed in Chapter 4 as well. This includes exterior derivatives of products, and the relationship between the exterior derivative and pulling differential forms back using differentiable mappings.

We shall sometimes be concerned with n-dimensional volumes of reasonably nice subsets of R^{n}, and integrals of continuous real-valued functions on them. The reader is not necessarily expected to be familiar with Lebesgue measure or integration, which permits one to deal with wider classes of sets and functions. See Frank Jones’ book [92], for instance.

The inverse function theorem is reviewed in Chapter 5, without getting into the proof, except for a few basic points. This includes some preliminary remarks about matrix-valued functions and invertible matrices. Diffeomorphisms between open subsest of R^{n} are discussed as well. Some related aspects of analysis on Euclidean spaces are mentioned too, which will also be helpful later. Among the various topics are some constructions of smooth real-valued functions of one or more variables, and cells in R^{n}.

Chapter 6 is largely concerned with integration of differential forms. This means integration of differential n-forms on R^{n}, and on suitable n-dimensional objects in higher-dimensional Euclidean spaces. We shall also look at the role of the injectivity of the differential of a continuously-differentiable mapping from an open subset of R^{n} into R^{m} when m > n. The implicit function theorem is related to the surjectivity of the differential of a continuously-differentiable mapping from an open subset of R^{m} into R^{n} when m > n. We shall consider whether a diffeomorphism between open subsets of R^{n} preserves or reverses orientations as well.

Chapter 7 deals with n-simplices in R^{m} parameterized by various types of mappings, starting with affine mappings. We also consider n-chains, which correspond to formal sums of finitely many parameterized n-simplices with integer coefficients. The boundary of a parameterized n-simplex can be defined as an (n−1)-chain, for instance. Under suitable conditions, Stokes’ theorem says that the integral of a differential (n − 1)-form ω over the boundary of an n-chain Γ is equal to the integral of the exterior derivative of ω over Γ. This is discussed in the first five sections of Chapter 8.

Vector fields are discussed briefly in Chapter 4, and we return to this in Chapter 8. In particular, we consider Lie brackets of vector fields with suitable regularity on open sets in R^{n}, which corresponds to commutators of the associated first-order differential operators. A version of the Poincaré lemma is discussed as well.

Lie derivatives of functions and vector fields are discussed in Section 8.8, and we continue with Lie derivatives of some other tensor fields in Chapter 9. More precisely, tensor fields of type (0, k) were discussed in Sections 4.1 and 4.2, and vector fields may be considered as tensor fields of type (1, 0), as in Section 4.3. We shall also consider tensor fields of type (1, k) on open sets in R^{n}. Interior products of multilinear forms on a vector space by elements of the vector space are discussed as well.

Some related topics involving linear and abstract algebra, Lie algebras, and complex analysis are discussed in the appendices. A number of books and other references may be found in the bibliography with more information and various perspectives. Of course, there are many other texts dealing with questions like these, and adjacent areas of mathematics.

There are also some places along the way with some elaborations or other matters that may not be neeeded for the moment, and that are related to the broader subject. The reader may want to skip some of these parts, at least temporarily, or simply not dwell on them too much at the beginning.

Contents

1 Some helpful algebra 1 1.1 The binomial theorem . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Polynomials on R^{n} . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2.1 Homogeneous polynomials . . . . . . . . . . . . . . . . . . 2 1.3 More on polynomials . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3.1 More on homogeneous polynomials . . . . . . . . . . . . . 3 1.4 Even and odd permutations . . . . . . . . . . . . . . . . . . . . . 4 1.5 More on permutations . . . . . . . . . . . . . . . . . . . . . . . . 5

1.5.1 An interesting polynomial . . . . . . . . . . . . . . . . . . 6 1.6 Multilinear forms on R^{n} . . . . . . . . . . . . . . . . . . . . . . . 6

1.6.1 Characterizing multilinear forms on R^{n} . . . . . . . . . . 7 1.6.2 A basis for M_{k}(R^{n}) . . . . . . . . . . . . . . . . . . . . . 8

1.7 Bilinear forms on R^{n} . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.7.1 Associated quadratic polynomials . . . . . . . . . . . . . . 9 1.8 Multilinear forms and homogeneous polynomials . . . . . . . . . 10 1.9 Symmetric multilinear forms . . . . . . . . . . . . . . . . . . . . 11

1.9.1 Symmetrizing multilinear forms . . . . . . . . . . . . . . . 12 1.10 More on symmetric multilinear forms . . . . . . . . . . . . . . . . 13

1.10.1 The space SM_{k}(R^{n}) . . . . . . . . . . . . . . . . . . . . . 14 1.11 Alternating multilinear forms . . . . . . . . . . . . . . . . . . . . 14

1.11.1 The space AM_{k}(R^{n}) . . . . . . . . . . . . . . . . . . . . 15 1.12 More on alternating multilinear forms . . . . . . . . . . . . . . . 15

1.12.1 The dimension of AM_{k}(R^{n}) . . . . . . . . . . . . . . . . 16 1.13 Alternatizing multilinear forms . . . . . . . . . . . . . . . . . . . 17 1.14 The multinomial theorem . . . . . . . . . . . . . . . . . . . . . . 18 1.15 Matrices and determinants . . . . . . . . . . . . . . . . . . . . . . 19

2 More on multilinear forms 22

2.1 Some direct sums . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.1.1 Direct sums and linear mappings . . . . . . . . . . . . . . 23 2.2 Multilinear mappings . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2.1 Some basic properties and examples . . . . . . . . . . . . 24 2.3 Some compositions . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.4 Compositions and permutations . . . . . . . . . . . . . . . . . . . 25

2.4.1 Symmetrizations and alternatizations . . . . . . . . . . . 27 2.5 Compositions and determinants . . . . . . . . . . . . . . . . . . . 27

2.5.1 n-Dimensional vector spaces . . . . . . . . . . . . . . . . . 28 2.6 Subgroups of Sym(k) . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.6.1 Associated symmetrizations and alternatizations . . . . . 30 2.7 Permutations on subsets . . . . . . . . . . . . . . . . . . . . . . . 31 2.8 Products of multilinear forms . . . . . . . . . . . . . . . . . . . . 31

2.8.1 Polynomials, associativity, and linear mappings . . . . . . 32 2.8.2 Products and permutations . . . . . . . . . . . . . . . . . 33

2.9 Symmetry and products . . . . . . . . . . . . . . . . . . . . . . . 34

2.9.1 Some symmetrizations of products . . . . . . . . . . . . . 35 2.10 Alternatizing products . . . . . . . . . . . . . . . . . . . . . . . . 36

2.10.1 The ∧_{0} product . . . . . . . . . . . . . . . . . . . . . . . . 36 2.11 The wedge product . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.12 More on wedge products . . . . . . . . . . . . . . . . . . . . . . . 38

2.12.1 Linear mappings and AM(W) . . . . . . . . . . . . . . . 39 2.13 Some more subgroups of Sym(k) . . . . . . . . . . . . . . . . . . 40 2.14 Subgroups and products of forms . . . . . . . . . . . . . . . . . . 42 2.15 Some related shuffles . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.15.1 The set of shuffles . . . . . . . . . . . . . . . . . . . . . . 44 2.15.2 Symmetrization, alternatization, and shuffles . . . . . . . 44

3 Some geometry and analysis 46 3.1 Metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.2 Norms on vector spaces . . . . . . . . . . . . . . . . . . . . . . . 47 3.3 Inner product spaces . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.4 Continuous functions on R^{n} . . . . . . . . . . . . . . . . . . . . . 49

3.4.1 Open balls and open sets . . . . . . . . . . . . . . . . . . 51 3.5 Partial derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.5.1 Continuous differentiability . . . . . . . . . . . . . . . . . 52 3.5.2 Second derivatives . . . . . . . . . . . . . . . . . . . . . . 52

3.6 Additional regularity . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.6.1 More on r-times continuous differentiability . . . . . . . . 54 3.7 Smooth functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.7.1 Compositions with functions on R . . . . . . . . . . . . . 55 3.7.2 A smooth function on R . . . . . . . . . . . . . . . . . . . 56

3.8 Differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.8.1 Directional derivatives . . . . . . . . . . . . . . . . . . . . 57 3.8.2 Some properties of the differential . . . . . . . . . . . . . 57

3.9 The chain rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.9.1 Additional regularity of compositions . . . . . . . . . . . . 59 3.10 A basis for AM_{k}(R^{n}) . . . . . . . . . . . . . . . . . . . . . . . . 60 3.11 A basis for AM(R^{n}) . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.12 Differential forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.12.1 Continuous differential forms . . . . . . . . . . . . . . . . 65 3.13 Additional regularity of differential forms . . . . . . . . . . . . . 66 3.14 Differentials as 1-forms . . . . . . . . . . . . . . . . . . . . . . . . 68 3.15 Isometric mappings . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4 Pull-backs and exterior differentiation 71

4.1 Tensors of type (0, k) . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.1.1 Continuous tensor fields . . . . . . . . . . . . . . . . . . . 72 4.2 More on tensor fields . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.3 Vector fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.3.1 Associated differential operators . . . . . . . . . . . . . . 74 4.4 Pulling some tensors back . . . . . . . . . . . . . . . . . . . . . . 75

4.4.1 Some properties of pull-backs . . . . . . . . . . . . . . . . 76 4.5 More on pull-backs . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.5.1 Pull-backs and composiitons . . . . . . . . . . . . . . . . 77 4.5.2 Regularity of pull-backs . . . . . . . . . . . . . . . . . . . 78

4.6 Exterior differentiation . . . . . . . . . . . . . . . . . . . . . . . . 78 4.7 The anti-derivation property . . . . . . . . . . . . . . . . . . . . 79

4.7.1 The cases where k, l ≥ 1 . . . . . . . . . . . . . . . . . . . 80 4.8 The second exterior derivative . . . . . . . . . . . . . . . . . . . . 81 4.9 Differentiating pull-backs . . . . . . . . . . . . . . . . . . . . . . 82

4.9.1 Some basic cases . . . . . . . . . . . . . . . . . . . . . . . 83 4.9.2 The rest of the proof . . . . . . . . . . . . . . . . . . . . . 84

4.10 Connected sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.10.1 Convex sets . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.10.2 Locally constant functions . . . . . . . . . . . . . . . . . . 85

4.11 Path-connected sets . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.11.1 Path-connected components . . . . . . . . . . . . . . . . . 87 4.12 Volumes and determinants . . . . . . . . . . . . . . . . . . . . . . 87

4.12.1 Orthogonal transformations . . . . . . . . . . . . . . . . . 88 4.13 Admissible volumes . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.13.1 Admissible volumes and linear mappings . . . . . . . . . . 90 4.14 Parallelepipeds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.14.1 Parallelepipeds and admissible volumes . . . . . . . . . . 91 4.14.2 Admissible volumes and alternating forms . . . . . . . . . 91

4.15 Volumes and inner products . . . . . . . . . . . . . . . . . . . . . 92

5 Diffeomorphisms and cells 93

5.1 Matrix-valued functions . . . . . . . . . . . . . . . . . . . . . . . 93 5.1.1 Products of matrix-valued functions . . . . . . . . . . . . 94 5.1.2 C^{r} Matrix-valued functions . . . . . . . . . . . . . . . . . 94

5.2 General linear groups . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.2.1 GL(m, R) And the determinant . . . . . . . . . . . . . . . 95 5.3 Some spaces of linear mappings . . . . . . . . . . . . . . . . . . . 96

5.3.1 C^{r} L(R^{m_{2}}, R^{m_{1}})-Valued functions . . . . . . . . . . . . . 97 5.4 More on linear mappings . . . . . . . . . . . . . . . . . . . . . . . 97

5.4.1 Linear mappings on R^{m} . . . . . . . . . . . . . . . . . . . 98 5.5 Continuity of the differential . . . . . . . . . . . . . . . . . . . . . 99 5.6 The inverse function theorem . . . . . . . . . . . . . . . . . . . . 100 5.7 Closed sets and limit points . . . . . . . . . . . . . . . . . . . . . 101 5.8 Compact sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.8.1 Continuity and compactness . . . . . . . . . . . . . . . . . 104 5.9 More on continuous mappings . . . . . . . . . . . . . . . . . . . . 105

5.9.1 Homeomorphisms . . . . . . . . . . . . . . . . . . . . . . . 105 5.10 Homeomorphisms and open mappings . . . . . . . . . . . . . . . 106

5.10.1 A sufficient condition . . . . . . . . . . . . . . . . . . . . 107 5.11 C^{1} Diffeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.11.1 The definition of a diffeomorphism . . . . . . . . . . . . . 109 5.11.2 The n = 1 case . . . . . . . . . . . . . . . . . . . . . . . . 109

5.12 Diffeomorphisms with more regularity . . . . . . . . . . . . . . . 110 5.12.1 Regularity of the inverse mapping . . . . . . . . . . . . . 111 5.12.2 C^{r} And C^{∞} diffeomorphisms . . . . . . . . . . . . . . . . 111

5.13 Cells and supports of functions . . . . . . . . . . . . . . . . . . . 112 5.13.1 Integrals over cells . . . . . . . . . . . . . . . . . . . . . . 113 5.13.2 Supports of functions . . . . . . . . . . . . . . . . . . . . 114

5.14 Some smooth functions on R . . . . . . . . . . . . . . . . . . . . 114 5.15 Some smooth functions on R^{n} . . . . . . . . . . . . . . . . . . . . 116

6 Integration and n-surfaces 118 6.1 Some partitions of unity . . . . . . . . . . . . . . . . . . . . . . . 118 6.2 The standard simplex in R^{n} . . . . . . . . . . . . . . . . . . . . . 119

6.2.1 More on integration over Q^{n} . . . . . . . . . . . . . . . . 121 6.3 Jacobians and changes of variables . . . . . . . . . . . . . . . . . 122 6.4 Another version of changing variables . . . . . . . . . . . . . . . 123 6.5 Some remarks about linear independence . . . . . . . . . . . . . 124

6.5.1 Considering L as a graph . . . . . . . . . . . . . . . . . . 126 6.6 Some nice local graphs . . . . . . . . . . . . . . . . . . . . . . . . 126

6.6.1 Using the local inverse . . . . . . . . . . . . . . . . . . . . 127 6.7 Injective linear mappings . . . . . . . . . . . . . . . . . . . . . . 128 6.8 Embeddings and immersions . . . . . . . . . . . . . . . . . . . . 129

6.8.1 Some remarks about manifolds . . . . . . . . . . . . . . . 130 6.9 Surjective linear mappings . . . . . . . . . . . . . . . . . . . . . . 131 6.10 The implicit function theorem . . . . . . . . . . . . . . . . . . . . 132

6.10.1 The zero set of ϕ . . . . . . . . . . . . . . . . . . . . . . . 133 6.11 Injectivity and alternating forms . . . . . . . . . . . . . . . . . . 134

6.11.1 Injectivity of the differential . . . . . . . . . . . . . . . . . 136 6.12 Integrating n-forms on R^{n} . . . . . . . . . . . . . . . . . . . . . . 136

6.12.1 A family of integrals . . . . . . . . . . . . . . . . . . . . . 137 6.13 More on changes of variables . . . . . . . . . . . . . . . . . . . . 138

6.13.1 Integrals and diffeomorphisms . . . . . . . . . . . . . . . . 139 6.14 Diffeomorphisms and orientations . . . . . . . . . . . . . . . . . . 140

6.14.1 Another version . . . . . . . . . . . . . . . . . . . . . . . . 141 6.15 n-Surfaces in R^{m} . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

6.15.1 Integration over n-surfaces . . . . . . . . . . . . . . . . . 142 6.15.2 Changing variables in R^{n} . . . . . . . . . . . . . . . . . . 143 6.15.3 Mapping V into R^{p} . . . . . . . . . . . . . . . . . . . . . 144

7 Simplices and chains 145 7.1 Affine mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 7.2 Some remarks about determinants . . . . . . . . . . . . . . . . . 146

7.2.1 The case where τ(0) ≠ 0 . . . . . . . . . . . . . . . . . . . 147 7.3 Affine simplices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 7.4 Some affine mappings on R^{n} . . . . . . . . . . . . . . . . . . . . 150

7.4.1 Some affine mappings from permutations . . . . . . . . . 150 7.5 Integrals over affine simplices . . . . . . . . . . . . . . . . . . . . 151

7.5.1 Orientations of affine simplices . . . . . . . . . . . . . . . 152 7.5.2 0-Simplices . . . . . . . . . . . . . . . . . . . . . . . . . . 153

7.6 Affine chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 7.6.1 Integrals over affine chains . . . . . . . . . . . . . . . . . . 154 7.6.2 Defining ≃ for affine chains . . . . . . . . . . . . . . . . . 154

7.7 Boundaries of affine simplices . . . . . . . . . . . . . . . . . . . . 155

7.7.1 Boundaries of affine chains . . . . . . . . . . . . . . . . . 156 7.8 Boundaries and permutations . . . . . . . . . . . . . . . . . . . . 157

7.8.1 Permuting vertices of affine simplices . . . . . . . . . . . . 158 7.9 C^{r} Simplices and chains . . . . . . . . . . . . . . . . . . . . . . . 159

7.9.1 C^{r} chains . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 7.10 Simplices and compositions . . . . . . . . . . . . . . . . . . . . . 161

7.10.1 Chains and compositions . . . . . . . . . . . . . . . . . . 161 7.11 Boundaries of C^{r} simplices . . . . . . . . . . . . . . . . . . . . . 162

7.11.1 C^{r} Simplices as compositions . . . . . . . . . . . . . . . . 162 7.11.2 Boundaries of C^{r} chains . . . . . . . . . . . . . . . . . . . 164

7.12 Compositions and affine simplices . . . . . . . . . . . . . . . . . . 164 7.12.1 Defining ≃ for C^{r} simplices . . . . . . . . . . . . . . . . . 165 7.12.2 Defining ≃ for C^{r} chains . . . . . . . . . . . . . . . . . . 165

7.13 Some remarks about n-chains . . . . . . . . . . . . . . . . . . . . 165 7.13.1 Linear functionals on differential forms . . . . . . . . . . . 166 7.13.2 Defining ≅ for C^{1} n-chains . . . . . . . . . . . . . . . . . 166 7.13.3 Comparing ≃ and ≅ . . . . . . . . . . . . . . . . . . . . . 167

7.14 An associativity property . . . . . . . . . . . . . . . . . . . . . . 167

7.14.1 Associativity and chains . . . . . . . . . . . . . . . . . . . 168 7.15 Compositions and boundaries . . . . . . . . . . . . . . . . . . . . 168

7.15.1 Boundaries of boundaries . . . . . . . . . . . . . . . . . . 169

8 Stokes’ theorem and other matters 170

8.1 Stokes’ theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

8.1.1 Stokes’ theorem and ≅ . . . . . . . . . . . . . . . . . . . . 171 8.2 More on Stokes’ theorem . . . . . . . . . . . . . . . . . . . . . . . 171 8.3 Stokes’ theorem for Φ_{0} . . . . . . . . . . . . . . . . . . . . . . . . 173

8.3.1 The integral over ∂Φ_{0} . . . . . . . . . . . . . . . . . . . . 173 8.3.2 The integral over Φ_{0} . . . . . . . . . . . . . . . . . . . . . 174 8.3.3 Matching a pair of terms . . . . . . . . . . . . . . . . . . 175 8.3.4 Matching the other two terms . . . . . . . . . . . . . . . . 176

8.4 A related argument . . . . . . . . . . . . . . . . . . . . . . . . . . 177 8.5 Stokes’ theorem on cells . . . . . . . . . . . . . . . . . . . . . . . 178 8.6 More on vector fields . . . . . . . . . . . . . . . . . . . . . . . . . 180

8.6.1 Commutators and Lie brackets . . . . . . . . . . . . . . . 181 8.7 Some more compositions . . . . . . . . . . . . . . . . . . . . . . . 182

8.7.1 The Jacobi identity for brackets . . . . . . . . . . . . . . . 182 8.8 Lie derivatives of functions and vector fields . . . . . . . . . . . . 183 8.9 ϕ-Related vector fields . . . . . . . . . . . . . . . . . . . . . . . . 184

8.9.1 Brackets of ϕ-related vector fields . . . . . . . . . . . . . . 185 8.10 Polynomial vector fields . . . . . . . . . . . . . . . . . . . . . . . 186 8.11 Cell-nice sets in R^{n} . . . . . . . . . . . . . . . . . . . . . . . . . . 186 8.12 Integrating some functions . . . . . . . . . . . . . . . . . . . . . . 188

8.12.1 Some properties of these functions . . . . . . . . . . . . . 189 8.13 A version of Poincaré’s lemma . . . . . . . . . . . . . . . . . . . . 190

8.13.1 Invariance under diffeomorphisms . . . . . . . . . . . . . . 190 8.14 Some initial steps . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

8.14.1 The induction hypothesis . . . . . . . . . . . . . . . . . . 192 8.15 The induction step . . . . . . . . . . . . . . . . . . . . . . . . . . 192

8.15.1 Using the induction hypothesis . . . . . . . . . . . . . . . 193

9 Lie derivatives and tensor fields 195

9.1 Tensor fields and multilinearity . . . . . . . . . . . . . . . . . . . 195 9.1.1 Multilinearity over C(E, R) . . . . . . . . . . . . . . . . . 196 9.1.2 Multilinearity over C^{r}(U, R) . . . . . . . . . . . . . . . . 196

9.2 Spaces of multilinear mappings . . . . . . . . . . . . . . . . . . . 197

9.2.1 Multilinear mappings and R^{n} . . . . . . . . . . . . . . . . 198 9.3 Tensor fields of type (1, k) . . . . . . . . . . . . . . . . . . . . . . 198 9.4 Some more multilinearity conditions . . . . . . . . . . . . . . . . 200 9.5 Some more Lie derivatives . . . . . . . . . . . . . . . . . . . . . . 201 9.6 Using nonconstant vector fields . . . . . . . . . . . . . . . . . . . 202 9.7 Another family of Lie derivatives . . . . . . . . . . . . . . . . . . 203

9.7.1 Using nonconstant vector fields again . . . . . . . . . . . 204 9.8 Lie derivatives as representations . . . . . . . . . . . . . . . . . . 205 9.9 Some more representations . . . . . . . . . . . . . . . . . . . . . 206 9.10 Some properties of Lie derivatives . . . . . . . . . . . . . . . . . . 207

9.10.1 Lie derivatives of products . . . . . . . . . . . . . . . . . . 208 9.10.2 Lie derivatives and exterior differentiation . . . . . . . . . 209 9.10.3 Tensors of type (1, k) . . . . . . . . . . . . . . . . . . . . . 209

9.11 Some interior products . . . . . . . . . . . . . . . . . . . . . . . . 210

9.11.1 Connection with products of forms . . . . . . . . . . . . . 211 9.12 Interior products and symmetrizations . . . . . . . . . . . . . . . 211

9.12.1 Connection with symmetric products . . . . . . . . . . . . 213 9.13 Interior products and polynomials . . . . . . . . . . . . . . . . . 213 9.14 Interior multiplication on AM(W) . . . . . . . . . . . . . . . . . 214

9.14.1 Checking the anti-derivation property . . . . . . . . . . . 214 9.15 More on interior multiplication . . . . . . . . . . . . . . . . . . . 216

9.15.1 Some related linear mappings . . . . . . . . . . . . . . . . 216 9.15.2 Connections with Lie derivatives . . . . . . . . . . . . . . 217

Appendices 218

A Some linear and abstract algebra 218 A.1 Some remarks about vector spaces . . . . . . . . . . . . . . . . . 218 A.2 Algebras over the real numbers . . . . . . . . . . . . . . . . . . . 219

A.2.1 Algebra homomorphisms . . . . . . . . . . . . . . . . . . . 219 A.3 More on vector spaces . . . . . . . . . . . . . . . . . . . . . . . . 220

A.3.1 Dual linear mappings . . . . . . . . . . . . . . . . . . . . 221 A.3.2 Direct sums and dual spaces . . . . . . . . . . . . . . . . 221

A.4 Some direct sums and products . . . . . . . . . . . . . . . . . . . 222

A.4.1 Sums, products, and dual spaces . . . . . . . . . . . . . . 223 A.5 Some remarks about linear mappings . . . . . . . . . . . . . . . . 224

A.5.1 Linear mappings and direct sums . . . . . . . . . . . . . . 224 A.5.2 Linear mappings and direct products . . . . . . . . . . . . 225 A.5.3 Some more direct sums . . . . . . . . . . . . . . . . . . . 225

A.6 The algebra of multilinear forms . . . . . . . . . . . . . . . . . . 226 A.6.1 The definition of SM(W) . . . . . . . . . . . . . . . . . . 227 A.6.2 Polynomials and M(R^{n}) . . . . . . . . . . . . . . . . . . 227

A.7 More on AM(W) . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 A.8 More on inner products . . . . . . . . . . . . . . . . . . . . . . . 229 A.9 Inner products and adjoints . . . . . . . . . . . . . . . . . . . . . 230

A.9.1 Some properties of the adjoint . . . . . . . . . . . . . . . 231 A.10 Nonnegativity and self-adjointness . . . . . . . . . . . . . . . . . 232

A.10.1 A family of examples . . . . . . . . . . . . . . . . . . . . . 233 A.11 Adjoints and volumes . . . . . . . . . . . . . . . . . . . . . . . . 233

A.11.1 Another inner product space . . . . . . . . . . . . . . . . 234 A.12 Traces and transposes of matrices . . . . . . . . . . . . . . . . . . 235

A.12.1 Traces of linear mappings . . . . . . . . . . . . . . . . . . 236

B Lie algebras 238

B.1 The definition and some examples . . . . . . . . . . . . . . . . . 238

B.1.1 Some examples of Lie algebras . . . . . . . . . . . . . . . 239 B.2 Subalgebras of Lie algebras . . . . . . . . . . . . . . . . . . . . . 240 B.3 Some linear mappings . . . . . . . . . . . . . . . . . . . . . . . . 241 B.4 Some more Lie subalgebras . . . . . . . . . . . . . . . . . . . . . 243 B.5 Some multiplication operators . . . . . . . . . . . . . . . . . . . . 244

B.5.1 Multiplication operators on associative algebras . . . . . . 245 B.6 Ideals in algebras over R . . . . . . . . . . . . . . . . . . . . . . . 246

B.6.1 Ideals in Lie algebras . . . . . . . . . . . . . . . . . . . . . 247 B.7 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

B.7.1 Representations of associative algebras . . . . . . . . . . . 248

B.7.2 Right modules over associative algebras . . . . . . . . . . 249 B.8 More on representations . . . . . . . . . . . . . . . . . . . . . . . 249

B.8.1 Subrepresentations . . . . . . . . . . . . . . . . . . . . . . 249 B.8.2 Direct sums of representations . . . . . . . . . . . . . . . 250 B.8.3 Representations and dual spaces . . . . . . . . . . . . . . 251 B.8.4 Commuting representations of Lie algebras . . . . . . . . 252

B.9 Representations and multilinear mappings . . . . . . . . . . . . . 252

B.9.1 Representations on M_{k}(V, Z) . . . . . . . . . . . . . . . . 253 B.10 Homomorphisms between representations . . . . . . . . . . . . . 254

B.10.1 Modules over associative algebras . . . . . . . . . . . . . . 255 B.11 Invariant elements . . . . . . . . . . . . . . . . . . . . . . . . . . 256

B.11.1 Invariant linear mappings . . . . . . . . . . . . . . . . . . 256 B.11.2 Invariant bilinear forms . . . . . . . . . . . . . . . . . . . 257

C Complex numbers and complex analysis 258

C.1 Complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 258

C.1.1 Complex vector spaces . . . . . . . . . . . . . . . . . . . . 259 C.2 Complex derivatives . . . . . . . . . . . . . . . . . . . . . . . . . 260

C.2.1 More on holomorphic functions . . . . . . . . . . . . . . . 260 C.2.2 Some related differential operators . . . . . . . . . . . . . 261

C.3 Complex differential forms . . . . . . . . . . . . . . . . . . . . . . 262 C.3.1 Complex differential forms on C . . . . . . . . . . . . . . 262 C.3.2 Some closed forms on C . . . . . . . . . . . . . . . . . . . 263

Bibliography 264

Index 276

Chapter 1

Some helpful algebra

1.1 The binomial theorem

Let n be a positive integer, and let x, y be real numbers. It is well known that

(x + y)^{n} = ∑_{j=0}^{n} \binom{n}{j} x^{j} y^{n−j}, (1.1.1)

where the binomial coefficients \binom{n}{j} are given by

\binom{n}{j} = \frac{n!}{j! (n − j)!}. (1.1.2)

Of course, if l is a positive integer, then

l! = ∏_{k=1}^{l} k, (1.1.3)

which is interpreted as being equal to 1 when l = 0.

It is easy to see that (x + y)^{n} can be expressed as in the right side of (1.1.1) for some coefficients \binom{n}{j}, which are positive integers. The expression (1.1.2) for these coefficients can be verified using induction. This can also be obtained using calculus. One can check that these are the only values of the coefficients for which (1.1.1) holds for all x, y. Note that \binom{n}{j} is the same as the number of ways of choosing j elements from a set with n elements.

1.2 Polynomials on R^{n}

The real line is denoted R, as usual. Let n be a positive integer, and let R^{n} be the space of n-tuples x = (x_{1}, …, x_{n}) of real numbers. If x, y ∈ R^{n}, then x + y ∈ R^{n} is defined by adding the coordinates of x and y, so that

x + y = (x_{1} + y_{1}, …, x_{n} + y_{n}). (1.2.1)

Similarly, if x ∈ R^{n} and t ∈ R, then t x ∈ R^{n} is defined by multiplying the coordinates of x by t, so that

t x = (t x_{1}, …, t x_{n}). (1.2.2)

It is well known that R^{n} is a vector space over the real numbers with respect to these operations of addition and scalar multiplication.

In order to talk about polynomials on R^{n}, it will be helpful to introduce some additional notation and terminology. By a multi-index we mean an n-tuple α = (α_{1}, …, α_{n}) of nonnegative integers. If α is a multi-index and x ∈ R^{n}, then we put

x^{α} = ∏_{j=1}^{n} x_{j}^{α_{j}}. (1.2.3)

Here x_{j}^{α_{j}} is interpreted as being equal to 1 when α_{j} = 0, even if x_{j} = 0. We may refer to x^{α} as the monomial associated to α.

A polynomial on R^{n} may be expressed as a linear combination of finitely many monomials. This will be discussed further in the next section.

1.2.1 Homogeneous polynomials

Let d be a nonnegative integer, and suppose for the moment that

∑_{j=1}^{n} α_{j} = d. (1.2.4)

Put f_{α}(x) = x^{α} (1.2.5)

for each x ∈ R^{n}, for convenience. Observe that

f_{α}(t x) = t^{d} f_{α}(x) (1.2.6)

for every t ∈ R and x ∈ R^{n}. Thus f_{α} is homogeneous of degree d on R^{n}. A polynomial on R^{n} is homogeneous of degree d if it can be expressed as a linear combination of monomials x^{α} corresponding to multi-indices α that satisfy (1.2.4).

It is well known that the number of multi-indices α satisfying (1.2.4) is equal to

\binom{n + d − 1}{n − 1}. (1.2.7)

This is the same as the number of ways of choosing n − 1 elements from a set of n + d − 1 elements, as mentioned in the previous section. More precisely, suppose that we have a set of n + d − 1 elements that is linearly ordered, like the set of positive integers from 1 to n + d − 1. If we choose n − 1 elements from this set, then the other d elements of the set can be partitioned into n subsets in a natural way, some of which may be empty. The numbers of the elements of these n subsets correspond exactly to a multi-index α that satisfies (1.2.4).

1.3 More on polynomials

Let n be a positive integer, and let N be a nonnegative integer. A polynomial on R^{n} of degree less than or equal to N with real coefficients may be expressed as

p(x) = ∑_{α_{1}+···+α_{n}≤N} c_{α} x^{α}. (1.3.1)

More precisely, the sum is taken over all multi-indices α with

α_{1} + ⋯ + α_{n} ≤ N, (1.3.2)

and it is easy to see that there are only finitely many such multi-indices. The coefficient c_{α} is supposed to be a real number for each of these multi-indices α, so that (1.3.1) defines a real-valued function on R^{n}.

It is well known that the coefficients c_{α} are uniquely determined by the corresponding function p(x) on R^{n}. This can be obtained by considering p and its derivatives at 0.

Let P(R^{n}) be the space of all polynomials on R^{n} with real coefficients. This is a linear subspace of the space of all real-valued functions on R^{n}, so that P(R^{n}) may be considered as a vector space over the real numbers with respect to pointwise addition and scalar multiplication of functions. Of course, pointwise addition and scalar multiplication of polynomials on R^{n} corresponds exactly to termwise addition and scalar multiplications of sums as in the right side of (1.3.1).

1.3.1 More on homogeneous polynomials

Let d be a nonnegative integer, and suppose that a polynomial p on R^{n} can be expressed as

p(x) = ∑_{α_{1}+···+α_{n}=d} c_{α} x^{α}, (1.3.3)

where more precisely the sum is taken over all multi-indices α with

α_{1} + ⋯ + α_{n} = d, (1.3.4)

and the coefficients c_{α} are real numbers. This implies that p is homogeneous of degree d on R^{n}, in the sense that

p(t x) = t^{d} p(x) (1.3.5)

for every t ∈ R and x ∈ R^{n}, as in Subsection 1.2.1. Conversely, if a polynomial p on R^{n} is homogeneous of degree d, then one can check that p can be expressed as in (1.3.3).

Of course, homogeneous polynomials on R^{n} of degree 0 are the same as constant functions on R^{n}. A homogeneous polynomial of degree 1 on R^{n} is the same as a linear functional on R^{n}, which is to say a linear mapping from R^{n} into R.

Let P_{d}(R^{n}) be the space of polynomials on R^{n} with real coefficients that are homogeneous of degree d. This is a linear subspace of P(R^{n}). The collection of monomials x^{α}, where α is a multi-index that satisfies (1.3.4), is a basis for P_{d}(R^{n}), as a vector space over the real numbers. Thus

(1.3.6) the dimension of P_{d}(R^{n}) is equal to (1.2.7),

as a vector space over the real numbers.

Let p_{1}, p_{2} be polynomials on R^{n} with real coefficients of degrees ≤ N_{1}, N_{2}, respectively. It is easy to see that their product

p_{1}(x) p_{2}(x) (1.3.7)

is a polynomial on R^{n} of degree ≤ N_{1}+N_{2}. If p_{1}, p_{2} are homogeneous of degrees d_{1}, d_{2}, respectively, then

(1.3.8) p_{1}(x) p_{2}(x) is homogeneous of degree d_{1} + d_{2}.

1.4 Even and odd permutations

If t is a real number, then let us use sign(t) to denote the sign of t, which is 1 when t > 0, −1 when t < 0, and 0 when t = 0. Let n be a positive integer, and let σ be a mapping from the set {1, …, n} of positive integers from 1 to n into itself. Put

sgn(σ) = ∏_{1≤j<l≤n} sign(σ(l) − σ(j)), (1.4.1)

which may be called the signum of σ, as on p319 of [20]. More precisely, the product is taken over all pairs of integers j, l satisfying the conditions indicated. If σ is not one-to-one on {1, …, n}, then

sgn(σ) = 0. (1.4.2)

Suppose that σ is one-to-one on {1, …, n}, so that

sgn(σ) = 1 or − 1. (1.4.3)

In this case, σ maps {1, …, n} onto itself, basically because {1, …, n} is a finite set. We say that σ is a permutation on {1, …, n}, and the inverse mapping σ^{−1} is a permutation on {1, …, n} as well. One can verify that

sgn(σ) = sgn(σ^{−1}). (1.4.4)

Note that sgn(σ) = 1 when σ is the identity mapping on {1, …, n}.

Sometimes sgn(σ) is only defined for permutations σ, and this definition for all mappings σ from {1, …, n} is used in Definition 9.33 on p232 of [155]. This will also be helpful for considering some properties of related mappings. If σ, τ are mappings from {1, …, n} into itself, then their composition τ ∘ σ is the mapping from {1, …, n} into itself that sends j to τ(σ(j)). This is a permutation on {1, …, n} when σ and τ are both permuations. Note that

(1.4.5) there are n^{n} mappings from {1, …, n} into itself,

and (1.4.6) there are n! permutations on {1, …, n}.

A permutation τ on {1, …, n} is said to be a transposition if it interchanges two elements of {1, …, n}, and maps every other element of {1, …, n} to itself. One can check that sgn(τ) = −1 (1.4.7)

when τ is a transposition. A permutation σ on {1, …, n} is said to be even when sgn(σ) = 1, and odd when sgn(σ) = −1. It is well known that every permutation on {1, …, n} can be expressed as the composition of finitely many transpositions.

If σ, τ are mappings from {1, …, n} into itself, then it is well known that

(1.4.8) sgn(τ ∘ σ) = sgn(τ) sgn(σ).

This is easy to see when either σ or τ is not one-to-one on {1, …, n}, which implies that τ ∘ σ is not one-to-one on {1, …, n}, so that both sides of (1.4.8) are equal to 0. A proof of (1.4.8) when σ and τ are both permutations will be given in the next section.

1.5 More on permutations

Let n be a positive integer, and let σ be a mapping from {1, …, n} into itself. If x ∈ R^{n}, then let T_{σ}(x) be the element of R^{n} whose jth coordinate is equal to (T_{σ}(x))_{j} = x_{σ(j)} (1.5.1)

for each j = 1, …, n. This defines a linear mapping from R^{n} into itself. If σ is not one-to-one on {1, …, n}, then T_{σ} is not one-to-one on R^{n}. If σ is a permutation on {1, …, n}, then T_{σ} is a one-to-one mapping from R^{n} onto itself, with (T_{σ})^{−1} = T_{σ^{−1}}. (1.5.2)

Let τ be another mapping from {1, …, n} into itself. If x ∈ R^{n}, then the jth coordinate of T_{σ}(T_{τ} (x)) is equal to

(T_{σ}(T_{τ} (x)))_{j} = (T_{τ} (x))_{σ(j)} = x_{τ(σ(j))} = (T_{τ∘σ}(x))_{j} (1.5.3)

for each j = 1, …, n. This means that

T_{σ} ∘ T_{τ} = T_{τ∘σ}. (1.5.4)

1.5.1 An interesting polynomial

Consider the polynomial P (x) defined on R^{n} by

P (x) = ∏_{1≤j<l≤n}(x_{l} − x_{j}). (1.5.5)

This is a homogeneous polynomial of degree n (n − 1)/2, which is the number of factors in the product on the right side. Observe that

P (T_{σ}(x)) = ∏_{1≤j<l≤n}((T_{σ}(x))_{l} − (T_{σ}(x))_{j}) = ∏_{1≤j<l≤n}(x_{σ(l)} − x_{σ(j)}) (1.5.6)

for every x ∈ R^{n}. One can use this to get that

P (T_{σ}(x)) = sgn(σ) P (x) (1.5.7)

for every x ∈ R^{n}. In particular, (1.5.6) is equal to 0 when σ is not one-to-one on {1, …, n}.

On p153 of [20], a polynomial is defined which is the same as (1.5.5), except for a possible factor of −1, depending on the dimension. This possible extra factor is not important for (1.5.7). This polynomial is used to define even and odd permuations, and sgn(σ) is defined for a permutation using that on p319 of [20]. Note that sgn(σ) is uniquely determined by (1.5.7).

One can get (1.4.8) from (1.5.4) and (1.5.7). This corresponds to the properties of compositions of even and odd permuations in (12) on p153 of [20]. This also corresponds to Theorem 17 on p66 of [125], with somewhat different notation.

1.6 Multilinear forms on R^{n}

Let k and n be positive integers. In this section, we shall be concerned with real-valued functions on the Cartesian product

R^{n} × ⋯ × R^{n}, (1.6.1)

with k factors of R^{n}. This may also be expressed as (R^{n})^{k}, the space of k-tuples of elements of R^{n}. A real-valued function on this space may be expressed as μ(x(1), …, x(k)), where x(1), …, x(k) are elements of R^{n}. The jth coordinate of x(l) may be expressed as x_{j}(l) for each j = 1, …, n and l = 1, …, k.

We say that μ(x(1), …, x(k)) is multilinear on (R^{n})^{k} if it is linear in x(l) for each l = 1, …, k, when x(r) is fixed for r ≠ l. One may also say that μ is k-linear in this case, to emphasize the role of k. One may call μ a multilinear form or k-linear form on R^{n} as well, to emphasize the role of n and perhaps k too, and the fact that μ takes values in R. If k = 1, then this is the same as a linear mapping from R^{n} into R, which is to say a linear functional on R^{n}. If k = 2, then this is the same as a bilinear form on R^{n}.

1.6. MULTILINEAR FORMS ON R^{N} 7

Let λ_{1}, …, λ_{k} be linear functionals on R^{n}. It is easy to see that

(x(1), …, x(k)) ↦ ∏_{l=1}^{k} λ_{l}(x(l)) (1.6.2)

defines a k-linear form on R^{n}. This k-linear form may be denoted λ_{1} ⊗ ⋯ ⊗ λ_{k}, so that

(λ_{1} ⊗ ⋯ ⊗ λ_{n})(x(1), …, x(k)) = ∏_{l=1}^{k} λ_{l}(x(l)). (1.6.3)

Let M_{k}(R^{n}) (1.6.4)

be the space of all k-linear forms on R^{n}. If μ, ν ∈ M_{k}(R^{n}) and t ∈ R, then it is easy to see that μ + ν ∈ M_{k}(R^{n}) (1.6.5)

and t μ ∈ M_{k}(R^{n}). (1.6.6)

Thus M_{k}(R^{n}) is a linear subspace of the space of all real-valued functions on (R^{n})^{k}, as a vector space over the real numbers with respect to pointwise addition and scalar multiplication. In particular, M_{k}(R^{n}) is a vector space over the real numbers with respect to pointwise addition and scalar multiplication.

1.6.1 Characterizing multilinear forms on R^{n}

Let e_{1}, …, e_{n} be the standard basis for R^{n}, so that the jth coordinate of e_{m} is equal to 1 when j = m, and to 0 otherwise. Thus

x = ∑_{m=1}^{n} x_{m} e_{m} (1.6.7)

for every x ∈ R^{n}. Let μ be a k-linear form on R^{n}. If x(1), …, x(k) ∈ R^{n}, then

x(l) = ∑_{m=1}^{n} x_{m}(l) e_{m} (1.6.8)

for each l = 1, …, k, as in (1.6.7). Using the linearity of μ(x(1), …, x(k)) in x(l) for each l = 1, …, k, we get that

μ(x(1), …, x(k)) = ∑_{m_{1}=1}^{n} ⋯ ∑_{m_{k}=1}^{n} x_{m_{1}}(1) ⋯ x_{m_{k}}(k) μ(e_{m_{1}}, …, e_{m_{k}}). (1.6.9)

This shows that μ is uniquely determined on (R^{n})^{k} by the real numbers

μ(e_{m_{1}}, …, e_{m_{k}}), 1 ≤ m_{1}, …, m_{k} ≤ n. (1.6.10)

8 CHAPTER 1. SOME HELPFUL ALGEBRA One can also get a k-linear form μ on R^{n} as in (1.6.9) for any family of real numbers as in (1.6.10). It follows that (1.6.11) the dimension of M_{k}(R^{n}) is equal to n^{k}, as a vector space over the real numbers.

1.6.2 A basis for M_{k}(R^{n})

Let θ_{j} be the linear functional on R^{n} defined for each j = 1, …, n by θ_{j}(x) = x_{j} (1.6.12) for every x ∈ R^{n}. If j_{1}, …, j_{k} are positive integers less than or equal to n, then θ_{j_{1}} ⊗ ⋯ ⊗ θ_{j_{k}} (1.6.13) defines a k-linear form on R^{n}, as in (1.6.3). More precisely,

(θ_{j_{1}} ⊗ ⋯ ⊗ θ_{j_{k}})(x(1), …, x(k)) = ∏_{l=1}^{k} x_{j_{l}}(l). (1.6.14)

Thus

(θ_{j_{1}} ⊗ ⋯ ⊗ θ_{j_{k}})(e_{m_{1}}, …, e_{m_{k}}) = 1 when j_{l} = m_{l} for each l = 1, …, k
(1.6.15)  = 0 otherwise.

The collection of k-linear forms on R^{n} as in (1.6.13) is a basis for M_{k}(R^{n}), as a vector space over the real numbers.

1.7 Bilinear forms on R^{n}

Let n be a positive integer, and let b(x, y) be a bilinear form on R^{n}. If e_{1}, …, e_{n} are the standard basis vectors in R^{n}, then

b(x, y) = ∑_{m_{1}=1}^{n} ∑_{m_{2}=1}^{n} x_{m_{1}} y_{m_{2}} b(e_{m_{1}}, e_{m_{2}}) (1.7.1)

for all x, y ∈ R^{n}, as in Subsection 1.6.1. Thus b is uniquely determined by the family of real numbers b(e_{m_{1}}, e_{m_{2}}), 1 ≤ m_{1}, m_{2} ≤ n, (1.7.2) and any family of real numbers of this type determines a bilinear form on R^{n} as in (1.7.1).

If b(x, y) = b(y, x) (1.7.3)

1.7. BILINEAR FORMS ON R^{N} 9 for all x, y ∈ R^{n}, then b is said to be a symmetric bilinear form on R^{n}. It is easy to see that this happens if and only if b(e_{m_{1}}, e_{m_{2}}) = b(e_{m_{2}}, e_{m_{1}}) (1.7.4) for all m_{1}, m_{2} = 1, …, n.

If b(x, y) = −b(y, x) (1.7.5) for all x, y ∈ R^{n}, then b is said to be an antisymmetric bilinear form on R^{n}. This happens if and only if b(e_{m_{1}}, e_{m_{2}}) = −b(e_{m_{2}}, e_{m_{1}}) (1.7.6) for all m_{1}, m_{2} = 1, …, n. In particular, this implies that b(e_{m}, e_{m}) = 0 (1.7.7) for every m = 1, …, n.

If b is any bilinear form on R^{n}, then b(x + y, x + y) = b(x, x) + b(x, y) + b(y, x) + b(y, y). (1.7.8) If b is antisymmetric, then b(w, w) = 0 (1.7.9) for every w ∈ R^{n}. Conversely, if (1.7.9) holds for every w ∈ R^{n}, then one can check that b is antisymmetric, using (1.7.8).

If b is any bilinear form on R^{n} again, then (1/2) (b(x, y) + b(y, x)) (1.7.10) is a symmetric bilinear form on R^{n}, and (1/2) (b(x, y) − b(y, x)) (1.7.11) is an antisymmetric bilinear form on R^{n}. Of course, b(x, y) is equal to the sum of (1.7.10) and (1.7.11). If b is both symmetric and antisymmetric, then b(x, y) = 0 (1.7.12) for every x, y ∈ R^{n}.

1.7.1 Associated quadratic polynomials

If b is any bilinear form on R^{n}, then

P_{b}(x) = b(x, x) = ∑_{m_{1}=1}^{n} ∑_{m_{2}=1}^{n} x_{m_{1}} x_{m_{2}} b(e_{m_{1}}, e_{m_{2}}) (1.7.13)

defines a homogeneous polynomial on R^{n} of degree 2, with real coefficients. It is easy to see that every homogeneous polynomial on R^{n} of degree 2 with real coefficients corresponds to a bilinear form b on R^{n} in this way.

Note that the polynomial corresponding to b as in (1.7.13) is the same as the polynomial corresponding to the symmetric part (1.7.10). This implies that every homogeneous polynomial on R^{n} of degree 2 with real coefficients corresponds to a symmetric bilinear form on R^{n} in the same way as before, which could also be verified more directly.

If b is a symmetric bilinear form on R^{n}, then one can check that b is uniquely determined by P_{b}, using (1.7.8).

1.8 Multilinear forms and homogeneous polynomials

Let k and n be positive integers, and let μ be a k-linear form on R^{n}. Under these conditions, P_{μ}(x) = μ(x, …, x) (1.8.1)

defines a homogeneous polynomial on R^{n} of degree k. To be more precise, if e_{1}, …, e_{n} are the standard basis vectors in R^{n}, then

P_{μ}(x) = ∑_{m_{1}=1}^{n} ⋯ ∑_{m_{k}=1}^{n} x_{m_{1}} ⋯ x_{m_{k}} μ(e_{m_{1}}, …, e_{m_{k}}), (1.8.2)

as in Subsection 1.6.1. Note that

μ ↦ P_{μ} (1.8.3)

is a linear mapping from the space M_{k}(R^{n}) of all k-linear forms on R^{n} into the space P_{k}(R^{n}) of all homogeneous polynomials on R^{n} of degree k with real coefficients.

Let λ_{1}, …, λ_{k} be k linear functionals on R^{n}, and suppose for the moment that μ = λ_{1} ⊗ ⋯ ⊗ λ_{k}, (1.8.4)

where the right side is as in Section 1.6. In this case,

P_{μ}(x) = ∏_{l=1}^{k} λ_{l}(x) (1.8.5)

for every x ∈ R^{n}.

Let θ_{j} be the linear functional on R^{n} defined by θ_{j}(x) = x_{j} for each j = 1, …, n, as before. Suppose for the moment again that

μ = θ_{j_{1}} ⊗ ⋯ ⊗ θ_{j_{k}}, (1.8.6) where j_{1}, …, j_{k} are positive integers less than or equal to n. This implies that

P_{μ}(x) = ∏_{l=1}^{k} x_{j_{k}} (1.8.7)

for every x ∈ R^{n}.

Using this, it is easy to see that (1.8.3) maps M_{k}(R^{n}) onto P_{k}(R^{n}). This was mentioned in Subsection 1.7.1 when k = 2. If k = 1, then (1.8.3) is one-to-one.

1.9 Symmetric multilinear forms

If X is any nonempty set, then we let Sym(X) be the space of all one-to-one mappings from X onto itself. This is a group with respect to composition of mappings, which is known as the symmetric group on X. If k is a positive integer and X = {1, …, k}, then we may use the notation Sym(k) for Sym(X).

Let k and n be positive integers, and let μ be a k-linear form on R^{n}. If σ ∈ Sym(k), then let μ^{σ} be the k-linear form on R^{n} defined by

μ^{σ}(x(1), …, x(k)) = μ(x(σ(1)), …, x(σ(k))). (1.9.1)

Of course, this is the same as μ when σ is the identity mapping on {1, …, k}. Note that μ ↦ μ^{σ} (1.9.2)

is a onto-to-one linear mapping from the space M_{k}(R^{n}) of k-linear forms on R^{n} onto itself. If σ is any mapping from {1, …, k} into itself, then (1.9.1) defines a real-valued function on (R^{n})^{k}, but it is not necessarily k-linear.

Let τ be another element of Sym(k), so that τ ∘σ ∈ Sym(k) as well. Observe that

μ^{τ∘σ}(x(1), …, x(k)) = μ(x(τ(σ(1))), …, x(τ(σ(k))))
(1.9.3)  = μ^{σ}(x(τ(1)), …, x(τ(k))) = (μ^{σ})^{τ} (x(1), …, x(k)).

We say that μ is a symmetric k-linear form on R^{n} if

μ^{σ} = μ (1.9.4)

for every σ ∈ Sym(k). This holds trivially when k = 1, and it is equivalent to the definition of a symmetric bilinear form on R^{n} in Section 1.7 when k = 2. In order to verify that μ is symmetric for any k, it suffices to check that (1.9.4) holds when σ is a transposition on {1, …, k}, because every element of Sym(k) may be expressed as a composition of transpositions, as mentioned in Section 1.4.

Let e_{1}, …, e_{n} be the standard basis vectors in R^{n}, as usual. If (1.9.4) holds for some σ ∈ Sym(k), then

μ(e_{m_{σ(1)}}, …, e_{m_{σ(k)}}) = μ(e_{m_{1}}, …, e_{m_{k}}) (1.9.5) for all positive integers m_{1}, …, m_{k} ≤ n. Conversely, if (1.9.5) holds for all such m_{1}, …, m_{k}, then (1.9.4) holds, as in Subsection 1.6.1. Thus μ is symmetric if and only if (1.9.5) holds for all σ ∈ Sym(k) and positive integers m_{1}, …, m_{k} ≤ n. As before, it suffices to check that (1.9.5) holds for all transpositions σ on {1, …, k} and 1 ≤ m_{1}, …, m_{k} ≤ n, to get that μ is symmetric.

1.9.1 Symmetrizing multilinear forms

If μ is any k-linear form on R^{n}, then put

S(μ) = \frac{1}{k!} ∑_{σ∈Sym(k)} μ^{σ}, (1.9.6)

which is another k-linear form on R^{n}. Note that S is a linear mapping from M_{k}(R^{n}) into itself, which may also be denoted S_{k}, to indicate the role of k. If μ is symmetric, then

S(μ) = \frac{1}{k!} ∑_{σ∈Sym(k)} μ = μ, (1.9.7)

because Sym(k) has k! elements, as mentioned in Section 1.4. Let us check that

(1.9.8) S(μ) is symmetric

for every k-linear form μ on R^{n}. One may describe S(μ) as the symmetrization of μ.

If τ ∈ Sym(k), then

S(μ)^{τ} = \frac{1}{k!} ∑_{σ∈Sym(k)}(μ^{σ})^{τ} = \frac{1}{k!} ∑_{σ∈Sym(k)} μ^{τ∘σ}, (1.9.9)

using (1.9.3) in the second step. Observe that

σ ↦ τ ∘ σ (1.9.10)

is a one-to-one mapping from Sym(k) onto itself. It follows that

S(μ)^{τ} = S(μ), (1.9.11)

so that (1.9.8) holds.

Similarly,

S(μ^{τ}) = \frac{1}{k!} ∑_{σ∈Sym(k)}(μ^{τ})^{σ} = \frac{1}{k!} ∑_{σ∈Sym(k)} μ^{σ∘τ} . (1.9.12)

We also have that σ ↦ σ ∘ τ (1.9.13)

is a one-to-one mapping from Sym(k) onto itself. This implies that

S(μ^{τ}) = S(μ) (1.9.14)

for every τ ∈ Sym(k).

1.10 More on symmetric multilinear forms

Let k and n be positive integers again, and let μ be a k-linear form on R^{n}. Also let P_{μ}(x) = μ(x, …, x) be the homogeneous polynomial on R^{n} of degree k associated to μ as in Section 1.8. If σ ∈ Sym(k), then

P_{μ^{σ}} = P_{μ}. (1.10.1)

This implies that

P_{S(μ)} = \frac{1}{k!} ∑_{σ∈Sym(k)} P_{μ^{σ}} = \frac{1}{k!} ∑_{σ∈Sym(k)} P_{μ} = P_{μ}. (1.10.2)

Remember that every homogeneous polynomial on R^{n} of degree k with real coefficients is of the form P_{μ} for some k-linear form μ on R^{n}, as in Section 1.8. Using this and (1.10.2), we get that every such polynomial on R^{n} is of the form P_{μ} for some symmetric k-linear form μ on R^{n}.

Let {1, …, n}^{k} be the set of all k-tuples of positive integers less than or equal to n. If (m_{1}, …, m_{k}) is an element of this set, then put

α_{j} = #{1 ≤ l ≤ k : m_{l} = j}, (1.10.3)

for each j = 1, …, n, where more precisely the right side is the number of positive integers l ≤ k such that m_{l} = j. This defines a multi-index α = (α_{1}, …, α_{n}) with

∑_{j=1}^{n} α_{j} = k. (1.10.4)

Note that every multi-index α that satisfies (1.10.4) corresponds to an element of {1, …, n}^{k} in this way.

Suppose that a multi-index α corresponds to (m_{1}, …, m_{k}) ∈ {1, …, n}^{k} as in (1.10.3). If σ ∈ Sym(k), then

(m_{σ(1)}, …, m_{σ(k)}) (1.10.5)

is another element of {1, …, n}^{k}, which corresponds to α in the same way. Conversely, every element of {1, …, n}^{k} that corresponds to α in this way is of the form (1.10.5) for some σ ∈ Sym(k).

Remember that a k-linear form μ on R^{n} is symmetric if and only if (1.9.5) holds for all σ ∈ Sym(k) and positive integers m_{1}, …, m_{k} ≤ n. This condition holds if and only if

(1.10.6) μ(e_{m_{1}}, …, e_{m_{k}}) only depends on the multi-index α corresponding to (m_{1}, …, m_{k}),

by the remarks in the preceding paragraph.

1.10.1 The space SM_{k}(R^{n})

Let SM_{k}(R^{n}) (1.10.7)

be the space of symmetric k-linear forms on R^{n}. It is easy to see that this is a linear subspace of the space M_{k}(R^{n}) of all k-linear forms on R^{n}. One can check that the dimension of SM_{k}(R^{n}), as a vector space over the real numbers, is equal to the number of multi-indices α that satisfy (1.10.4), using the remarks in the previous paragraphs.

This is the same as the dimension of the space P_{k}(R^{n}) of all homogeneous polynomials on R^{n} of degree k with real coefficients, as mentioned in Subsection 1.3.1. It follows that

(1.10.8) μ ↦ P_{μ} defines a one-to-one mapping from SM_{k}(R^{n}) onto P_{k}(R^{n}).

1.11 Alternating multilinear forms

Let k and n be positive integers, and let μ be a k-linear form on R^{n} again. We say that μ is an alternating k-linear form if

μ^{σ} = sgn(σ) μ (1.11.1)

for every σ ∈ Sym(k). This condition holds trivially when k = 1, and it reduces to the definition of an antisymmetric bilinear form on R^{n} in Section 1.7 when k = 2. In order to show that μ is alternating for any k, it suffices to check that

μ^{σ} = −μ (1.11.2)

when σ is a transposition on {1, …, k}, because every element of Sym(k) may be expressed as a product of transpositions, as mentioned in Section 1.4.

If μ is an alternating k-linear form, then

μ(x(1), …, x(k)) = 0 (1.11.3)

whenever x(l_{1}) = x(l_{2}) (1.11.4)

for some 1 ≤ l_{1} < l_{2} ≤ k. This follows from (1.11.2), by taking σ to be the transposition that interchanges l_{1} and l_{2}.

Conversely, if (1.11.3) holds when (1.11.4) holds, then (1.11.2) holds, with σ equal to the transposition just mentioned. To see this, let x(l) ∈ R^{n} be given for l ≠ l_{1}, l_{2}, so that μ(x(1), …, x(k)) may be considered as a bilinear form on R^{n}, as a function of x(l_{1}) and x(l_{2}). In this case, the hypothesis that (1.11.3) holds when (1.11.4) holds implies that this bilinear form is antisymmetric, as in Section 1.7.

If (1.11.3) holds when (1.11.4) holds for any 1 ≤ l_{1} < l_{2} ≤ n, then we get that (1.11.2) holds for every transposition σ. This implies that μ is alternating, as before.

Let e_{1}, …, e_{n} be the standard basis vectors in R^{n} again, and let σ ∈ Sym(k) be given. Of course, (1.11.1) implies that

μ(e_{m_{σ(1)}}, …, e_{m_{σ(k)}}) = sgn(σ) μ(e_{m_{1}}, …, e_{m_{k}}) (1.11.5)

for all positive integers m_{1}, …, m_{k} ≤ n. Conversely, if (1.11.5) holds for all 1 ≤ m_{1}, …, m_{k} ≤ n, then (1.11.1) holds, as in Subsection 1.6.1. This means that μ is alternating if and only if (1.11.5) holds for all σ ∈ Sym(k) and positive integers m_{1}, …, m_{k} ≤ n.

Similarly, (1.11.2) holds for some transposition σ if and only if

μ(e_{m_{σ(1)}}, …, e_{m_{σ(k)}}) = −μ(e_{m_{1}}, …, e_{m_{k}}) (1.11.6)

for all positive integers m_{1}, …, m_{k} ≤ n. It follows that μ is alternating if and only if (1.11.6) holds for all transpositions σ and 1 ≤ m_{1}, …, m_{k} ≤ n. In particular, if μ is alternating, then

μ(e_{m_{1}}, …, e_{m_{k}}) = 0 (1.11.7)

when m_{l_{1}} = m_{l_{2}} (1.11.8)

for some 1 ≤ l_{1} < l_{2} ≤ k, as in (1.11.3).

If k > n, then any family of k positive integers m_{1}, …, m_{k} ≤ n satisfies (1.11.8) for some 1 ≤ l_{1} < l_{2} ≤ k. If μ is alternating, then we get that μ = 0 in this case, as in Subsection 1.6.1.

1.11.1 The space AM_{k}(R^{n})

Let AM_{k}(R^{n}) (1.11.9)

be the space of alternating k-linear forms on R^{n}. This is a linear subspace of the space M_{k}(R^{n}) of all k-linear forms on R^{n}. If k > n, then

AM_{k}(R^{n}) = {0}, (1.11.10)

as in the preceding paragraph.

1.12 More on alternating multilinear forms

Let k and n be positive integers, and let e_{1}, …, e_{n} be the standard basis for R^{n}, as usual. If μ is any k-linear form on R^{n}, then μ is uniquely determined by the family of real numbers

μ(e_{m_{1}}, …, e_{m_{k}}), (1.12.1)

where m_{1}, …, m_{k} are positive integers less than or equal to n, as in Subsection 1.6.1. If μ is alternating, then (1.12.1) is equal to 0 unless the m_{l}’s are distinct, as in (1.11.7). In particular, this always happens when k > n, and so we suppose now that k ≤ n.

Thus, if μ is alternating, then μ is uniquely determined by the family of real numbers (1.12.1), where m_{1}, …, m_{k} ≤ n are distinct. We also have that (1.12.1) determines μ(e_{m_{σ(1)}}, …, e_{m_{σ(k)}}) (1.12.2)

for every σ ∈ Sym(k), as in (1.11.5). This implies that μ is uniquely determined by the family of real numbers (1.12.1), with

1 ≤ m_{1} < ⋯ < m_{k} ≤ n. (1.12.3)

The family of real numbers (1.12.1) with m_{1}, …, m_{k} as in (1.12.3) for an alternating k-linear form μ may be arbitrary too. More precisely, if (1.12.1) is given when m_{1}, …, m_{k} satisfy (1.12.3), then one can determine (1.12.1) when m_{1}, …, m_{k} are distinct positive integers less than or equal to n using (1.11.5). If we set (1.12.1) equal to 0 when the m_{l}’s are not distinct, then (1.12.1) has been defined for all positive integers m_{1}, …, m_{k} ≤ n. This determines a k-linear form μ on R^{n} as in Subsection 1.6.1. One would like to check that

(1.12.4) μ is alternating

under these conditions.

To do this, it suffices to verify that (1.11.5) holds for all σ ∈ Sym(k) and positive integers m_{1}, …, m_{k} ≤ n. If the m_{l}’s are not distinct, then the m_{σ(l)}’s are not distinct either, and both sides of (1.11.5) are equal to 0, by construction. If the m_{l}’s satisfy (1.12.3), then (1.11.5) holds by construction again.

If m_{1}, …, m_{k} satisfy (1.12.3) and τ ∈ Sym(k), then m_{τ(1)}, …, m_{τ(k)} are distinct positive integers less than or equal to k, and any sequence of k distinct positive integers less than or equal to n can be expressed in this way. If σ is an element of Sym(k), then we would like to have that

μ(e_{m_{σ(τ(1))}}, …, e_{m_{σ(τ(k))}}) = sgn(σ) μ(e_{m_{τ(1)}}, …, e_{m_{τ(k)}}). (1.12.5)

By construction, the left side is equal to

sgn(σ ∘ τ) μ(e_{m_{1}}, …, e_{m_{k}}), (1.12.6)

and the right side is equal to

sgn(σ) sgn(τ) μ(e_{m_{1}}, …, e_{m_{k}}). (1.12.7)

Thus (1.12.5) follows from the fact that sgn(σ∘τ) = sgn(σ) sgn(τ), as in Section 1.4 and Subsection 1.5.1.

1.12.1 The dimension of AM_{k}(R^{n})

This shows that the dimension of AM_{k}(R^{n}), as a vector space over the real numbers, is equal to the number of subsets of {1, …, n} with exactly k elements. This means that

dim AM_{k}(R^{n}) = \binom{n}{k} (1.12.8)

when k ≤ n.

In particular, dim AM_{n}(R^{n}) = 1. (1.12.9)

It is well known that the determinant of an n × n matrix corresponds to an element of AM_{n}(R^{n}), which will be discussed further in Section 1.15. Indeed, the determinant is uniquely determined by this and the fact that the determinant of the identity matrix is equal to 1. Any other element of AM_{n}(R^{n}) can be expressed as a real number times the element of AM_{n}(R^{n}) corresponding to the determinant, because of (1.12.9).

1.13 Alternatizing multilinear forms

Let k and n be positive integers, and let μ be a k-linear form on R^{n}. Remember that if σ ∈ Sym(k), then μ^{σ} may be defined as a k-linear form on R^{n} as in Section 1.9. Put

A(μ) = \frac{1}{k!} ∑_{σ∈Sym(k)} sgn(σ) μ^{σ}. (1.13.1)

This defines a linear mapping from the space M_{k}(R^{n}) of k-linear forms on R^{n} into itself, which may also be denoted A_{k}, to indicate the role of k.

If μ is alternating, then

A(μ) = \frac{1}{k!} ∑_{σ∈Sym(k)} sgn(σ)^{2} μ. (1.13.2)

This implies that

A(μ) = \frac{1}{k!} ∑_{σ∈Sym(k)} μ = μ, (1.13.3)

because sgn(σ)^{2} = 1 for every σ ∈ Sym(k), and Sym(k) has k! elements.

Let μ be any k-linear form on R^{n} again, and let τ ∈ Sym(k) be given. Observe that

(1.13.4) A(μ)^{τ} = \frac{1}{k!} ∑_{σ∈Sym(k)} sgn(σ) (μ^{σ})^{τ} = \frac{1}{k!} ∑_{σ∈Sym(k)} sgn(σ) μ^{τ∘σ},

where the second step is as in Section 1.9. This means that

(1.13.5) A(μ)^{τ} = \frac{sgn(τ)}{k!} ∑_{σ∈Sym(k)} sgn(τ ∘ σ) μ^{τ∘σ},

because sgn(τ ∘ σ) = sgn(τ) sgn(σ) and sgn(τ)^{2} = 1. It follows that

A(μ)^{τ} = sgn(τ) A(μ), (1.13.6)

because σ ↦ τ ∘ σ is a one-to-one mapping from Sym(k) onto itself. Thus

(1.13.7) A(μ) is alternating, and A(μ) may be described as the alternatization of μ.

Similarly,

(1.13.8) A(μ^{τ}) = \frac{1}{k!} ∑_{σ∈Sym(k)} sgn(σ) (μ^{τ})^{σ} = \frac{1}{k!} ∑_{σ∈Sym(k)} sgn(σ) μ^{σ∘τ} .

This implies that

(1.13.9) A(μ^{τ}) = \frac{sgn(τ)}{k!} ∑_{σ∈Sym(k)} sgn(σ ∘ τ) μ^{σ∘τ} ,

for essentially the same reasons as before. Using this, we get that

A(μ^{τ}) = sgn(τ) A(μ), (1.13.10)

because σ ↦ σ ∘ τ is a one-to-one mapping from Sym(k) onto itself.

1.14 The multinomial theorem

Let n be a positive integer, and let α be a multi-index. If ∑_{j=1}^{n} α_{j} = k, then put

\binom{k}{α} = \binom{k}{α_{1} ⋯ α_{n}} = \frac{k!}{α_{1}! ⋯ α_{n}!}, (1.14.1)

which is the multinomial coefficient corresponding to k and α. If n = 2, then this is the same as the binomial coefficients \binom{k}{α_{1}} and \binom{k}{α_{2}}.

If k is a positive integer and x ∈ R^{n}, then it is well known then

(x_{1} + ⋯ + x_{n})^{k} = ∑_{α_{1}+···+α_{n}=k} \binom{k}{α} x^{α}, (1.14.2)

where the sum is taken over all multi-indices α with ∑_{j=1}^{n} α_{j} = k. This is trivial when n = 1, and the n = 2 case is the same as the binomial theorem. The k = 1 case is also easy, for any n.

As for the binomial theorem, it is clear that the left side of (1.14.2) can be expanded into a linear combination of monomials x^{α} of degree k whose coefficients are positive integers. One can get (1.14.2) directly using induction on k, but if one already knows the binomial theorem, then it is easier to use induction on n. One can also obtain (1.14.2) using calculus, by taking derivatives of both sides of order k, which shows that these are the only coefficients such that (1.14.2) holds for every x ∈ R^{n}.

The coefficient of x^{α} in (1.14.2) is the same as the number of ways of partitioning {1, …, k} into n pairwise-disjoint sets, where the jth set has exactly α_{j} elements for each j. Thus the number of these partitions is the same as the multinomial coefficient (1.14.1). Alternatively, any permutation on {1, …, k} sends a partition of this type to another such partition, and every partiition of this type may be obtained from any other partition of this type in this way. There are k! permutations on {1, …, k}, and the number of permutations that send a particular partition of this type to itself is

α_{1}! ⋯ α_{n}!. (1.14.3)

This is another way to see that the number of these partitions is equal to (1.14.1).

A partition of {1, …, k} into n pairwise-disjoint sets corresponds exactly to a mapping from {1, …, k} into {1, …, n}. A mapping of this type also corresponds to a k-tuple (m_{1}, …, m_{k}) of positive integers less than or equal to n. Of course,

(x_{1} + ⋯ + x_{n})^{k} = ∑_{m_{1}=1}^{n} ⋯ ∑_{m_{k}=1}^{n} x_{m_{1}} ⋯ x_{m_{k}} (1.14.4)

for every x ∈ R^{n}. A monomial of the form

x_{m_{1}} ⋯ x_{m_{k}} (1.14.5)

is of the form x^{α} exactly when

α_{j} = #{1 ≤ l ≤ k : m_{l} = j} (1.14.6)

for each j = 1, …, n.

Thus (1.14.1) is the same as the number of k-tuples (m_{1}, …, m_{k}) of positive integers less than or equal to n that satisfy (1.14.6) for each j. This is related to some of the remarks in Section 1.10.

If we take x_{j} = 1 for each j in (1.14.2), then we get that

n^{k} = ∑_{α_{1}+···+α_{n}=k} \binom{k}{α}. (1.14.7)

Of course, the set {1, …, n}^{k} of k-tuples of positive integers less than or equal to n has exactly n^{k} elements. The right side of (1.14.7) expresses this as the sum over all multi-indices α with ∑_{j=1}^{n} α_{j} = k of the number of k-tuples (m_{1}, …, m_{k}) in {1, …, n}^{k} that satisfy (1.14.6) for each j.

1.15 Matrices and determinants

Let k and n be positive integers, and let x(1), …, x(k) be k elements of R^{n}. Thus x_{j}(l) defines an n × k matrix of real numbers.

Let u_{1}, …, u_{k} be the standard basis for R^{k}, so that the lth coordinate of u_{r} is equal to 1 when l = r, and to 0 when l ≠ r. There is a unique linear mapping from R^{k} into R^{n} such that T (u_{l}) = x(l) (1.15.1) for each l = 1, …, k. More precisely, if w ∈ R^{k}, then

w = ∑_{l=1}^{k} w_{l} u_{l}, (1.15.2)

and

T (w) = ∑_{l=1}^{k} w_{l} T (u_{l}) = ∑_{l=1}^{k} w_{l} x(l). (1.15.3)

This means that the jth coordinate of T (w) is given by

(T (w))_{j} = ∑_{l=1}^{k} x_{j}(l) w_{l} (1.15.4)

for each j = 1, …, n. Of course, this is the same as saying that T corresponds to the matrix x_{j}(l) in the usual way.

Suppose from now on in this section that k = n, so that x_{j}(l) is an n × n matrix of real numbers. Put

(1.15.5) μ_{det}(x(1), …, x(n)) = ∑_{σ∈Sym(n)} sgn(σ) ∏_{j=1}^{n} x_{j}(σ(j)).

The right side is the same as the determinant of the matrix x_{j}(l). Equivalently,

(1.15.6) μ_{det}(x(1), …, x(n)) = ∑_{τ∈Sym(n)} sgn(τ) ∏_{l=1}^{n} x_{τ(l)}(l).

This corresponds to taking τ = σ^{−1} in (1.15.5), using the fact that sgn(σ) = sgn(σ^{−1}).

Clearly μ_{det} is an n-linear form on R^{n}. More precisely, μ_{det} is an alternating n-linear form on R^{n}, and in fact it can be expressed as an alternatization in a nice way. Put θ_{j}(y) = y_{j} for each j = 1, …, n and y ∈ R^{n}, so that

(θ_{1} ⊗ ⋯ ⊗ θ_{n})(y(1), …, y(n)) = ∏_{j=1}^{n} y_{j}(j) (1.15.7)

defines an n-linear form on R^{n}, as in Section 1.6. If σ ∈ Sym(n), then

(θ_{1} ⊗ ⋯ ⊗ θ_{n})^{σ}(x(1), …, x(n)) = (θ_{1} ⊗ ⋯ ⊗ θ_{n})(x(σ(1)), …, x(σ(n)))
(1.15.8)  = ∏_{j=1}^{n} x_{j}(σ(j)),

using the notation in Section 1.9 in the first step. This means that

μ_{det} = ∑_{σ∈Sym(n)} sgn(σ) (θ_{1} ⊗ ⋯ ⊗ θ_{n})^{σ} = n! A(θ_{1} ⊗ ⋯ ⊗ θ_{n}), (1.15.9)

using the notation in Section 1.13 in the second step.

It is well known that the determinant of the identity matrix is equal to 1. This corresponds to μ_{det}(e_{1}, …, e_{n}) = 1 (1.15.10)

in the present notation, where e_{1}, …, e_{n} are the standard basis vectors in R^{n}. Indeed, (θ_{1} ⊗ ⋯ ⊗ θ_{n})(e_{1}, …, e_{n}) = 1, (1.15.11)

and

(θ_{1} ⊗ ⋯ ⊗ θ_{n})^{σ}(e_{1}, …, e_{n}) = (θ_{1} ⊗ ⋯ ⊗ θ_{n})(e_{σ(1)}, …, e_{σ(n)}) = 0 (1.15.12)

when σ ∈ Sym(n) is not the identity mapping on {1, …, n}. Note that (1.15.6) corresponds to the fact that the determinant of a matrix is equal to the determinant of its transpose. Some other properties of determinants will be discussed in Section 2.5.

More on multilinear forms

2.1 Some direct sums

Let N be a positive integer, and let V_{1}, …, V_{N} be N vector spaces over the real numbers. Consider the Cartesian product

∏_{j=1}^{N} V_{j} = V_{1} × ⋯ × V_{N}, (2.1.1)

which is the set of N-tuples (v_{1}, …, v_{N}) with v_{j} ∈ V_{j} for each j = 1, …, N. This is also a vector space over the real numbers, with respect to coordinatewise addition and scalar multiplication. This is the direct sum of V_{1}, …, V_{N}, which may be denoted

⨁_{j=1}^{N} V_{j} or V_{1} ⨁ ⋯ ⨁ V_{N}. (2.1.2)

If v_{j} ∈ V_{j} for j = 1, …, N, then we may use the notation v_{1} ⊕ ⋯ ⊕ v_{N} for the element of the direct sum corresponding to (v_{1}, …, v_{N}).

There is an obvious mapping ι_{l} from V_{l} into the direct sum for each l = 1, …, N. Namely, if v_{l} ∈ V_{l}, then let ι_{l}(v_{l}) be the element of the direct sum whose lth coordinate is equal to v_{l}, and whose other coordinates are equal to 0. This is a one-to-one linear mapping from V_{l} into the direct sum for each l. Sometimes one may wish to identify V_{l} with its image ι_{l}(V_{l}) in the direct sum under ι_{l}. Every element of the direct sum corresponds to a sum of elements of ι_{l}(V_{l}), 1 ≤ l ≤ N, in a unique way, by construction.

Let V be another vector space over the real numbers, and suppose for the moment that V_{l} is a linear subspace of V for each l = 1, …, N. Suppose also that every element of V can be expressed in a unique way as a sum of elements of V_{1}, …, V_{N}. This leads to a one-to-one linear mapping from the direct sum of V_{1}, …, V_{N} onto V. We may say that V corresponds to the direct sum of V_{1}, …, V_{N}, as a vector space over the real numbers, under these conditions.

If V_{j} has finite dimension for each j = 1, …, N, then the direct sum has finite dimension as well, with

dim ⨁_{j=1}^{N} V_{j} = ∑_{j=1}^{N} dim V_{j}. (2.1.3)

This may be seen by choosing a basis for each V_{l}, and considering the images of the basis vectors in the direct sum under ι_{l}. The combination of the images of these bases of the V_{l}’s is a basis for the direct sum.

2.1.1 Direct sums and linear mappings

Let W be another vector space over the real numbers, and suppose that T_{l} is a linear mapping from V_{l} into W for each l = 1, …, N. Observe that

T (v_{1} ⊕ ⋯ ⊕ v_{N}) = ∑_{l=1}^{N} T_{l}(v_{l}) (2.1.4)

defines a linear mapping from ⨁_{j=1}^{N} V_{j} into W. Conversely, any linear mapping T from ⨁_{j=1}^{N} V_{j} into W determines a unique linear mapping T_{l} from V_{l} into W for each l = 1, …, N such that (2.1.4) holds. More precisely, one can take

T_{l} = T ∘ ι_{l} (2.1.5)

for each l = 1, …, n.

2.2 Multilinear mappings

Let k be a positive integer, let V_{1}, …, V_{k} be k vector spaces over the real numbers, and let Z be another vector space over the real numbers. Also let ϕ be a function defined on the Cartesian product

∏_{l=1}^{k} V_{l} = V_{1} × ⋯ × V_{k} (2.2.1)

with values in Z. We say that ϕ is multilinear on (2.2.1) if ϕ(v_{1}, …, v_{k}) is linear in v_{l} for each l = 1, …, k, when v_{r} is fixed for r ≠ l. This is the same as in Section 1.6 when V_{l} = R^{n} for each l = 1, …, k and Z = R. As before, we may say that ϕ is k-linear to emphasize the role of k.

Of course, if k = 1, then this is the same as a linear mapping from V_{1} into Z. If k = 2, then this is the same as a bilinear mapping from V_{1} × V_{2} into Z.

Remember that (2.2.1) may be considered as a vector space over the real numbers with respect to coordinatewise addition and scalar multiplication, as in the previous section. Note that k-linear mappings on (2.2.1) are not the same as linear mappings when k ≥ 2.

Suppose for the moment that Z = R, considered as a vector space over itself. A real-valued multilinear function on (2.2.1) may be called a multilinear form or k-linear form, as in Section 1.6 again. This is the same as a linear functional on V_{1} when k = 1, and it may be called a bilinear form when k = 2, as before.

2.2.1 Some basic properties and examples

Let λ_{l} be a linear functional on V_{l} for each l = 1, …, k, which is to say a linear mapping from V_{l} into R. Observe that

(λ_{1} ⊗ ⋯ ⊗ λ_{k})(v_{1}, …, v_{k}) = ∏_{l=1}^{k} λ_{l}(v_{l}) (2.2.2)

defines a k-linear form on (2.2.1). This was mentioned in Section 1.6 when V_{l} = R^{n} for each l = 1, …, k.

Let Z be any vector space over the real numbers again, and let z ∈ Z be given. Also let ψ be a k-linear form on (2.2.1). Under these conditions,

ϕ(v_{1}, …, v_{k}) = ψ(v_{1}, …, v_{k}) z (2.2.3)

defines a k-linear mapping from (2.2.1) into Z.

If X is a nonempty set, then the space of all Z-valued functions on X is a vector space over the real numbers, with respect to pointwise addition and scalar multiplication of functions. In particular, the space of all Z-valued functions on (2.2.1) is a vector space over the real numbers in this way. It is easy to see that the space of all k-linear mappings from (2.2.1) into Z is a linear subspace of the space of all Z-valued functions on (2.2.1).

Let V be a vector space over the real numbers, and suppose that we take V_{l} = V for each l = 1, …, k. In this case, (2.2.1) is the same as the Cartesian product V × ⋯ × V, (2.2.4)

with k factors of V. This is the same as the space V^{k} of k-tuples of elements of V. The space of all k-linear forms on V may be denoted

M_{k}(V), (2.2.5)

as in Section 1.6. This is a linear subspace of the space of all real-valued functions on V^{k}, as in the preceding paragraph.

2.3 Some compositions

Let V and W be vector spaces over the real numbers, let k be a positive integer, and let T_{1}, …, T_{k} be k linear mappings from V into W. If μ is a k-linear form on W, then μ(T_{1}(v_{1}), …, T_{k}(v_{k})) (2.3.1) defines a k-linear form on V.

If T is a linear mapping from V into W, then we can apply the preceding remark with T_{l} = T for each l. We may use the notation

(T^{∗}(μ))(v_{1}, …, v_{k}) = μ(T (v_{1}), …, T (v_{k})) (2.3.2)

for the k-linear form on V obtained in this way, or the notation (T)_{k}^{∗}(μ) to indicate the role of k. Note that this defines a linear mapping from M_{k}(W) into M_{k}(V). The notation δT may be used for this linear mapping as well, as in 2.13 on p62 of [184]. If k = 1, then (2.3.2) is the same as saying that

T^{∗}(μ) = (T)_{1}^{∗}(μ) = μ ∘ T. (2.3.3)

Let Y be another vector space over the real numbers, and let R be a linear mapping from W into Y. Also let ν be a k-linear form on Y, so that R^{∗}(ν) defines a k-linear form on W, as before. Similarly, T^{∗}(R^{∗}(ν)) and (R ∘ T)^{∗}(ν) define k-linear forms on V, using the fact that R ∘ T is a linear mapping from V into Y in the second case. If v_{1}, …, v_{k} ∈ V, then

(2.3.4)  (T^{∗}(R^{∗}(ν)))(v_{1}, …, v_{k}) = (R^{∗}(ν))(T (v_{1}), …, T (v_{k}))
= ν(R(T (v_{1})), …, R(T (v_{k})))
= ((R ∘ T)^{∗}(ν))(v_{1}, …, v_{k}).

This means that (R ∘ T)^{∗} = T^{∗} ∘ R^{∗} (2.3.5)

as linear mappings from M_{k}(Y) into M_{k}(V).

If T is a one-to-one linear mapping from V onto W, then it is easy to see that

(2.3.6) T^{∗} = (T)_{k}^{∗} is a one-to-one linear mapping from M_{k}(W) onto M_{k}(V).

More precisely, ((T)_{k}^{∗})^{−1} = (T^{−1})_{k}^{∗}, (2.3.7)

as in (2.3.5).

If W has dimension n for some positive integer n, then there is a one-to-one linear mapping T from R^{n} onto W. In fact, if e_{1}, …, e_{n} is the standard basis for R^{n}, then one can choose T so that T (e_{1}), …, T (e_{n}) is any basis for W.

2.4 Compositions and permutations

Let V and W be vector spaces over the real numbers again, and let T be a linear mapping from V into W. Also let k be a positive integer, and let μ be a k-linear form on W. If σ is a permutation on {1, …, k}, then μ^{σ} may be defined as a k-linear form on W as in Section 1.9, which is to say that

μ^{σ}(w_{1}, …, w_{k}) = μ(w_{σ(1)}, …, w_{σ(k)}) (2.4.1) for all w_{1}, …, w_{k} ∈ W. If τ is another permutation on {1, …, k}, then

μ^{τ∘σ} = (μ^{σ})^{τ} , (2.4.2)

as before. Of course, we can do the same with k-linear forms on V. In particular, we can do this for T^{∗}(μ). Observe that

(T^{∗}(μ))^{σ}(v_{1}, …, v_{k}) = (T^{∗}(μ))(v_{σ(1)}, …, v_{σ(k)})
(2.4.3)  = μ(T (v_{σ(1)}), …, T (v_{σ(k)}))
= μ^{σ}(T (v_{1}), …, T (v_{k})) = (T^{∗}(μ^{σ}))(v_{1}, …, v_{k})

for all v_{1}, …, v_{k} ∈ V. Thus

(T^{∗}(μ))^{σ} = T^{∗}(μ^{σ}), (2.4.4)

as k-linear forms on V.

We say that μ is a symmetric k-linear form on W if

μ^{σ} = μ (2.4.5)

for every permutation σ on {1, …, k}, as in Section 1.9. Similarly, we say that μ is an alternating k-linear form on W if

μ^{σ} = sgn(σ) μ (2.4.6)

for every σ ∈ Sym(k), as in Section 1.11. In both cases, it suffices to check that the condition holds when σ is a transposition on {1, …, k}, as before. The analogous conditions may be used for k-linear forms on V as well. If μ is a symmetric or alternating k-linear form on W, then T^{∗}(μ) has the same property as a k-linear form on V, because of (2.4.4).

Let SM_{k}(W) and AM_{k}(W) (2.4.7)

be the spaces of symmetric and alternating k-linear forms on W, respectively. These are linear subspaces of M_{k}(W), as in the case where W = R^{n} for some positive integer n, as in Subsections 1.10.1 and 1.11.1. Using the same notation for V, we have that T^{∗}(SM_{k}(W)) ⊆ SM_{k}(V) (2.4.8)

and T^{∗}(AM_{k}(W)) ⊆ AM_{k}(V), (2.4.9)

as in the preceding paragraph. If T is a one-to-one linear mapping from V onto W, then we can combine these statements with their analogous for T^{−1}, to get that T^{∗}(SM_{k}(W)) = SM_{k}(V) (2.4.10)

and T^{∗}(AM_{k}(W)) = AM_{k}(V). (2.4.11)

2.4.1 Symmetrizations and alternatizations

If μ is any k-linear form on W again, then the symmetrization and alternatization of μ are defined by

S(μ) = S_{k}(μ) = \frac{1}{k!} ∑_{σ∈Sym(k)} μ^{σ} (2.4.12)

and

A(μ) = A_{k}(μ) = \frac{1}{k!} ∑_{σ∈Sym(k)} sgn(σ) μ^{σ}, (2.4.13)

respectively, as in Subsection 1.9.1 and Section 1.13. As before, if μ is already symmetric or alternating, then

S(μ) = μ (2.4.14)

or A(μ) = μ, (2.4.15)

as appropriate. If μ is any k-linear form on W, then

(2.4.16) S(μ) is symmetric

and (2.4.17) A(μ) is alternating,

for the same reasons as before. If τ ∈ Sym(k), then

S(μ^{τ}) = S(μ) (2.4.18)

and A(μ^{τ}) = sgn(τ) A(μ), (2.4.19)

as before. It is easy to see that

T^{∗}(S(μ)) = S(T^{∗}(μ)) (2.4.20)

and T^{∗}(A(μ)) = A(T^{∗}(μ)), (2.4.21)

using (2.4.4) and the analogues of S and A for V on the right sides of the equations.

2.5 Compositions and determinants

Let n be a positive integer, let T be a linear mapping from R^{n} into itself, and let e_{1}, …, e_{n} be the standard basis for R^{n}. The determinant of T is normally defined as the determinant of the matrix associated to T using e_{1}, …, e_{n}. This means that det T = μ_{det}(T (e_{1}), …, T (e_{n})), (2.5.1)

28 CHAPTER 2. MORE ON MULTILINEAR FORMS

where μ_{det} is the alternating n-linear form on R^{n} associated to the determinant as in Section 1.15.

If ν is any alternating n-form on R^{n}, then ν = ν(e_{1}, …, e_{n}) μ_{det}. (2.5.2) To see this, remember that the space AM_{n}(R^{n}) of alternating n-linear forms on R^{n} has dimension 1 as a vector space over the real numbers, as in Subsection 1.12.1. This implies that ν is equal to a real number times μ_{det}. This real number is equal to ν(e_{1}, …, e_{n}), because μ_{det}(e_{1}, …, e_{n}) = 1.

Note that T^{∗}(μ_{det}) is an alternating n-linear form on R^{n}, as in the previous section. We also have that (2.5.3) T^{∗}(μ_{det}) = (T^{∗}(μ_{det}))(e_{1}, …, e_{n}) μ_{det} = (det T) μ_{det}, using (2.5.2) in the first step, and (2.5.1) in the second step. This implies that T^{∗}(ν) = (det T) ν (2.5.4) for every alternating n-linear form ν on R^{n}, because ν is a constant multiple of μ_{det}.

Let R be another linear mapping from R^{n} into itself, so that det R = μ_{det}(R(e_{1}), …, R(e_{n})), (2.5.5) as in (2.5.1). Similarly, det(T ∘ R) = μ_{det}(T (R(e_{1})), …, T (R(e_{n}))). (2.5.6) Observe that (2.5.7) (T^{∗}(μ_{det}))(R(e_{1}), …, R(e_{n})) = (det T) μ_{det}(R(e_{1}), …, R(e_{n})), by (2.5.3). Of course, the right side of (2.5.6) is the same as the left side of (2.5.7). It follows that (2.5.8) det(T ∘ R) = (det T) (det R), by (2.5.5).

2.5.1 n-Dimensional vector spaces

Let W be a vector space over the real numbers of dimension n, and let L be a one-to-one linear mapping from R^{n} onto W. If M is a linear mapping from W into itself, then one can define the determinant of M by det M = det(L^{−1} ∘ M ∘ L), (2.5.9) where the right side is the determinant of L^{−1} ∘ M ∘ L, as a linear mapping from R^{n} into itself. One can check that this does not depend on the choice of L, using (2.5.8).

2.6. SUBGROUPS OF SYM(K) 29

Using L, we get a one-to-one linear mapping L^{∗} = (L)_{k}^{∗} from M_{k}(W) onto M_{k}(R^{n}) for each positive integer k, as in Section 2.3. We also get that (L)_{k}^{∗} sends SM_{k}(W), AM_{k}(W) onto SM_{k}(R^{n}), AM_{k}(R^{n}), respectively, as in the previous section. This implies that M_{k}(W), SM_{k}(W), and AM_{k}(W) have the same dimensions as M_{k}(R^{n}), SM_{k}(R^{n}), and AM_{k}(R^{n}), respectively.

In particular, dim AM_{n}(W) = 1. (2.5.10)

One can verify that (2.5.4) holds for every linear mapping T from W into itself and alternating n-linear form ν on W, by reducing to the analogous statement for R^{n} using L.

2.6 Subgroups of Sym(k)

Let X be a nonempty set, and remember that Sym(X) is the space of one-to-one mappings from X onto itself, as in Section 1.9. A subset Σ of Sym(X) is said to be a subgroup if it satisfies the following three conditions. First, the identity mapping on X is an element of Σ. Second, if σ ∈ Σ, then

σ^{−1} ∈ Σ. (2.6.1)

Third, if σ, τ ∈ Σ, then σ ∘ τ ∈ Σ. (2.6.2)

It is well known that Sym(X) is an example of a group, with respect to composition of mappings on X. The definition of a subgroup of Sym(X) in the preceding paragraph corresponds exactly to the definition of a subgroup of an arbitrary group. Some basic examples of subgroups of Sym(X) can be obtained by considering the permutations that map cerain elements of X to themselves, or certain subsets of X to themselves. Note that Sym(X) is a subgroup of itself, and that the subset of Sym(X) consisting only of the identity mapping on X is a subgroup of Sym(X) too.

Let k be a positive integer, and let us now take X = {1, …, k}. Thus we let Σ be a subgroup of Sym(k). Also let W be a vector space over the real numbers, and let μ be a k-linear form on W. Remember that μ^{σ} may be defined as a k-linear form on W for every σ ∈ Sym(k) as in Section 2.4. Let us say that μ is symmetric with respect to Σ if μ^{σ} = μ (2.6.3)

for every σ ∈ Σ. Similarly, we say that μ is alternating with respect to Σ if

μ^{σ} = sgn(σ) μ (2.6.4)

for every σ ∈ Σ. If Σ = Sym(k), then these definitions are the same as the previous definitions of symmetric and alternating k-forms on W in Section 2.4.

30 CHAPTER 2. MORE ON MULTILINEAR FORMS

2.6.1 Associated symmetrizations and alternatizations

The symmetrization and alternatization with respect to Σ of any k-linear form μ on W may be defined by

S_{Σ}(μ) = \frac{1}{#Σ} ∑_{σ∈Σ} μ^{σ} (2.6.5)

and

A_{Σ}(μ) = \frac{1}{#Σ} ∑_{σ∈Σ} sgn(σ) μ^{σ}, (2.6.6)

respectively, where #Σ is the number of elements of Σ. These are the same as the usual symmetrization and alternatization of μ, as in Subsection 2.4.1, when Σ = Sym(k). It is easy to see that S_{Σ}(μ) = μ (2.6.7) when μ is symmetric with respect to Σ, and that A_{Σ}(μ) = μ (2.6.8) when μ is alternating with respect to Σ. One can check that (2.6.9) S_{Σ}(μ) is symmetric with respect to Σ and (2.6.10) A_{Σ}(μ) is alternating with respect to Σ for any k-linear form μ on W, using the same type of arguments as in Subsection 1.9.1 and Section 1.13. If τ ∈ Σ, then S_{Σ}(μ^{τ}) = S_{Σ}(μ) (2.6.11) and A_{Σ}(μ^{τ}) = sgn(τ) A_{Σ}(μ), (2.6.12) for essentially the same reasons as before.

Observe that

(2.6.13) S(S_{Σ}(μ)) = S(\frac{1}{#Σ} ∑_{τ∈Σ} μ^{τ} ) = \frac{1}{#Σ} ∑_{τ∈Σ} S(μ^{τ})
= \frac{1}{#Σ} ∑_{τ∈Σ} S(μ) = S(μ),

because S(μ^{τ}) = S(μ) for every τ ∈ Sym(k), as in Subsection 2.4.1. Similarly,

A(A_{Σ}(μ)) = A(\frac{1}{#Σ} ∑_{τ∈Σ} sgn(τ) μ^{τ} ) = \frac{1}{#Σ} ∑_{τ∈Σ} sgn(τ) A(μ^{τ})
(2.6.14)  = \frac{1}{#Σ} ∑_{τ∈Σ} sgn(τ)^{2} A(μ) = A(μ),

because A(μ^{τ}) = sgn(τ) A(μ) for every τ ∈ Sym(k), as before.

2.7 Permutations on subsets

Let X be a nonempty set, and let E be a nonempty subset of X. Consider the spaces Sym(X) and Sym(E) of one-to-one mappings from X and E onto themselves, respectively, as before. If σ ∈ Sym(E), then let σ̂ be the mapping from X into itself which is equal to σ on E, and equal to the identity mapping on X \ E. It is easy to see that σ̂ is an element of Sym(X), which is the identity mapping on X when σ is the identity mapping on E. We also have that

ˆ (σ ∘ τ) = σ̂ ∘ τ̂ (2.7.1)

for every σ, τ ∈ Sym(X).

This means that σ ↦ σ̂ (2.7.2)

is a homomorphism from Sym(E) into Sym(X), as groups with respect to compositions of mappings. Note that (2.7.2) is a one-to-one mapping from Sym(E) into Sym(X). Put Σ_{E} = {σ̂ : σ ∈ Sym(E)}, (2.7.3)

which is a subgroup of Sym(X), as in the previous section. This uses the fact that ˆ (σ^{−1}) = (σ̂)^{−1} (2.7.4)

for every σ ∈ Σ.

Now let k be a positive integer, and let us take X = {1, …, k} again. If σ ∈ Sym(E), then one can check that

sgn(σ) = sgn(σ̂). (2.7.5)

More precisely, one can define sgn(σ) by identifying σ with an element of Sym(#E), by listing the elements of E in order. In fact, one could list the elements of E in any order and get the same result for sgn(σ), but we do not really need that now.

It is easier to verify (2.7.5) when the elements of E are consecutive in {1, …, k}, which holds in many examples of interest. Otherwise, there can be an even number of additional fectors of −1 in the definition of sgn(σ̂).

Alternatively, if σ is a transposition on E, then σ̂ is a transposition on {1, …, k}. Thus (2.7.5) is clear in this case, and otherwise one can express σ as the composition of finitely many transposition.

2.8 Products of multilinear forms

Let W be a vector space over the real numbers, and let r be a positive integer. Also let k_{1}, …, k_{r} be r positive integers, and put

k = ∑_{m=1}^{r} k_{m}. (2.8.1)

32 CHAPTER 2. MORE ON MULTILINEAR FORMS

If 1 ≤ m ≤ r, then we put

L(m) = ∑_{i=1}^{m−1} k_{i}, (2.8.2)

which is interpreted as being equal to 0 when m = 1. Note that every positive integer less than or equal to k can be expressed in a unique way as L(m) + l, (2.8.3) where 1 ≤ m ≤ r and 1 ≤ l ≤ k_{m}.

Suppose that μ_{m} is a k_{m}-linear form on W for each m = 1, …, r. If w_{1}, …, w_{k} are k elements of W, then put

(μ_{1} ⊗ ⋯ ⊗ μ_{r})(w_{1}, …, w_{k}) = ∏_{m=1}^{r} μ_{m}(w_{L(m)+1}, …, w_{L(m)+k_{m}}). (2.8.4)

This defines a k-linear form μ = μ_{1} ⊗ ⋯ ⊗ μ_{r} (2.8.5) on W. This is related to some remarks in Section 1.6 and Subsection 2.2.1 when k_{m} = 1 for each m = 1, …, r. Note that (μ_{1}, …, μ_{r}) ↦ μ_{1} ⊗ ⋯ ⊗ μ_{r} (2.8.6) defines an r-linear mapping from

∏_{m=1}^{r} M_{k_{m}}(W) = M_{k_{1}}(W) × ⋯ × M_{k_{r}}(W) (2.8.7)

into M_{k}(W).

If r = 2, then (2.8.4) is the same as saying that (μ_{1} ⊗ μ_{2})(w_{1}, …, w_{k_{1}+k_{2}}) = μ_{1}(w_{1}, …, w_{k_{1}}) μ_{2}(w_{k_{1}+1}, …, w_{k_{1}+k_{2}}). (2.8.8) Similarly, if r = 3, then (2.8.4) is the same as saying that (μ_{1} ⊗ μ_{2} ⊗ μ_{3})(w_{1}, …, w_{k_{1}+k_{2}+k_{3}}) (2.8.9)

= μ_{1}(w_{1}, …, w_{k_{1}}) μ_{2}(w_{k_{1}+1}, …, w_{k_{1}+k_{2}}) μ_{3}(w_{k_{1}+k_{2}+1}, …, w_{k_{1}+k_{2}+k_{3}}).

2.8.1 Polynomials, associativity, and linear mappings

Suppose for the moment that W = R^{n} for some positive integer n. Remember that a multilinear form ν on R^{n} corresponds to a homogeneous polynomial P_{ν} on R^{n} as in Section 1.8. If μ is as in (2.8.5), then it is easy to see that

P_{μ} = ∏_{m=1}^{r} P_{μ_{m}}. (2.8.10)

Suppose for the moment again that r = 3. One can define μ_{1} ⊗ μ_{2} as a (k_{1} + k_{2})-linear form on W as before, and use that to define

(μ_{1} ⊗ μ_{2}) ⊗ μ_{3} (2.8.11)

as a (k_{1} + k_{2} + k_{3})-linear form on W. Similarly, one can define μ_{2} ⊗ μ_{3} as a (k_{2} + k_{3})-linear form on W, and use that to define

μ_{1} ⊗ (μ_{2} ⊗ μ_{3}) (2.8.12)

as a (k_{1} + k_{2} + k_{3})-linear form on W. It is easy to see that (2.8.11) and (2.8.12) are the same as μ_{1} ⊗ μ_{2} ⊗ μ_{3}. (2.8.13)

Let V be another vector space over the real numbers, and let T be a linear mapping from V into W. It is easy to see that

T^{∗}(μ_{1} ⊗ ⋯ ⊗ μ_{r}) = T^{∗}(μ_{1}) ⊗ ⋯ ⊗ T^{∗}(μ_{r}), (2.8.14)

as k-linear forms on V. Of course, this uses the mappings induced on multilinear forms by T as in Section 2.3.

2.8.2 Products and permutations

Let ϕ be a permutation on {1, …, r}, so that k_{ϕ(1)}, …, k_{ϕ(r)} are r positive integers with

∑_{m=1}^{r} k_{ϕ(m)} = k. (2.8.15)

Thus ν = μ_{ϕ(1)} ⊗ ⋯ ⊗ μ_{ϕ(r)} (2.8.16)

is a k-linear form on W, as before. If W = R^{n}, then

P_{ν} = ∏_{m=1}^{r} P_{μ_{ϕ(m)}} = P_{μ}. (2.8.17)

Put

L_{ϕ}(m) = ∑_{i=1}^{m−1} k_{ϕ(i)} (2.8.18)

for each m = 1, …, r, as before. Every positive integer less than or equal to k can be expressed in a unique way as

L_{ϕ}(m) + l (2.8.19)

for some m = 1, …, r and l = 1, …, k_{ϕ(m)}, as before.

Consider the mapping σ_{ϕ} from {1, …, k} into itself defined by

σ_{ϕ}(L_{ϕ}(m) + l) = L(ϕ(m)) + l (2.8.20) for each m = 1, …, r and l = 1, …, k_{ϕ(m)}. More precisely, this defines a permutation on {1, …, k}. One can check that

ν = μ^{σ_{ϕ}}, (2.8.21)

where the right side is defined as in Section 2.4.

If r = 2, ϕ(1) = 2, and ϕ(2) = 1, then

ν = μ_{2} ⊗ μ_{1}. (2.8.22)

Note that L(1) = 0, L(2) = k_{1} and L_{ϕ}(1) = 0, L_{ϕ}(2) = k_{2} in this case. This means that σ_{ϕ}(l) = k_{1} + l for l = 1, …, k_{2} = k_{ϕ(1)} (2.8.23)

and σ_{ϕ}(k_{2} + l) = l for l = 1, …, k_{1} = k_{ϕ(2)}. (2.8.24)

2.9 Symmetry and products

Let us continue with the same notation and hypotheses as in the previous section. If μ_{1}, …, μ_{r} are symmetric multilinear forms on W, then we may prefer to consider a product that is a symmetric multilinear form as well. In this case, we may put μ_{1} ⊙ ⋯ ⊙ μ_{r} = S(μ), (2.9.1)

where the right side is as in Subsection 2.4.1. This defines an r-linear mapping

(μ_{1}, …, μ_{r}) ↦ μ_{1} ⊙ ⋯ ⊙ μ_{r} (2.9.2)

from ∏_{m=1}^{r} SM_{k_{m}}(W) = SM_{k_{1}}(W) × ⋯ × SM_{k_{r}}(W) into SM_{k}(W). If W = R^{n}, then

P_{S(μ)} = P_{μ} = ∏_{m=1}^{r} P_{μ_{m}}, (2.9.3)

where the first step is as in Section 1.10. This determines S(μ) uniquely, as in Subsection 1.10.1.

Let ϕ be a permutation on {1, …, r} again, and let ν be as in (2.8.16). Observe that S(ν) = S(μ^{σ_{ϕ}}) = S(μ), (2.9.4)

using (2.8.21) in the first step, and where the second step is as in Subsection 2.4.1. This means that

μ_{ϕ(1)} ⊙ ⋯ ⊙ μ_{ϕ(r)} = μ_{1} ⊙ ⋯ ⊙ μ_{r}. (2.9.5)

2.9.1 Some symmetrizations of products

Suppose for the moment that r = 2. We would like to check that

S_{k_{1}+k_{2}}(S_{k_{1}}(μ_{1}) ⊗ μ_{2}) = S_{k_{1}+k_{2}}(μ_{1} ⊗ μ_{2}), (2.9.6)

where S_{k_{1}}, S_{k_{1}+k_{2}} are the symmetrization operators associated to k_{1} and k_{1} +k_{2} as in Subsection 2.4.1. If τ ∈ Sym(k_{1}), then let τ̂ be the element of Sym(k_{1} +k_{2}) that is equal to τ on {1, …, k_{1}}, and is the identity mapping on {k_{1}+1, …, k_{1}+ k_{2}}, as in Section 2.7. It is easy to see that

μ_{1}^{τ} ⊗ μ_{2} = (μ_{1} ⊗ μ_{2})_{τ}, (2.9.7)

using the notation from Section 2.4.

Remember that Σ = {τ̂: τ ∈ Sym(k_{1})} (2.9.8)

is a subgroup of Sym(k_{1} +k_{2}), as in Sections 2.6 and 2.7. Of course, the number of elements of Σ is the same as the number of elements of Sym(k_{1}), which is k_{1}!. One can check that

S_{k_{1}}(μ_{1}) ⊗ μ_{2} = S_{Σ}(μ_{1} ⊗ μ_{2}), (2.9.9)

where S_{Σ} is as in Subsection 2.6.1, using (2.9.7). This implies that

S_{k_{1}+k_{2}}(S_{k_{1}}(μ_{1}) ⊗ μ_{2}) = S_{k_{1}+k_{2}}(S_{Σ}(μ_{1} ⊗ μ_{2})) = S_{k_{1}+k_{2}}(μ_{1} ⊗ μ_{2}), (2.9.10)

where the second step is as in Subsection 2.6.1.

Alternatively, if W = R^{n}, then the polynomial associated to the right side of (2.9.6) is as in (2.9.3). One can verify that this is the same as the polynomial associated to the left side of (2.9.6).

Similarly, S_{k_{1}+k_{2}}(μ_{1} ⊗ S_{k_{2}}(μ_{2})) = S_{k_{1}+k_{2}}(μ_{1} ⊗ μ_{2}). (2.9.11)

It follows that

(2.9.12)  S_{k_{1}+k_{2}}(S_{k_{1}}(μ_{1}) ⊗ S_{k_{2}}(μ_{2})) = S_{k_{1}+k_{2}}(μ_{1} ⊗ S_{k_{2}}(μ_{2}))
= S_{k_{1}+k_{2}}(μ_{1} ⊗ μ_{2}).

One could also look at this in terms of the subgroup of Sym(k_{1} + k_{2}) corresponding to the product of Sym(k_{1}) and Sym(k_{2}), as in Section 2.13.

Suppose now that r = 3 again. Observe that

S_{k}(S_{k_{1}+k_{2}}(μ_{1} ⊗ μ_{2}) ⊗ μ_{3}) = S_{k}(μ_{1} ⊗ μ_{2} ⊗ μ_{3}), (2.9.13)

where k = k_{1} + k_{2} + k_{3}, as in (2.9.6). Similarly,

S_{k}(μ_{1} ⊗ S_{k_{2}+k_{3}}(μ_{2} ⊗ μ_{3})) = S_{k}(μ_{1} ⊗ μ_{2} ⊗ μ_{3}), (2.9.14)

as in (2.9.11). It follows that

((μ_{1} ⊙ μ_{2}) ⊙ μ_{3}) = μ_{1} ⊙ μ_{2} ⊙ μ_{3} = μ_{1} ⊙ (μ_{2} ⊙ μ_{3}) (2.9.15)

when μ_{1}, μ_{2}, μ_{3} are symmetric, in the notation of (2.9.1).

2.10 Alternatizing products

Let W be a vector space over the real numbers, let k_{1}, k_{2} be positive integers, and let μ_{1}, μ_{2} be k_{1}, k_{2}-linear forms on W, respectively. We would like to check that A_{k_{1}+k_{2}}(A_{k_{1}}(μ_{1}) ⊗ μ_{2}) = A_{k_{1}+k_{2}}(μ_{1} ⊗ μ_{2}), (2.10.1)

where A_{k_{1}} and A_{k_{1}+k_{2}} are the alternatization operators associated to k_{1} and k_{1} + k_{2} as in Subsection 2.4.1.

If τ ∈ Sym(k_{1}), then let τ̂ ∈ Sym(k_{1} + k_{2}) be as in Subsection 2.9.1, and let Σ be as in (2.9.8) again. Remember that sgn(τ) = sgn(τ̂) for every τ ∈ Sym(k_{1}), as in Section 2.7. One can use this and (2.9.7) to get that

A_{k_{1}}(μ_{1}) ⊗ μ_{2} = A_{Σ}(μ_{1} ⊗ μ_{2}), (2.10.2)

where A_{Σ} is as in Subsection 2.6.1. This implies that

A_{k_{1}+k_{2}}(A_{k_{1}}(μ_{1}) ⊗ μ_{2}) = A_{k_{1}+k_{2}}(A_{Σ}(μ_{1} ⊗ μ_{2})) = A_{k_{1}+k_{2}}(μ_{1} ⊗ μ_{2}), (2.10.3)

where the second step is as in Subsection 2.6.1.

Similarly, A_{k_{1}+k_{2}}(μ_{1} ⊗ A_{k_{2}}(μ_{2})) = A_{k_{1}+k_{2}}(μ_{1} ⊗ μ_{2}). (2.10.4)

Thus

(2.10.5) A_{k_{1}+k_{1}}(A_{k_{1}}(μ_{1}) ⊗ A_{k_{2}}(μ_{2})) = A_{k_{1}+k_{2}}(μ_{1} ⊗ A_{k_{2}}(μ_{2}))
= A_{k_{1}+k_{2}}(μ_{1} ⊗ μ_{2}).

As before, one can look at this more directly in terms of the appropriate subgroup of Sym(k_{1} + k_{2}), as in Section 2.13.

Now let k_{1}, k_{2}, and k_{3} be positive integers, and put k = k_{1} + k_{2} + k_{3} again. If μ_{1}, μ_{2}, and μ_{3} are k_{1}, k_{2}, and k_{3}-linear forms on W, respectively, then

A_{k}(A_{k_{1}+k_{2}}(μ_{1} ⊗ μ_{2}) ⊗ μ_{3}) = A_{k}(μ_{1} ⊗ μ_{2} ⊗ μ_{3}), (2.10.6)

as in (2.10.1). Similarly,

A_{k}(μ_{1} ⊗ A_{k_{1}+k_{2}}(μ_{2} ⊗ μ_{3})) = A_{k}(μ_{1} ⊗ μ_{2} ⊗ μ_{3}), (2.10.7)

as in (2.10.4).

2.10.1 The ∧_{0} product

If μ_{1}, μ_{2} are alternating k_{1}, k_{2}-linear forms on W, respectively, then put

μ_{1} ∧_{0} μ_{2} = A_{k_{1}+k_{2}}(μ_{1} ⊗ μ_{2}) (2.10.8)

which is an alternating (k_{1} + k_{2})-linear form on W. This corresponds to

μ_{1} ∧_{β} μ_{2} (2.10.9) in the notation used in (3) on p60 of [184], and we shall say more about this in the next section. Note that

(μ_{1}, μ_{2}) ↦ μ_{1} ∧_{0} μ_{2} (2.10.10)

defines a bilinear mapping from AM_{k_{1}}(W) × AM_{k_{2}}(W) into AM_{k_{1}+k_{2}}(W). If μ_{3} is an alternating k_{3}-linear form on W, then we have that

(μ_{1} ∧_{0} μ_{2}) ∧_{0} μ_{3} = μ_{1} ∧_{0} (μ_{2} ∧_{0} μ_{3}) = A_{k_{1}+k_{2}+k_{3}}(μ_{1} ⊗ μ_{2} ⊗ μ_{3}), (2.10.11)

by (2.10.6) and (2.10.7).

Now let r be a positive integer, let k_{1}, …, k_{r} be r positive integers, and put k = ∑_{m=1}^{r} k_{m}, as in Section 2.8. If μ_{m} is an alternating k_{m}-linear form on W for each m = 1, …, r, then

μ_{1} ∧_{0} ⋯ ∧_{0} μ_{r} = A_{k}(μ_{1} ⊗ ⋯ ⊗ μ_{r}) (2.10.12)

is an alternating k-linear form on W, where the right side is as in Subsection 2.4.1. More precisely, the left side may be defined by combining alternating multilinear forms two at a time using ∧_{0} as in the preceding paragraph, and one gets the same answer however this is done, because of (2.10.11). One can also check that the result is equal to the right side of (2.10.12), using (2.10.11). This defines an r-linear mapping from

∏_{m=1}^{r} AM_{k_{m}}(W) = AM_{k_{1}}(W) × ⋯ × AM_{k_{r}}(W) (2.10.13)

into AM_{k}(W).

This is a nice way to “multiply” alternating multilinear forms on W, but it is customary to use a different normalization. This will be discussed in the next section.

2.11 The wedge product

Let W be a vector space over the real numbers, and let k_{1}, k_{2} be positive integers. If μ_{1}, μ_{2} are alternating k_{1}, k_{2}-linear forms on W, respectively, then put

μ_{1} ∧ μ_{2} = \frac{(k_{1} + k_{2})!}{k_{1}! k_{2}!} (μ_{1} ∧_{0} μ_{2}) = \frac{(k_{1} + k_{2})!}{k_{1}! k_{2}!} A_{k_{1}+k_{2}}(μ_{1} ⊗ μ_{2}), (2.11.1)

which is an alternating (k_{1} + k_{2})-linear form on W. This corresponds to

μ_{1} ∧_{α} μ_{2} (2.11.2)

in the notation used in (4) on p60 of [184]. Of course,

(μ_{1}, μ_{2}) ↦ μ_{1} ∧ μ_{2} (2.11.3) is a bilinear mapping from AM_{k_{1}}(W) × AM_{k_{2}}(W) into AM_{k_{1}+k_{2}}(W).

Let k_{3} be another positive integer, and let μ_{3} be an alternating k_{3}-linear form on W. Using (2.10.11), one can check that

(2.11.4) (μ_{1} ∧ μ_{2}) ∧ μ_{3} = μ_{1} ∧ (μ_{2} ∧ μ_{3})
= \frac{(k_{1} + k_{2} + k_{3})!}{k_{1}! k_{2}! k_{3}!} A_{k_{1}+k_{2}+k_{3}}(μ_{1} ⊗ μ_{2} ⊗ μ_{3}).

Similarly, let r be a positive integer, let k_{1}, …, k_{r} be r positive integers, and put k = ∑_{m=1}^{r} k_{m}, as before. If μ_{m} is an alternating k_{m}-linear form on W for each m = 1, …, r, then

(2.11.5)  μ_{1} ∧ ⋯ ∧ μ_{r} = \frac{k!}{k_{1}! ⋯ k_{r}!} (μ_{1} ∧_{0} ⋯ ∧_{0} μ_{r})
= \frac{k!}{k_{1}! ⋯ k_{r}!} A_{k}(μ_{1} ⊗ ⋯ ⊗ μ_{r})

is an alternating k-linear form, where the second step is as in (2.10.12). As before, the left side may be defined using the wedge product of alternating multilinear forms two at a time, and one gets the same answer however this is done, because of (2.11.4). To get the first step in (2.11.5), one should verify the coefficient on the right side. Note that this defines an r-linear mapping from ∏_{m=1}^{r} AM_{k_{m}}(W) into AM_{k}(W).

Suppose for the moment that k_{m} = 1 for each m = 1, …, r, so that k = r. In this case, (2.11.5) implies that

(2.11.6)  μ_{1} ∧ ⋯ ∧ μ_{k} = k! A_{k}(μ_{1} ⊗ ⋯ ⊗ μ_{k})
= ∑_{σ∈Sym(k)} sgn(σ) (μ_{1} ⊗ ⋯ ⊗ μ_{k})^{σ},

using the definition of A_{k} in Subsection 2.4.1 in the second step.

Suppose for the moment again that W = R^{n} for some positive integer n, and let θ_{1}, …, θ_{n} be the usual coordinate functions on R^{n}, as in Subsection 1.6.2. Using (2.11.6), we get that

θ_{1} ∧ ⋯ ∧ θ_{n} = μ_{det}, (2.11.7)

where μ_{det} is as in Section 1.15.

2.12 More on wedge products

Let k_{1}, k_{2} be positive integers, and let ψ_{k_{1},k_{2}} be the mapping from

{1, …, k_{1} + k_{2}} (2.12.1)

into itself defined by

(2.12.2)  ψ_{k_{1},k_{2}}(l) = l + k_{1} when 1 ≤ l ≤ k_{2}
= l − k_{2} when k_{2} + 1 ≤ l ≤ k_{1} + k_{2},

It is easy to see that this defines a permutation on (2.12.1). One can check that

sgn(ψ_{k_{1},k_{2}}) = (−1)^{k_{1} k_{2}}. (2.12.3)

Let W be a vector space over the real numbers, and let μ_{1}, μ_{2} be k_{1}, k_{2}-linear forms on W, respectively. One can verify that

μ_{2} ⊗ μ_{1} = (μ_{1} ⊗ μ_{2})^{ψ_{k_{1},k_{2}}} , (2.12.4)

where the right side is as defined in Section 2.4. This corresponds to a remark in Subsection 2.8.2.

This implies that

(2.12.5)  A_{k_{1}+k_{2}}(μ_{2} ⊗ μ_{1}) = A_{k_{1}+k_{2}}((μ_{1} ⊗ μ_{2})^{ψ_{k_{1},k_{2}}} )
= sgn(ψ_{k_{1},k_{2}}) A_{k_{1}+k_{2}}(μ_{1} ⊗ μ_{2})
= (−1)^{k_{1} k_{2}} A_{k_{1}+k_{2}}(μ_{1} ⊗ μ_{2}),

where the second step is as in Subsection 2.4.1. If μ_{1}, μ_{2} are alternating k_{1}, k_{2}-linear forms on W, then this is the same as saying that

μ_{2} ∧_{0} μ_{1} = (−1)^{k_{1} k_{2}} (μ_{1} ∧_{0} μ_{2}). (2.12.6)

Equivalently, this means that

μ_{2} ∧ μ_{1} = (−1)^{k_{1} k_{2}} (μ_{1} ∧ μ_{2}). (2.12.7)

It is convenient to put M_{0}(W) = R, (2.12.8)

so that a real number is considered as a 0-linear form on W. We can extend the previous definition of products of multilinear forms in Section 2.8 to include 0-linear forms, using scalar multiplication of multilinear forms by real numbers, and ordinary multiplication on R.

Let us consider 0-linear forms on W to be both symmetric and alternating, so that AM_{0}(W) = SM_{0}(W) = R (2.12.9)

as well. The symmetrization and alternatization operators S_{k} and A_{k} defined in Subsection 2.4.1 may be extended to k = 0 using the identity mapping on R. Wedge products of alternating multilinear forms may be extended to include 0linear forms too, using scalar multiplication by real numbers again, and ordinary multiplication on R.

2.12.1 Linear mappings and AM(W)

Let V be another vector space over the real numbers, and let T be a linear mapping from V into W. If μ_{1}, μ_{2} are alternating k_{1}, k_{2}-linear forms on W again, then

T^{∗}(μ_{1} ∧_{0} μ_{2}) = T^{∗}(A_{k_{1}+k_{2}}(μ_{1} ⊗ μ_{2})) = A_{k_{1}+k_{2}}(T^{∗}(μ_{1} ⊗ μ_{2}))
(2.12.10)  = A_{k_{1}+k_{2}}(T^{∗}(μ_{1}) ⊗ T^{∗}(μ_{2})) = T^{∗}(μ_{1}) ∧_{0} T^{∗}(μ_{2}).

More precisely, the second step here is as in Subsection 2.4.1, and the third step is as in Subsection 2.8.1. It follows that

T^{∗}(μ_{1} ∧ μ_{2}) = T^{∗}(μ_{1}) ∧ T^{∗}(μ_{2}). (2.12.11)

Suppose now that W has dimension n for some positive integer n, as a vector space over the real numbers. Remember that

AM_{k}(W) = {0} when k > n (2.12.12)

in this case, as in Subsection 1.11.1 and Section 2.4. This implies that

μ_{1} ∧ μ_{2} = 0 when k_{1} + k_{2} > n. (2.12.13)

Let us define AM(W) initially as a vector space over the real numbers by

AM(W) = ⨁_{k=0}^{n} AM_{k}(W), (2.12.14)

where the right side is as in Section 2.1. We shall normally identify elements of AM_{l}(W) with their images in AM(W) under the embeddings mentioned earlier, so that every element of AM(W) corresponds to a sum of elements of AM_{l}(W), 0 ≤ l ≤ n.

The wedge product has a natural extension to a bilinear mapping from AM(W) × AM(W) into AM(W). This makes AM(W) into an associative algebra over the real numbers. Note that R corresponds to a subalgebra of AM(W), because of (2.12.9), and that 1 is also the multiplicative identity element of AM(W).

2.13 Some more subgroups of Sym(k)

In this and the next two sections, we shall consider some additional subgroups of Sym(k), and related properties of k-linear forms. This will be used to give another way to look at wedge products in Subsection 2.15.2. The reader may not want to dwell on these three sections too much at first.

Let r be a positive integer, let k_{1}, …, k_{r} be r positive integers, and put k = ∑_{m=1}^{r} k_{m}, as before. If 1 ≤ m ≤ r, then put L(m) = ∑_{i=1}^{m−1} k_{i}, which is interpreted as being equal to 0 when m = 1, as in Section 2.8. Also put

E_{m} = {L(m) + 1, …, L(m) + k_{m}}, (2.13.1)

and note that

⋃_{m=1}^{r} E_{m} = {1, …, k}, (2.13.2)

and that the E_{m}’s are pairwise-disjoint, as before.

2.13. SOME MORE SUBGROUPS OF SYM(K) 41

If σ_{m} ∈ Sym(k_{m}), then let σ̂_{m} be the mapping from {1, …, k} into itself defined by σ̂_{m}(L(m) + l) = L(m) + σ(l) (2.13.3) for l = 1, …, k_{m}, and taking σ̂ to be the identity mapping on the complement of E_{m}. It is easy to see that this defines an element of Sym(k), and that ˆ (σ_{m} ∘ τ_{m}) = σ̂_{m} ∘ τ̂_{m} (2.13.4) for every σ_{m}, τ_{m} ∈ Sym(k_{m}). This basically corresponds to some remarks in Section 2.7, by identifying Sym(k_{m}) with Sym(E_{m}) using the mapping l ↦ L(m) + l (2.13.5) from {1, …, k_{m}} onto E_{m}.

As before, σ_{m} ↦ σ̂_{m} (2.13.6) is a one-to-one mapping from Sym(k_{m}) into Sym(k), and a group homomorphism. We also have that sgn(σ_{m}) = sgn(σ̂_{m}) (2.13.7) for every σ_{m} ∈ Sym(k_{m}), as before.

Suppose for the moment that 1 ≤ m_{1} ≠ m_{2} ≤ r, so that E_{m_{1}} ∩ E_{m_{2}} = ∅. (2.13.8) If σ_{m_{1}} ∈ Sym(k_{m_{1}}) and σ_{m_{2}} ∈ Sym(k_{m_{2}}), then it is easy to see that σ̂_{m_{1}} ∘ σ̂_{m_{2}} = σ̂_{m_{2}} ∘ σ̂_{m_{1}}. (2.13.9)

Consider the subset Σ of Sym(k) defined by (2.13.10) Σ = {σ̂_{1} ∘ ⋯ ∘ σ̂_{r} : σ_{m} ∈ Sym(k_{m}) for each m = 1, …, r}. One can check that this is a subgroup of Sym(k), as in Section 2.6. Equivalently, Σ consists of the elements of Sym(k) that map E_{m} onto itself for each m = 1, …, r.

Consider the Cartesian product

(2.13.11)  ∏_{m=1}^{r} Sym(k_{m}) = Sym(k_{1}) × ⋯ × Sym(k_{r}),

which is the set of r-tuples (σ_{1}, …, σ_{r}) with σ_{m} ∈ Sym(k_{m}) for each m = 1, …, r. By construction, (σ_{1}, …, σ_{r}) ↦ σ̂_{1} ∘ ⋯ ∘ σ̂_{r} (2.13.12) maps this Cartesion product onto Σ. One can check that this mapping is oneto-one as well. This implies that the number of elements of Σ is given by

#Σ = ∏_{m=1}^{r}(k_{m}!). (2.13.13)

In fact, (2.13.11) may be considered as a group, where the group operations are defined coordinatewise. This is an example of a direct product of groups. One can verify that (2.13.12) is a homomorphism from this group onto Σ, using (2.13.9). More precisely, (2.13.12) is an isomorphism from (2.13.11) onto Σ, because this mapping is one-to-one, as in the preceding paragraph.

2.14 Subgroups and products of forms

Let us continue with the same notation and hypotheses as in the previous section. Also let W be a vector space over the real numbers, and let μ_{m} be a k_{m}-linear form on W for each m = 1, …, r. Thus μ = μ_{1} ⊗ ⋯ ⊗ μ_{r} defines a k-linear form on W, as in Section 2.8.

Let σ_{m} ∈ Sym(k_{m}) be given for each m = 1, …, r, and put

σ = σ̂_{1} ∘ ⋯ ∘ σ̂_{r}. (2.14.1)

It is easy to see that μ^{σ} = μ_{1}^{σ_{1}} ⊗ ⋯ ⊗ μ_{m}^{σ_{m}}, (2.14.2)

using the notation from Section 2.4. If μ_{m} is a symmetric k_{m}-linear form on W for each m = 1, …, r, then it follows that μ is symmetric with respect to Σ, as in Section 2.6.

Suppose for the moment that μ_{m} is an alternating k_{m}-linear form on W for each m = 1, …, r. This implies that

μ^{σ} = ( ∏_{m=1}^{r} sgn(σ_{m})) μ, (2.14.3)

because of (2.14.2). Observe that

(2.14.4)  ∏_{m=1}^{r} sgn(σ_{m}) = ∏_{m=1}^{r} sgn(σ̂_{m}) = sgn(σ),

using (2.13.7) in the first step. Thus

μ^{σ} = sgn(σ) μ, (2.14.5)

so that μ is alternating with respect to Σ, as in Section 2.6.

Remember that S_{Σ}(μ) may be defined as in Subsection 2.6.1. If μ_{m} is any k_{m}-linear form on W for each m again, then one can check that

S_{Σ}(μ) = S_{k_{1}}(μ_{1}) ⊗ ⋯ ⊗ S_{k_{r}}(μ_{r}), (2.14.6)

2.15. SOME RELATED SHUFFLES 43

using (2.14.2).

Similarly, A_{Σ}(μ) may be defined as in Subsection 2.6.1. One can verify that A_{Σ}(μ) = A_{k_{1}}(μ_{1}) ⊗ ⋯ ⊗ A_{k_{r}}(μ_{r}), (2.14.7) using (2.14.2) and (2.14.4).

2.15 Some related shuffles

Let us continue with the same notation and hypotheses as in the previous two sections. Let us say that τ ∈ Sym(k) is a shuffle with respect to k_{1}, …, k_{r} if for each m = 1, …, r, (2.15.1) the restriction of τ to E_{m} is increasing. This means that for each m = 1, …, r and 1 ≤ l_{1} < l_{2} ≤ k_{m}, we have that τ(L(m) + l_{1}) < τ(L(m) + l_{2}). (2.15.2) This is mentioned with r = 2 on p60 of [184]. If τ is a shuffle with respect to k_{1}, …, k_{r}, then for each m = 1, …, r, the restriction of τ to E_{m} is uniquely determined by τ(E_{m}).

Let ρ ∈ Sym(k) be given, and put C_{m} = ρ(E_{m}) (2.15.3) for each m = 1, …, r. Observe that #C_{m} = #E_{m} (2.15.4) for each m = 1, …, r,

⋃_{m=1}^{r} C_{m} = {1, …, k}, (2.15.5)

and (2.15.6) the C_{m}’s are pairwise-disjoint. If C_{1}, …, C_{r} is any sequence of r subsets of {1, …, k} that satisfies these three conditions, then it is easy to see that there is a unique shuffle τ ∈ Sym(k) with respect to k_{1}, …, k_{r} such that C_{m} = τ(E_{m}) (2.15.7) for every m = 1, …, r. Put σ = τ^{−1} ∘ ρ, so that σ ∈ Sym(k) and ρ = τ ∘ σ. (2.15.8) Clearly σ(E_{m}) = E_{m} (2.15.9) for each m = 1, …, r, by construction.

2.15.1 The set of shuffles

Remember that Σ is the subgroup of Sym(k) consisting of σ ∈ Sym(k) that satisfy (2.15.9), as in Section 2.13. Let Ξ be the set of τ ∈ Sym(k) such that τ is a shuffle with respect to k_{1}, …, k_{r}. Consider the Cartesian product

Ξ × Σ, (2.15.10)

which is the set of ordered pairs (τ, σ) with τ ∈ Ξ and σ ∈ Σ. Note that

(τ, σ) ↦ τ ∘ σ (2.15.11)

defines a mapping from (2.15.10) into Sym(k). In fact, this is a one-to-one mapping from (2.15.10) onto Sym(k), by the remarks in the preceding paragraph.

We also have that

#Ξ = \binom{k}{k_{1} ⋯ k_{r}} = \frac{k!}{k_{1}! ⋯ k_{r}!}. (2.15.12)

More precisely, the number of shuffles τ ∈ Sym(k) with respect to k_{1}, …, k_{r} is the same as the number of ways of partitioning {1, …, k} into r subsets C_{1}, …, C_{r}, where #C_{m} = k_{m} for each m = 1, …, r, as before. The number of these partitions is the same as the multinomial coefficient in (2.15.12), as in Section 1.14. Alternatively,

(#Ξ) (#Σ) = # Sym(k), (2.15.13)

as in the preceding paragraph. This implies (2.15.12), because # Sym(k) = k! and #Σ = ∏_{m=1}^{r}(k_{m}!).

2.15.2 Symmetrization, alternatization, and shuffles

If ν is a k-linear form on W, then

S_{k}(ν) = \frac{1}{k!} ∑_{ρ∈Sym(k)} ν^{ρ} = \frac{1}{k!} ∑_{τ∈Ξ} ∑_{σ∈Σ} ν^{τ∘σ} = \frac{1}{k!} ∑_{τ∈Ξ} ∑_{σ∈Σ}(ν^{σ})^{τ} , (2.15.14)

where the first and third steps are as in Section 2.4, and the second step uses the earlier remarks about (2.15.11). This implies that

S_{k}(ν) = \frac{1}{k!} ∑_{τ∈Ξ} ( ∑_{σ∈Σ} ν^{σ})^{τ} = \frac{1}{#Ξ} ∑_{τ∈Ξ} S_{Σ}(ν)^{τ} , (2.15.15)

using (2.15.13) and the definition of S_{Σ}(ν) in Subsection 2.6.1 in the second step. In particular, if ν is symmetric with respect to Σ, then

S_{k}(ν) = \frac{1}{#Ξ} ∑_{τ∈Ξ} ν^{τ} . (2.15.16)

Similarly,

A_{k}(ν) = \frac{1}{k!} ∑_{ρ∈Sym(k)} sgn(ρ) ν^{ρ} = \frac{1}{k!} ∑_{τ∈Ξ} ∑_{σ∈Σ} sgn(τ ∘ σ) ν^{τ∘σ}

(2.15.17)  = \frac{1}{k!} ∑_{τ∈Ξ} ∑_{σ∈Σ} sgn(τ) sgn(σ) (ν^{σ})^{τ} .

It follows that

(2.15.18) A_{k}(ν) = \frac{1}{k!} ∑_{τ∈Ξ} sgn(τ) ( ∑_{σ∈Σ} sgn(σ) ν^{σ})^{τ} = \frac{1}{#Ξ} ∑_{τ∈Ξ} sgn(τ) A_{Σ}(ν)^{τ} .

If ν is alternating with respect to Σ, then we get that

A_{k}(ν) = \frac{1}{#Ξ} ∑_{τ∈Ξ} sgn(τ) ν^{τ} . (2.15.19)

Suppose that μ_{m} is an alternating k_{m}-linear form on W for each m = 1, …, r, so that μ = μ_{1} ⊗ ⋯ ⊗ μ_{r} is alternating with respect to Σ, as in the previous section. Observe that

μ_{1} ∧_{0} ⋯ ∧_{0} μ_{r} = A_{k}(μ) = \frac{1}{#Ξ} ∑_{τ∈Ξ} sgn(τ) μ^{τ} , (2.15.20)

where the first step is as in Subsection 2.10.1, and the second step is as in (2.15.19). It follows that

μ_{1} ∧ ⋯ ∧ μ_{r} = ∑_{τ∈Ξ} sgn(τ) μ^{τ} , (2.15.21)

using (2.15.12) and a remark in Section 2.11. This corresponds to (2) on p60 of [184] when r = 2.

Some geometry and analysis

3.1 Metric spaces

Let X be a set. A nonnegative real-valued function d(x, y) defined for x, y ∈ X is said to be a metric on X if it satisfies the following three conditions. First, (3.1.1) d(x, y) = 0 is and only if x = y. Second, d(x, y) = d(y, x) (3.1.2) for every x, y ∈ X. Third, d(x, z) ≤ d(x, y) + d(y, z) (3.1.3) for every x, y, z ∈ X. This third condition is known as the triangle inequality. If d(x, y) is a metric on X, then (X, d(x, y)) is called a metric space.

If t is a real number, then the absolute value |t| of t is defined as usual by |t| = t when t ≥ 0, and |t| = −t when t ≤ 0. It is easy to see that |r + t| ≤ |r| + |t| (3.1.4) for all r, t ∈ R. One can use this to check that d(x, y) = |x − y| (3.1.5) defines a metric on the real line, which is the standard Euclidean metric on R.

If X is any set, then the discrete metric on X is defined by

(3.1.6)  d(x, y) = 1 when x ≠ y
= 0 when x = y.

One can check that this defines a metric on X.

Let (X, d(x, y)) be a metric space, and let E be a subset of X. It is easy to see that (3.1.7) the restriction of d(x, y) to x, y ∈ E

defines a metric on E.

We shall often be concerned here with metrics obtained from norms on vector spaces, as in the next section. More precisely, we shall often be concerned with norms obtained from inner products, as in Section 3.3.

3.2 Norms on vector spaces

Let V be a vector space over the real numbers. A nonnegative real-valued function N on V is said to be a norm if it satisfies the following three conditions. First, (3.2.1) N(v) = 0 if and only if v = 0.

Second, N(t v) = |t| N(v) (3.2.2)

for every t ∈ R and v ∈ V. Third,

N(v + w) ≤ N(v) + N(w) (3.2.3)

for every v, w ∈ V, which is the triangle inequality for a norm.

Let n be a positive integer, and remember that R^{n} is a vector space over the real numbers with respect to coordinatewise addition and scalar multiplication. The standard Euclidean norm is defined on R^{n} by

∥v∥_{2} = ( ∑_{j=1}^{n} v_{j}^{2})^{1/2}, (3.2.4)

using the nonnegative square root on the right side. It is easy to see that this satisfies the first two conditions in the definition of a norm, but the triangle inequality is more complicated when n ≥ 2. This will be discussed further in the next section.

One can check directly that

∥v∥_{1} = ∑_{j=1}^{n} |v_{j}| (3.2.5)

defines a norm on R^{n}. One can also verify that

∥v∥_{∞} = max_{1≤j≤n} |v_{j}| (3.2.6)

defines a norm on R^{n}. More precisely, to get that this satisfies the triangle inequality, one can observe that

∥v + w∥_{∞} = max_{1≤j≤n} |v_{j} + w_{j}| ≤ max_{1≤j≤n}(|v_{j}| + |w_{j}|) ≤ ∥v∥_{∞} + ∥w∥_{∞} (3.2.7)

48 CHAPTER 3. SOME GEOMETRY AND ANALYSIS for every v, w ∈ R^{n}.

Note that ∥v∥_{∞} ≤ ∥v∥_{1}, ∥v∥_{2} (3.2.8) for every v ∈ R^{n}. We also have that ∥v∥_{1} ≤ n ∥v∥_{∞} (3.2.9) and ∥v∥_{2} ≤ \sqrt{n} ∥v∥_{∞} (3.2.10) for every v ∈ R^{n}.

If v ∈ R^{n}, then

∥v∥_{2}^{2} = ∑_{j=1}^{n} v_{j}^{2} ≤ ∥v∥_{1} ∥v∥_{∞} ≤ ∥v∥_{1}^{2}, (3.2.11)

using (3.2.8) in the third step. This implies that ∥v∥_{2} ≤ ∥v∥_{1}. (3.2.12)

If N is a norm on any vector space V over the real numbers, then it is easy to see that d_{N}(v, w) = N(v − w) (3.2.13) defines a metric on V. The standard Euclidean metric on R^{n} is the metric d_{2}(v, w) = ∥v − w∥_{2} (3.2.14) associated to the standard Euclidean norm ∥ · ∥_{2} on R^{n}. Similarly, d_{1}(v, w) = ∥v − w∥_{1} (3.2.15) and d_{∞}(v, w) = ∥v − w∥_{∞} (3.2.16) define metrics on R^{n} too.

If W is a linear subspace of V and N is a norm on V, then (3.2.17) the restriction of N to W defines a norm on W as well.

3.3 Inner product spaces

Let V be a vector space over the real numbers, and let ⟨v, w⟩ be a symmetric bilinear form on V. If ⟨v, v⟩ > 0 (3.3.1)

3.4. CONTINUOUS FUNCTIONS ON R^{N} 49

for every v ∈ V with v ≠ 0, then ⟨·, ·⟩ is said to be an inner product on V. Under these conditions, (V, ⟨·, ·⟩) is called an inner product space.

If n is a positive integer, then the standard inner product on R^{n} is defined by

⟨v, w⟩ = ∑_{j=1}^{n} v_{j} w_{j}. (3.3.2)

In this case,

⟨v, v⟩ = ∑_{j=1}^{n} v_{j}^{2} = ∥v∥_{2}^{2} (3.3.3)

for every v ∈ R^{n}, which is strictly positive when v ≠ 0.

Let (V, ⟨·, ·⟩) be an inner product space, and put

∥v∥ = ⟨v, v⟩^{1/2} (3.3.4)

for every v ∈ V, using the nonnegative square root on the right side. Note that

∥t v∥ = ⟨t v, t v⟩^{1/2} = |t| ∥v∥ (3.3.5)

for every v ∈ V and t ∈ R. If v, w ∈ V, then it is well known that

|⟨v, w⟩| ≤ ∥v∥ ∥w∥, (3.3.6)

which is known as the Cauchy–Schwarz inequality. This can be obtained from the fact that ⟨v − t w, v − t w⟩ ≥ 0 (3.3.7)

for every t ∈ R.

One can use the Cauchy–Schwarz inequality to show that

∥v + w∥ ≤ ∥v∥ + ∥w∥ (3.3.8)

for every v, w ∈ V. This implies that

(3.3.9) ∥ · ∥ is a norm on V.

Of course, the standard Euclidean norm on R^{n} is the same as the norm associated to the standard inner product on R^{n}.

If W is a linear subspace of V, then

(3.3.10) the restriction of ⟨v, w⟩ to v, w ∈ W defines an inner product on W.

3.4 Continuous functions on R^{n}

Let n be a positive integer, and let E be a nonempty subset of R^{n}. Also let f be a real-valued function on E, and let x be an element of E. One can define what it means for f to be continuous at x as a function defined on E in a standard way. In fact, one can define what it means for a function defined on a subset of a metric space with values in another metric space to be continuous at a point in the set on which the function is defined in essentially the same way. Here we are using the standard Euclidean metrics on R^{n} and R.

We say that f is continuous on E if f is continuous at every point in E. Let

C(E) = C(E, R) (3.4.1)

be the space of all continuous real-valued functions on E. It is well known that this is a linear subspace of the space of all real-valued functions on E. In particular, C(E) is a vector space over the real numbers with respect to pointwise addition and scalar multiplication of functions on E.

If f, g are continuous real-valued functions on E, then it is well known that their product (3.4.2) f g is continuous on E

too. Using this, it is easy to see that polynomials are continuous on R^{n}. Thus the space P(R^{n}) of all polynomials on R^{n} with real coefficients may be considered as a linear subspace of C(R^{n}).

If f is a continuous real-valued function on E, and f(x) ≠ 0 for each x ∈ E, then it is well known that

(3.4.3) 1/f is continuous on E.

This implies that rational functions are continuous on sets where the denominator is nonzero.

Let m be a positive integer, and let f be a function on E with values in R^{m}. If x ∈ E, then f(x) can be expressed as

f(x) = (f_{1}(x), …, f_{m}(x)), (3.4.4)

where f_{1}(x), …, f_{m}(x) are real numbers. Thus f corresponds to m real-valued functions f_{1}, …, f_{m} on E. If x ∈ E, then one can define what it means for f to be continuous at x in a standard way, using the standard Euclidean metric on R^{m}. It is well known that this is equivalent to the continuity of f_{1}, …, f_{m} at x as real-valued functions on E.

As before, we say that f is continuous on E if f is continuous at every point in E. Let C(E, R^{m}) (3.4.5)

be the space of all continuous functions on E with values in R^{m}. The space of all R^{m}-valued functions on E is a vector space over the real numbers, with respect to pointwise addition and scalar multiplication. It is easy to see that C(E, R^{m}) is a linear subspace of the space of all R^{m}-valued functions on E. Thus C(E, R^{m}) is a vector space over the real numbers with respect to pointwise addition and scalar multiplication of functions on E.

Let A be a nonempty subset of R^{m}, and suppose that f(x) ∈ A for every x ∈ E, so that f(E) ⊆ A. (3.4.6)

Also let k be another positive integer, and let g be a function on A with values in R^{k}. Thus the composition g ∘ f of f and g may be defined as a function on E with values in R^{k}, with

(g ∘ f)(x) = g(f(x)) (3.4.7)

for every x ∈ E. If f is continuous at x, and g is continuous at f(x), then it is well known that g ∘ f is continuous at x. In particular, if f is continuous on E, and g is continuous on A, then g ∘ f is continuous on E.

3.4.1 Open balls and open sets

If x ∈ R^{n} and r is a positive real number, then the open ball in R^{n} centered at x with radius r with respect to the standard Euclidean metric is defined as usual by B(x, r) = {y ∈ R^{n} : ∥x − y∥_{2} < r}, (3.4.8)

where ∥ · ∥_{2} = ∥ · ∥_{2,R^{n}} is the standard Euclidean norm on R^{n}, as in Section 3.2. A subset U of R^{n} is said to be an open set with respect to the standard Euclidean metric if for every x ∈ U there is an r > 0 such that

B(x, r) ⊆ U. (3.4.9)

It is well known that open balls in R^{n} are open sets. This can be shown using the triangle inequality. More precisely, one can define open balls and open sets in any metric space, and one can show that open balls are open sets in the same way.

3.5 Partial derivatives

Let n be a positive integer, and let U be a nonempty open subset of R^{n}, with respect to the standard Euclidean metric. Also let f be a real-valued function on U, and let x ∈ U be given. The partial derivative of f at x in the jth variable, 1 ≤ j ≤ n, may be denoted

∂_{j}f(x) = \frac{∂f}{∂x_{j}} (x), (3.5.1)

when it exists.

Similarly, let m be a positive integer, and suppose now that f is a function on U with values in R^{m}. The partial derivative of f at x in the jth variable may be denoted as in (3.5.1) again, when it exists, which is an element of R^{m}. This happens exactly when the partial derivative of each of the components f_{1}, …, f_{m} at x in the jth variable exists, in which case

∂_{j}f(x) = (∂_{j}f_{1}(x), …, ∂_{j}f_{m}(x)). (3.5.2)

3.5.1 Continuous differentiability

We say that f is continuously differentiable on U if

(3.5.3) ∂_{j}f(x) exists at every point x ∈ U for each j = 1, …, n, and is continuous as an R^{m}-valued function on U.

Equivalently, this means that each of the components f_{1}, …, f_{m} of f is continuously differentiable on U in the same sense as a real-valued function on U. It is well known that this implies that f is continuous on U. Of course, if n = 1, then continuity at a point is implied by the existence of the derivative at that point.

Let C^{1}(U, R^{m}) (3.5.4)

be the space of continuously-differentiable R^{m}-valued functions on U. This is a linear subspace of the space C(U, R^{m}) of all continuous R^{m}-valued functions on U.

If f, g are continuously-differentiable real-valued functions on U, then one can check that (3.5.5) f g is continuously differentiable on U,

using the product rule. Note that polynomials are continuously differentiable on R^{n}.

If f is a continuously-differentiable real-valued function on U and f(x) ≠ 0 for every x ∈ U, then one can verify that

(3.5.6) 1/f is continuously differentiable on U.

It follows that rational functions are continuously differentiable on the complement of the set where the denominator is equal to 0.

3.5.2 Second derivatives

Let f be an R^{m}-valued function on U again, and suppose that the partial derivatives of f exist at each point in U and in each variable. The partial derivative of ∂_{l}f at x in the jth variable may be denoted

∂_{j}∂_{l}f(x) = \frac{∂^{2}f}{∂x_{j} ∂x_{l}} (x), (3.5.7)

when it exists. If

(3.5.8) the partial derivatives ∂_{1}f, …, ∂_{n}f of f are all continuously differentiable on U,

then f is said to be twice continuously differentiable on U. This implies that the partial derivatives of f are continuous on U, and that f is continuous on U, as before. Note that f is twice continuously differentiable on U if and only if the components f_{1}, …, f_{m} of f are all twice continuously differentiable as real-valued functions on U.

If f is twice continuously differentiable on U, then it is well known that

\frac{∂^{2}f}{∂x_{j} ∂x_{l}} = \frac{∂^{2}f}{∂x_{l} ∂x_{j}} (3.5.9)

on U for all j, l = 1, …, n. This is normally stated for real-valued functions, which implies the analogouus result for R^{m}-valued functions.

3.6 Additional regularity

Let m, n, and r be positive integers, and let U be a nonempty open subset of R^{n} again. If f is an R^{m}-valued function on U, then let us say that f is r-times continuously differentiable on U if

(3.6.1) all of the derivatives of f of order up to r exist and are continuous on U.

This is equivalent to continuous differentiability of f on U when r = 1, and to twice continuous differentiability of f on U when r = 2. It is sometimes convenient to interpret this with r = 0 as meaning that f is continuous on U.

More precisely, f is r-times continuously differentiable on U if

(3.6.2) all of the derivatives of f of order up to r exist on U, and the derivatives of f of order r are continuous on U.

This corresponds to the previous definitions of continuous differentiability and twice continuous differentiability of f when r = 1 and 2, respectively. If r ≥ 2, then this is the same as saying that

(3.6.3) all of the derivatives of f of order r − 1 are continuously differentiable on U.

In particular, this implies that all of the derivatives of f of order r − 1 are continuous on U, as before. One can repeat the process to get that the derivatives of f of all orders up to r are continuous on U.

Alternatively, if r ≥ 2, then f is r-times continuously differentiable on U if and only if

(3.6.4) the partial derivatives ∂_{1}f, …, ∂_{n}f exist on U and are (r − 1)-times continuously differentiable on U.

Note that f is r-times continuously differentiable on U if and only if each of the components f_{1}, …, f_{m} of f is r-times continuously differentiable as a real-valued function on U.

3.6.1 More on r-times continuous differentiability

Let C^{r}(U, R^{m}) (3.6.5) be the space of all R^{m}-valued functions on U that are r-times continuously differentiable on U. This may be interpreted as being the space C(U, R^{m}) of all continuous R^{m}-valued functions on U when r = 0, as before. It is easy to see that (3.6.6) C^{r}(U, R^{m}) is a linear subspace of C(U, R^{m}) for each r. We also have that

C^{r+1}(U, R^{m}) ⊆ C^{r}(U, R^{m}) (3.6.7)

for each r.

Let j_{1}, …, j_{r} be r positive integers less than or equal to n. Suppose that the derivatives of f on U up to order r exist, and consider the rth-order derivative

∂_{j_{1}} ⋯ ∂_{j_{r}}f (3.6.8)

of f on U. If σ is a permutation on {1, …, r}, then

∂_{j_{σ(1)}} ⋯ ∂_{j_{σ(n)}}f (3.6.9)

is an rth-order derivative of f on U as well. If f is r-times continuously differentiable on U, then it is well known that (3.6.8) is equal to (3.6.9). The r = 2 case was mentioned in Subsection 3.5.2.

If r > 2, then the previous case implies that (3.6.8) is equal to (3.6.9) when σ is a transposition of a pair of consecutive elements of {1, …, r}. To get that this holds for all permutations σ on {1, …, r}, one can show that every such permutation is the composition of finitely many transpositions of this type.

Let α be a multi-index with

∑_{j=1}^{n} α_{j} = r. (3.6.10)

We may use the notation

∂^{α}f = ∂_{1}^{α_{1}} ⋯ ∂_{n}^{α_{n}} f = \frac{∂^{r}f}{∂x_{1}^{α_{1}} ⋯ ∂x_{n}^{α_{n}}} (3.6.11)

for the corresponding derivative of f of other r. Of course, this means that no derivative is taken in the jth variable when α_{j} = 0. If f is r-times continuously differentiable on U, then the order in which the derivatives are taken does not matter, as in the previous two paragraphs, and every rth-order derivative of f is of this form for some α.

If f, g are real-valued functions on U that are r-times continuously differentiable for some r, then one can check that

(3.6.12) f g is r-times continuously differentiable on U.

If f(x) ≠ 0 for every x ∈ U, then one can verify that

(3.6.13) 1/f is r-times continuously differentiable on U.

3.7 Smooth functions

Let m and n be positive integers, let U be a nonempty open subset of R^{n}, and let f be an R^{m}-valued function on U again. Let us say that f is infinitely differentiable on U if

(3.7.1) f is r-times continuously differentiable on U for every r ≥ 1.

Sometimes f is simply said to be smooth on U in this case. As before, f is infinitely differentiable on U if and only if each of the components f_{1}, …, f_{m} is infinitely differentiable as a real-valued function on U. The space of infinitely differentiable R^{m}-valued functions on U is denoted

C^{∞}(U, R^{m}) = ⋂_{r=1}^{∞} C^{r}(U, R^{m}), (3.7.2)

which is a linear subspace of C^{r}(U, R^{m}) for each r.

If f and g are infinitely-differentiable real-valued functions on U, then

(3.7.3) f g is infinitely differentiable on U.

Of course, polynomials on R^{n} are infinitely differentiable. If f(x) ≠ 0 for every x ∈ U, then (3.7.4) 1/f is infinitely differentiable on U.

This implies that rational functions are infinitely differentiable on the complement of the set where the denominator is equal to 0.

3.7.1 Compositions with functions on R

Let V be an open set in the real line, and let ψ be a real-valued function on V. Suppose that (3.7.5) f takes values in V on U,

so that the composition ψ ∘ f is defined as a real-valued function on U. If the partial derivative of f at x ∈ U in the jth variable exists, and if

(3.7.6) ψ is differentiable at f(x) ∈ V,

then the partial derivative of ψ ∘ f in the jth variable exists at x, with

\frac{∂(ψ ∘ f)}{∂x_{j}}(x) = ψ′(f(x)) \frac{∂f}{∂x_{j}} (x). (3.7.7)

This follows from the ordinary chain rule for functions of one variable, by considering f and ψ ∘ f as functions of the jth variable, with the other variables kept fixed.

If f is continuously differentiable on U, and ψ is continuously differentiable on V, then it follows that

(3.7.8) ψ ∘ f is continuously differentiable on U.

If f is r-times continuously differentiable on U, and ψ is r-times continuously differentiable on V, then one can check that

(3.7.9) ψ ∘ f is r-times continuously differentiable on U.

If f is infinitely differentiable on U, and ψ is infinitely differentiable on V, then we get that (3.7.10) ψ ∘ f is infinitely differentiable on U.

3.7.2 A smooth function on R

Consider the real-valued function defined on the real line by

(3.7.11)  ϕ(x) = exp(−1/x) when x > 0
= 0  when x ≤ 0.

It is easy to see that ϕ is infinitely differentiable when x ≠ 0, using the remarks in the preceding paragraph when x > 0. It is well known and not too difficult to show that (3.7.12) ϕ is infinitely differentiable on R,

with derivatives of all orders of ϕ at 0 equal to 0.

3.8 Differentiability

Let m and n be positive integers, let U be an open subset of R^{n}, and let f be a mapping from U into R^{m}. We say that f is differentiable at a point x ∈ U if there is a linear mapping A from R^{n} into R^{m} such that

lim_{h→0} \frac{∥f(x + h) − f(x) − A(h)∥_{2,R^{m}}}{∥h∥_{2,R^{n}}} = 0. (3.8.1)

Here ∥ · ∥_{2,R^{m}} and ∥ · ∥_{2,R^{n}} are the standard Euclidean norms on R^{m} and R^{n}, respectively, as in Section 3.2. Note that the quotient whose limit is being considered is defined for h ∈ R^{n} with h ≠ 0 and ∥h∥_{2,R^{n}} sufficiently small.

If f is differentiable at x, then one can check that

(3.8.2) f is continuous at x.

One can also check directly that the linear mapping A is unique if this case, and another way to see this will be mentioned in Subsection 3.8.1. If n = 1, then this definition of differentiability reduces to the usual notion of differentiability of a function of one variable, where A corresponds to multiplying a real number by the ordinary derivative of f at x.

If f is differentiable at x, then we may put

f′(x) = A, (3.8.3) where A is as in (3.8.1). This may be called the differential of f at x. Although this is similar to the notation for the usual derivative when n = 1, one should keep in mind that this is a linear mapping from R^{n} into R^{m}. We may also use the notation df_{x} (3.8.4)

for the differential of f at x, when it exists.

3.8.1 Directional derivatives

If v ∈ R^{n}, then f(x + t v) (3.8.5)

may be considered as an R^{m}-valued function of t that is defined on an open set in the real line that contains 0. The directional derivative of f at x in the direction v is defined to be

\frac{d}{dt}f(x + t v) at t = 0, (3.8.6)

if the derivative exists, as usual. If f is differentiable at x and v ∈ R^{n}, then one can check that the directional derivative of f at x in the direction v exists, and is equal to f′(x)(v) = df_{x}(v). (3.8.7)

In particular, this means that the partial derivative of f at x in the jth variable exists for each j = 1, …, n, with

\frac{∂f}{∂x_{j}} (x) = f′(x)(e_{j}) = df_{x}(e_{j}), (3.8.8)

where e_{1}, …, e_{n} are the usual standard basis vectors for R^{n}. This can be used to get the uniqueness of the differential of f at x when it exists, as mentioned earlier.

3.8.2 Some properties of the differential

Of course, if f is constant on U, then f is differentiable at every x ∈ U, with

f′(x) = 0. (3.8.9)

If f is a linear mapping from R^{n} into R^{m}, then f is differentiable at every x ∈ R^{n}, with differential equal to f. If f is any continuously-differentiable mapping from U into R^{m}, then it is well known that

(3.8.10) f is differentiable at every x ∈ U.

If f is any mapping from U into R^{m} that is differentiable at x ∈ U, then

f′(x)(v) = df_{x}(v) = ∑_{j=1}^{n} v_{j} \frac{∂f}{∂x_{j}} (x) (3.8.11)

58 CHAPTER 3. SOME GEOMETRY AND ANALYSIS for every v ∈ R^{n}, by (3.8.8).

As usual, one can verify that f is differentiable at x if and only if its m components f_{1}, …, f_{m} are each differentiable at x as real-valued functions on U. In this case, if v ∈ R^{n}, then f′_{1}(x)(v), …, f′_{m}(x)(v) (3.8.12) are the m components of f′(x)(v).

If f is differentiable at x and t ∈ R, then it is easy to see that t f is differentiable at x too, as an R^{m}-valued function on U, with (t f)′(x) = t f′(x). (3.8.13) If g is another mapping from U into R^{m} that is differentiable at x, then one can verify that f + g is differentiable at x as an R^{m}-valued function on U, with (f + g)′(x) = f′(x) + g′(x). (3.8.14)

If f, g are real-valued functions on U that are differentiable at x, then it is not too difficult to show that f g is differentiable at x, with (f g)′(x) = g(x) f′(x) + f(x) g′(x). (3.8.15)

3.9 The chain rule

Let m, n, and p be positive integers, let U be an open subset of R^{n}, and let V be an open subset of R^{m}. Also let f be a mapping from U into V, and let g be a mapping from V into R^{p}. Thus the composition g ∘ f of f and g is defined as a mapping from U into R^{p}. Suppose that (3.9.1) f is differentiable at x ∈ U, and that (3.9.2) g is differentiable at f(x) ∈ V, so that the differentials f′(x) and g′(f(x)) of f and g at x and f(x), respectively, are defined as linear mappings from R^{n} and R^{m} into R^{m} and R^{p}, respectively. There is a version of the chain rule that states that (3.9.3) g ∘ f is differentiable at x, with (g ∘ f)′(x) = g′(f(x)) ∘ f′(x). (3.9.4) Equivalently, this means that d(g ∘ f)_{x} = dg_{f(x)} ∘ df_{x}, (3.9.5) as a linear mapping from R^{n} into R^{p}. It is easy to see that this reduces to the usual chain rule for real-valued functions of a single variable when m = n = p = 1.

Of course, a linear mapping from R^{n} into R^{m} corresponds to an m × n matrix of real numbers in a standard way. Similarly, a linear mapping from R^{m} into R^{p} corresponds to a p × m matrix. The composition of these two linear mappings corresponds to a p × n matrix, which can be expressed in terms of the other two matrices using matrix multiplication.

The entries of the matrices associated to f′(x) and g′(f(x)) are given by the partial derivatives of the components of f and g at x and f(x), respectively. The chain rule implies that

(3.9.6) the partial derivatives of the components of g ∘ f at x can be expressed as a sum of products of the partial derivatives of the components of f and g at x and f(x), respectively, using matrix multiplication.

If f and g are continuously differentiable on U and V, respectively, then one can use this to check that

(3.9.7) g ∘ f is continuously differentiable on U.

More precisely, this uses the fact that

(3.9.8) (∂_{l}g)(f(x)) is continuous on U

for each l = 1, …, m, because ∂_{l}g is continuous on V, and f is continuous on U. This also uses the fact that sums of products of continuous real-valued functions on U are continuous as well.

3.9.1 Additional regularity of compositions

Suppose that g is twice continuously differentiable on V, so that the partial derivatives of g are continuously differentiable on V. If f is continuously differentiable on U, then it follows that

(3.9.9) (∂_{l}g)(f(x)) is continuously differentiable on U

for each l = 1, …, m, as in the preceding paragraph. If f is twice continuously differentiable on U as well, then the partial derivatives of f are continuously differentiable on U. One can use this to get that

(3.9.10) the partial derivatives of g ∘ f are continuously differentiable on U,

because sums of products of continuously differentiable functions on U are continuously differentiable on U too. This means that g ∘ f is twice continuously differentiable on U.

60 CHAPTER 3. SOME GEOMETRY AND ANALYSIS

Similarly, if f and g are r-times continuously differentiable on U and V, respectively, for some positive integer r, then (3.9.11) g ∘ f is r-times continuously differentiable on U. This is the same as saying that (3.9.12) the partial derivatives of g ∘ f

are (r − 1)-times continuously differentiable on U.

The partial derivatives of f and g are (r − 1)-times continuously differentiable on U and V, respectively, by hypothesis. One can use induction to get that (3.9.13) (∂_{l}g)(f(x)) is (r − 1)-times continuously differentiable on U for each l = 1, …, m, because f is (r − 1)-times continuously differentiable on U. This implies (3.9.12), because sums of products of (r −1)-times continuously differentiable functions on U are (r − 1)-times continuously differentiable on U as well.

If f and g are infinitely differentiable on U and V, respectively, then we get that (3.9.14) g ∘ f is infinitely differentiable on U. This was mentioned in Subsection 3.7.1 when m = p = 1.

3.10 A basis for AM_{k}(R^{n})

Let k and n be positive integers, with k ≤ n, and let θ_{1}, …, θ_{n} be the standard coordinate functions on R^{n}, as in Subsection 1.6.2. Remember that the dimension of the space AM_{k}(R^{n}) of alternating k-linear forms on R^{n} is equal to \binom{n}{k}, as in Subsection 1.12.1. We would like to describe a particular basis for AM_{k}(R^{n}), as a vector space over the real numbers.

Let I be a subset of {1, …, n} with exactly k elements. Thus I = {j_{1}, …, j_{k}} (3.10.1) for some integers j_{1}, …, j_{k} with 1 ≤ j_{1} < ⋯ < j_{k} ≤ n. (3.10.2) Put θ^{I} = θ_{j_{1}} ∧ ⋯ ∧ θ_{j_{k}}, (3.10.3) where the right side is as in Section 2.11. This means that

θ^{I} = ∑_{σ∈Sym(k)} sgn(σ) (θ_{j_{1}} ⊗ ⋯ ⊗ θ_{j_{k}})^{σ}, (3.10.4)

3.10. A BASIS FOR AM_{K}(R^{N}) 61

as before. If x(1), …, x(k) are k elements of R^{n}, then we get that

θ^{I}(x(1), …, x(k)) = ∑_{σ∈Sym(k)} sgn(σ) (θ_{j_{1}} ⊗ ⋯ ⊗ θ_{j_{k}})^{σ}(x(1), …, x(k))
(3.10.5)  = ∑_{σ∈Sym(k)} sgn(σ) (θ_{j_{1}} ⊗ ⋯ ⊗ θ_{j_{k}})(x(σ(1)), …, x(σ(k)))

= ∑_{σ∈Sym(k)} sgn(σ) ∏_{l=1}^{k} θ_{j_{l}}(x(σ(l)))

= ∑_{σ∈Sym(k)} sgn(σ) ∏_{l=1}^{k} x_{j_{l}}(σ(l)),

where the second step is as in Section 1.9, and the third step is as in Section 1.6.

Let e_{1}, …, e_{n} be the usual standard basis vectors for R^{n}, and let m_{1}, …, m_{k} be k positive integers less than or equal to n. Note that

θ^{I}(e_{m_{1}}, …, e_{m_{k}}) = ∑_{σ∈Sym(k)} sgn(σ) ∏_{l=1}^{k} θ_{j_{l}}(e_{m_{σ(l)}}), (3.10.6)

as in (3.10.5). This implies that

θ^{I}(e_{m_{1}}, …, e_{m_{k}}) = 0 (3.10.7)

unless there is a τ ∈ Sym(k) such that

m_{τ(l)} = j_{l} for each l = 1, …, k. (3.10.8)

In particular, this means that m_{1}, …, m_{k} are distinct elements of {1, …, n}, because j_{1}, …, j_{k} are distinct elements of {1, …, n}. In this case, there is exactly one τ ∈ Sym(k) with this property, and we get that

θ^{I}(e_{m_{1}}, …, e_{m_{k}}) = sgn(τ). (3.10.9)

Of course, there is a τ ∈ Sym(k) such that (3.10.8) holds if and only if

{m_{1}, …, m_{k}} = {j_{1}, …, j_{k}} = I. (3.10.10)

We also have that m_{1}, …, m_{k} are k distinct positive integers less than or equal to n if and only if {m_{1}, …, m_{k}} (3.10.11)

is a subset of {1, …, n} with exactly k elements.

Remember that a k-linear form μ on R^{n} is uniquely determined by the real numbers μ(e_{m_{1}}, …, e_{m_{k}}), (3.10.12)

1 ≤ m_{1}, …, m_{k} ≤ n, and that every family of n^{k} real numbers corresponds to a k-linear form on R^{n} in this way, as in Subsection 1.6.1. If μ is an alternating k-linear form on R^{n}, then (3.10.12) is equal to 0 unless the m_{l}’s are distinct, as in Section 1.11. We also have that (3.10.12) determines

μ(e_{m_{σ(1)}}, …, e_{m_{σ(k)}}) (3.10.13)

for every σ ∈ Sym(k) in this case, as before. This implies that an alternating klinear form μ on R is uniqely determined by the family of real numbers (3.10.12) with 1 ≤ m_{1} < ⋯ < m_{k} ≤ n, (3.10.14)

as in Section 1.12. We have seen that any family of real numbers of this type corresponds to an alternating k-linear form on R^{n} too.

One can use this to get that

(3.10.15) the collection of θ^{I}, with I ⊆ {1, …, n} and #I = k, is a basis for AM_{k}(R^{n}).

More precisely, one can use the θ^{I}’s to get that that there is an alternating k-linear form μ on R^{n} for which (3.10.12) is any real number when (3.10.14) holds.

Remember that we take

AM_{0}(R^{n}) = R, (3.10.16)

as in Section 2.12. Let us put

θ^{I} = 1 when I = ∅, (3.10.17)

so that (3.10.15) holds when k = 0.

3.11 A basis for AM(R^{n})

Let n be a positive integer, and let us contunue with the notation used in the previous section. As in Subsection 2.12.1, we put

AM(R^{n}) = ⨁_{k=0}^{n} AM_{k}(R^{n}), (3.11.1)

and we shall normally identify elements of AM_{l}(R^{n}) with their images in AM(R^{n}) under the obvious embeddings for each l = 0, …, n. Using this identification, we get that

(3.11.2) the collection of θ^{I}, I ⊆ {1, …, n}, is a basis for AM(R^{n}),

as a vector space over the real numbers.

3.11. A BASIS FOR AM(R^{N}) 63

Let k be a positive integer, and let λ_{1}, …, λ_{k} be k linear functionals on R^{n}. If τ ∈ Sym(k) and x(1), …, x(k) ∈ R^{n}, then

(λ_{1} ⊗ ⋯ ⊗ λ_{k})^{τ} (x(1), …, x(k)) = (λ_{1} ⊗ ⋯ ⊗ λ_{k})(x(τ(1)), …, x(τ(k)))
(3.11.3)  = ∏_{l=1}^{k} λ_{l}(x(τ(l))),

using the notation in Section 1.9 in the first step, and the notation in Section 1.6 in the second step. It follows that

(λ_{1} ⊗ ⋯ ⊗ λ_{k})^{τ} (x(1), …, x(k)) = ∏_{p=1}^{k} λ_{τ^{−1}(p)}(x(p))
(3.11.4)  = (λ_{τ^{−1}(1)} ⊗ ⋯ ⊗ λ_{τ^{−1}(k)})(x(1), …, x(k)).

This means that

(λ_{1} ⊗ ⋯ ⊗ λ_{k})^{τ} = λ_{τ^{−1}(1)} ⊗ ⋯ ⊗ λ_{τ^{−1}(k)}. (3.11.5)

We can replace τ with τ^{−1} to get that

λ_{τ(1)} ⊗ ⋯ ⊗ λ_{τ(k)} = (λ_{1} ⊗ ⋯ ⊗ λ_{k})^{τ^{−1}}. (3.11.6)

This implies that

(3.11.7)  λ_{τ(1)} ∧ ⋯ ∧ λ_{τ(k)} = k! A_{k}(λ_{τ(1)} ⊗ ⋯ ⊗ λ_{τ(k)})
= k! A_{k}((λ_{1} ⊗ ⋯ ⊗ λ_{k})^{τ^{−1}}),

where the first step is as in Section 2.11. Thus

(3.11.8)  λ_{τ(1)} ∧ ⋯ ∧ λ_{τ(k)} = sgn(τ^{−1}) k! A_{k}(λ_{1} ⊗ ⋯ ⊗ λ_{k})
= sgn(τ) λ_{1} ∧ ⋯ ∧ λ_{k},

where the first step is as in Section 1.13. The second step uses the fact that sgn(τ^{−1}) = (sgn(τ))^{−1} = sgn(τ). In particular, if λ_{1}, λ_{2} are linear functionals on R^{n}, then λ_{2} ∧ λ_{1} = −λ_{1} ∧ λ_{2}, (3.11.9)

which also follows from a remark in Section 2.12. Using this, we get that

λ ∧ λ = 0 (3.11.10)

for every linear functional λ on R^{n}.

Let I_{1}, I_{2} be subsets of {1, …, n} with k_{1}, k_{2} elements, respectively, and put k = k_{1} + k_{2}. If I_{1} ∩ I_{2} ≠ ∅, (3.11.11)

then it is easy to see that θ^{I_{1}} ∧ θ^{I_{2}} = 0, (3.11.12) using the remarks in the preceding paragraph. Note that (3.11.11) holds automatically when k > n. Of course, we also know that AM_{k}(R^{n}) = {0} in this case, as in Subsection 1.11.1.

Suppose now that I_{1} ∩ I_{2} = ∅, (3.11.13)

and put I = I_{1} ∪ I_{2}. (3.11.14)

Let 1 ≤ j_{1} < ⋯ < j_{k} ≤ n be a list of the elements of I, in order. There is a unique τ ∈ Sym(k) such that

τ({1, …, k_{1}}) = I_{1}, τ({k_{1} + 1, …, k}) = I_{2}, (3.11.15)

and τ is increasing on each of {1, …, k_{1}} and {k_{1} + 1, …, k}. Note that τ is a shuffle, as in Section 2.15, with r = 2. We also have that

θ^{I_{1}} ∧ θ^{I_{2}} = θ_{j_{τ(1)}} ∧ ⋯ ∧ θ_{j_{τ(k)}} = sgn(τ) θ_{j_{1}} ∧ ⋯ ∧ θ_{j_{k}} = sgn(τ) θ^{I}, (3.11.16)

using (3.11.8) in the second step.

3.12 Differential forms

Let n be a positive integer, and let E be a nonempty subset of R^{n}. A differential form on E is (3.12.1) a function on E with values in AM(R^{n}).

Similarly, if k is a nonnegative integer, then a differential k-form on E is

(3.12.2) a function on E with values in AM_{k}(R^{n}).

This is automatically equal to 0 when k > n. A differential form on E may be considered as a sum of differential k-forms on E, 0 ≤ k ≤ n. Note that a differential 0-form on E is the same as a real-valued function on E.

A differential k-form on E may be expressed as

α = ∑_{#I=k} α_{I} θ^{I}, (3.12.3)

where the sum is taken over all subsets I of {1, …, n} with exactly k elements. This uses the fact that the θ^{I}’s, #I = k, form a basis for AM_{k}(R^{n}), as in Section 3.10. The α_{I}’s, #I = k, are real-valued functions on E.

Similarly, a differential form on E may be expressed as

β = ∑_{I} β_{I} θ^{I}, (3.12.4)

where now the sum is taken over all subsets I of {1, …, n}. The β_{I}’s are realvalued functions on E, as before.

The space of differential forms on E is a vector space over the real numbers with respect to pointwise addition and scalar multiplication. The space of differential k-forms on E may be considered as a linear subspace of the space of differential forms on E. If α_{1}, α_{2} are differential forms on E, then their wedge product α_{1} ∧ α_{2} (3.12.5)

is defined as a differential form on E as well, using the wedge product of alternating multilinear forms on R^{n} at each point. If α_{1}, α_{2} are differential k_{1}, k_{2}-forms on E for some nonnegative integers k_{1}, k_{2}, respectively, then

(3.12.6) α_{1} ∧ α_{2} is a differential (k_{1} + k_{2})-form on E.

Of course, this is equal to 0 when

k_{1} + k_{2} > n. (3.12.7)

3.12.1 Continuous differential forms

It is sometimes convenient to identify AM_{k}(R^{n}) with R^{\binom{n}{k}} when k ≤ n, using the basis for AM_{k}(R^{n}) consisting of the θ^{I}’s with #I = k. To do this, we should list the subsets of \{1, …, n\} with exactly k elements in a sequence, with \binom{n}{k} terms. The order in which these subsets of {1, …, n} are listed will normally not really matter. In particular, this leads to a metric on AM_{k}(R^{n}), corresponding to the standard Euclidean metric on R^{\binom{n}{k}}. This metric does not depend on the order in which the subsets of {1, …, n} with k elements are listed, because the standard Euclidean metric on R^{m} is invariant under permutations of the coordinates for any positive integer m.

Similarly, we may identify AM(R^{n}) with R^{2^{n}}, using the basis for AM(R^{n}) consisting of the θ^{I}’s corresponding to all subsets I of {1, …, n}. This involves listing the subsets of {1, …, n} in a sequence with 2^{n} terms, and the order in which the subsets of {1, …, n} are listed will not normally matter, as before. This leads to a metric on AM(R^{n}), corresponding to the standard Euclidean metric on R^{2^{n}}, that does not depend on the order in which the subsets of {1, …, n} are listed, as before.

Let us say that a differential form on E is continuous if it corresponds to a continuous mapping from E into R^{2^{n}} in this way. This does not depend on the order in which the subsets of {1, …, n} are listed, as usual. Equivalently, a differential form β on E as in (3.12.4) is continuous if and only if

(3.12.8) the coefficients β_{I} are continuous as real-valued functions on E

for all subsets I of {1, …, n}. The space

C(E, AM(R^{n})) = C(E, R^{2^{n}}) (3.12.9)

of all continuous differential forms on E is a linear subspace of the space of all differential forms on E.

Similarly, we say that a differential k-form on E is continuous if it corresponds to a continuous mapping from E into R^{\binom{n}{k}} when k ≤ n, which does not depend on the order in which the subsets of {1, …, n} with k elements are listed. This is the same as saying that a differential k-form α on E as in (3.12.3) is continuous if and only if

(3.12.10) the coefficients α_{I} are continuous as real-valued functions on E

for all subsets I of {1, …, n} with exactly k elements. The space

C(E, AM_{k}(R^{n})) = C(E, R^{\binom{n}{k}}) (3.12.11)

of all continuous differential k-forms on E is a linear subspace of the space of all differential k-forms on E. This may be considered as a linear subspace of the space of all continuous differential forms on E.

If α_{1}, α_{2} are continuous differential forms on E, then it is easy to see that

(3.12.12) α_{1} ∧ α_{2} is continuous as a differential form on E.

More precisely, if α_{1} ∧ α_{2} is expressed as in (3.12.4), then the corresponding coefficients of the θ^{I}’s may be expressed in terms of sums of products of the analogous coefficients of α_{1} and α_{2}.

3.13 Additional regularity of differential forms

Let n and r be positive integers, and let U be a nonempty open subset of R^{n}. Let us say that a differential form on U is r-times continuously differentiable on U if it corresponds to an r-times continuously-differentiable mapping from U into R^{2^{n}}, as in Subsection 3.12.1. This does not depend on the order in which the subsets of {1, …, n} are listed, as before. A differential form β on U as in (3.12.4) is r-times continuously differentiable if and only if

(3.13.1) the coefficients β_{I} are r-times continuously differentiable as real-valued functions on U

for all subsets I of {1, …, n}. The space

C^{r}(U, AM(R^{n})) = C^{r}(U, R^{2^{n}}) (3.13.2)

of all r-times continuously differentiable differential forms on U is a linear subspace of the space of all continuous differential forms on U.

Similarly, if k is a nonnegative integer less than or equal to n, then we say that a differential k-form on U is r-times continuously differentiable if it corresponds to an r-times continuously-differentiable mapping from U into R^{\binom{n}{k}}. This does not depend on the order in which the subsets of {1, …, n} with k elements are listed, as usual. A differential k-form α on U as in (3.12.3) is r-times continuously differentiable on U if and only if

(3.13.3) the coefficients α_{I} are r-times continuously differentiable as real-valued functions on U

for all subsets I of {1, …, n} with exactly k elements. The space

C^{r}(U, AM_{k}(R^{n})) = C^{r}(U, R^{\binom{n}{k}}) (3.13.4)

of all r-times continuously-differentiable differential k-forms on U is a linear subspace of the space of all continuous differential k-forms on U. This may be considered as a linear subspace of the space of all r-times continuouslydifferentiable differential forms on U.

Let us say that a differential form on U is infinitely differentiable or smooth if it corresponds to an infinitely differentiable mapping from U into R^{2^{n}}, which does not depend on the order in which the subsets of {1, …, n} are listed, as usual. A differential form β on U as in (3.12.4) is infinitely differentiable on U if and only if

(3.13.5) the coefficients β_{I} are infinitely differentiable as real-valued functions on U

for all subsets I of {1, …, n}. The space

C^{∞}(U, AM(R^{n})) = C^{∞}(U, R^{2^{n}}) (3.13.6)

of all infinitely-differentiable differential forms on U is a linear subspace of (3.13.2) for each r.

Similarly, a diiferential k-form on U is said to be infinitely differentiable or smooth if it corresponds to an infinitely-differentiable mapping from U into R^{\binom{n}{k}}, which does not depend on the order in which the subsets of \{1, …, n\} with k elements are listed. A differential k-form α on U as in (3.12.3) is infinitely differentiable on U if and only if

(3.13.7) the coefficients α_{I} are infinitely differentiable as real-valued functions on U

for all subsets I of {1, …, n} with exactly k elements. The space

C^{∞}(U, AM_{k}(R^{n})) = C^{∞}(U, R^{\binom{n}{k}}) (3.13.8)

of all infinite-differentiable differential k-forms on U is a linear subspace of (3.13.4) for each r. This may be considered as a linear subspace of the space of all infinitely-differentiable differential forms on U.

As before, it is sometimes convenient to interpret r-times continuous differentiability of a differential form as being the same as continuity when r = 0. If α_{1}, α_{2} are r-times continuously-differentiable differential forms on U, then one can check that

(3.13.9) α_{1} ∧ α_{2} is r-times continuously differentiable

as a differential form on U. If α_{1}, α_{2} are infinitely differentiable on U, then it follows that (3.13.10) α_{1} ∧ α_{2} is infinitely differentiable on U

as well.

3.14 Differentials as 1-forms

Let n be a positive integer, let U be a nonempty open subset of R^{n}, and let f be a real-valued function on U. Let us say that f is differentiable on U if

(3.14.1) f is differentiable at every x ∈ U,

in the sense of Section 3.8. In this case, the differential df of f defines a differential 1-form on U, because it defines a linear functional df_{x} on R^{n} at every x ∈ U.

It is customary to consider x_{j} as the jth coordinate function on R^{n} for each j = 1, …, n, which we have also denoted θ_{j}. The differential dx_{j} of x_{j} is the same as θ_{j} at every point in R^{n}. Thus one often uses dx_{j} for the differential 1-form on R^{n} that is equal to θ_{j} at every point.

If f is differentiable on U, then the differential of f may be expressed as

df = ∑_{j=1}^{n}(∂_{j}f) dx_{j} = ∑_{j=1}^{n} \frac{∂f}{∂x_{j}} dx_{j}, (3.14.2)

using this notation. This means that

df_{x} = ∑_{j=1}^{n}(∂_{j}f)(x) dx_{j} = ∑_{j=1}^{n} \frac{∂f}{∂x_{j}} (x) dx_{j} (3.14.3)

for each x ∈ U. If v ∈ R^{n}, then we get that

df_{x}(v) = ∑_{j=1}^{n}(∂_{j}f)(x) v_{j} = ∑_{j=1}^{n} \frac{∂f}{∂x_{j}} (x) v_{j}. (3.14.4)

If f is r-times continuously differentiable on U for some positive integer r, then

(3.14.5) df is (r − 1)-times continuously differentiable

as a differential 1-form on U. If f is infinitely differentiable on U, then df is infinitely differentiable as a differential 1-form on U.

Let I = {j_{1}, …, j_{k}} be a subset of {1, …, n} with k elements, where the j_{l}’s are strictly increasing. We may use the notation

dx^{I} = dx_{j_{1}} ∧ ⋯ ∧ dx_{j_{k}}, (3.14.6)

which is the differential k-form equal to θ^{I} at every point.

Thus a differential k-form on U may be expressed as

α = ∑_{#I=k} α_{I} dx^{I}, (3.14.7)

where the sum is taken over all subsets I of {1, …, n} with exactly k elements, and α_{I} is a real-valued function on U for each such I. Similarly, a differential form on U may be expressed as

β = ∑_{I} β_{I} dx^{I}, (3.14.8)

where the sum is taken over all subsets I of {1, …, n}, and β_{I} is a real-valued function on U for each such I.

3.15 Isometric mappings

Although the topics in this section may not be needed for the moment, they do fit nicely with the discussion of metrics, norms, and inner products at the beginning of the chapter. Of course, we are also concerned with various properties of mappings more broadly, and some related matters will be mentioned later.

Let (X, d_{X}) and (Y, d_{Y}) be metric spaces. A mapping f from X into Y is said to be an isometry if

d_{Y} (f(x), f(w)) = d_{X}(x, w) (3.15.1)

for every x, w ∈ X. Note that this implies that f is one-to-one on X. If f also maps X onto Y, then it is easy to see that the inverse mapping f^{−1} is an isometry from Y onto X.

Let (Z, d_{Z}) be another metric space. If f is an isometry from X into Y and g is an isometry from Y into Z, then it is easy to see that

(3.15.2) g ∘ f is an isometry from X into Z.

Let V, W be vector spaces over the real numbers with norms N_{V}, N_{W}, respectively. A linear mapping T from V into W is said to be an isometric linear mapping if N_{W} (T (v)) = N_{V} (v) (3.15.3)

for every v ∈ V. Of course, this implies that

N_{W} (T (u) − T (v)) = N_{W} (T (u − v)) = N_{V} (u − v) (3.15.4) for all u, v ∈ V, so that T is an isometry from V into W with respect to the metrics associated to N_{V}, N_{W}, respectively.

If T is an isometric linear mapping from V onto W, then the inverse mapping T^{−1} is an isometric linear mapping from W onto V. If T is a one-to-one linear mapping from V into W, and V, W have the same finite dimension, then it is well known that T maps V onto W.

Let Z be another vector space over the real numbers with a norm N_{Z}. If T is an isometric linear mapping from V into W, and R is an isometric linear mapping from W into Z, then

(3.15.5) R ∘ T is an isometric linear mapping from V into Z.

Let ⟨·, ·⟩_{V}, ⟨·, ·⟩_{W} be inner products on V, W, respectively. Observe that a linear mapping T from V into W is an isometric linear mapping with respect to the norms associated to these inner products if and only if

⟨T (v), T (v)⟩_{W} = ⟨v, v⟩_{V} (3.15.6)

for every v ∈ V. In this case, one can check that

⟨T (u), T (v)⟩_{W} = ⟨u, v⟩_{V} (3.15.7)

for every u, v ∈ V.

Some connections between isometries and volumes of sets will be mentioned in Section 4.12. We shall say a bit more about isometries between finitedimensional inner product spaces in Subsection A.9.1.

Pull-backs and exterior differentiation

4.1 Tensors of type (0, k)

Let k and n be positive integers, and let E be a nonempty subset of R^{n}. A tensor field of type (0, k) on E is

(4.1.1) a function on E with values in M_{k}(R^{n}).

If k = 1, then this is the same as a differential 1-form on E. If k = 0, then this is interpreted as being a real-valued function on E, as before. One may also consider tensor fields of type (r, s) for arbitrary nonnegative integers r, s, as in Definition 2.15 on p63 of [184], but we shall not pursue this here.

If j_{1}, …, j_{k} are positive integers less than or equal to k, then

θ_{j_{1}} ⊗ ⋯ ⊗ θ_{j_{k}} (4.1.2)

is an element of M_{k}(R^{n}), as in Section 1.6. The collection of these elements of M_{k}(R^{n}) forms a basis for M_{k}(R^{n}), as a vector space over the real numbers, as before. A tensor field of type (0, k) on E may be expressed as

a = ∑_{{1,...,n}^{k}} a_{j_{1},...,j_{k}} θ_{j_{1}} ⊗ ⋯ ⊗ θ_{j_{k}}, (4.1.3)

where the sum is taken over the set {1, …, n}^{k} of all k-tuples of positive integers less than or equal to n. Here the coefficients a_{j_{1},...,j_{k}} are real-valued functions on E for all such k-tuples. The space of tensor fields of type (0, k) on E is a vector space over the real numbers with respect to pointwise addition and scalar multiplication.

72 CHAPTER 4. PULL-BACKS AND EXTERIOR DIFFERENTIATION

4.1.1 Continuous tensor fields

It is sometimes convenient to identiy M_{k}(R^{n}) with R^{n^{k}}, using the basis mentioned in the preceding paragraph. This involves listing the elements of

{1, …, n}^{k} (4.1.4)

in a sequence with n^{k} terms, and the order in which the elements of (4.1.4) are listed will not normally matter, as usual. This leads to a metric on M_{k}(R^{n}), corresponding to the standard Euclidean metric on R^{n^{k}}. This metric on M_{k}(R^{n}) does not depend on the order in which the elements of (4.1.4) are listed.

Let us say that a tensor field of type (0, k) on E is continuous if it corresponds to a continuous mapping from E into R^{n^{k}} in this way. This does not depend on the order in which the elements of (4.1.4) are listed. Equivalently, a tensor field a of type (0, k) on E as in (4.1.3) is continuous if and only if

(4.1.5) the coefficients a_{j_{1},...,j_{k}} are continuous as real-valued functions on E

for all j_{1}, …, j_{k}. The space

C(E, M_{k}(R^{n})) = C(E, R^{n^{k}}) (4.1.6)

of all continuous tensor fields of type (0, k) on E is a linear subspace of the space of all tensor fields of type (0, k) on E.

If a, b are tensor fields of types (0, k_{1}), (0, k_{2}) on E, respectively, for some nonnegative integers k_{1}, k_{2}, then their product

a ⊗ b (4.1.7)

may be defined as a (k_{1} +k_{2})-linear form on R^{n} at each point in E, as in Section 2.8. This defines (4.1.7) as a tensor field of type (0, k_{1} + k_{2}) on E. If a, b are continuous as tensor fields on E, then one can check that

(4.1.8) a ⊗ b is continuous as a tensor field on E.

More precisely, the coefficients of (4.1.7) with respect to the usual basis for M_{k_{1}+k_{2}}(R^{n}) may be expressed in terms of products of the coefficients of a and b with respect to the usual bases for M_{k_{1}}(R^{n}) and M_{k_{2}}(R^{n}), respectively.

4.2 More on tensor fields

Let n be a positive integer, let U be a nonempty open subset of R^{n}, and let k, r be nonnegative integers. A tensor field of type (0, k) on U is said to be r-times continuously differentiable on U if it corresponds to an r-times continuouslydifferentiable mapping from U into R^{n^{k}} as in Subsection 4.1.1. This does not depend on the order in which the elements of {1, …, n}^{k} are listed, as usual. Remember that r-times continuous differentiability is interpreted as being the same as continuity when r = 0.

A tensor field a of type (0, k) on U as in (4.1.3) is r-times continuously differentiable if and only if

(4.2.1) the coefficients a_{j_{1},...,j_{k}} are r-times continuously differentiable as real-valued functions on U

for all j_{1}, …, j_{k}. The space

C^{r}(U, M_{k}(R^{n})) = C^{r}(U, R^{n^{k}}) (4.2.2)

of all r-times continuously-differentiable tensor fields of type (0, k) on U is a linear subspace of the space of all continuous tensor fields of type (0, k) on U.

A tensor field of type (0, k) on U is said to be infinitely differentiable or smooth if it corresponds to an infinitely-differentiable mapping from U into R^{n^{k}}, which does not depend on the order in which the elements of {1, …, n}^{k} are listed. A tensor field a of type (0, k) on U as in (4.1.3) is infinitely differentiable on U if and only if

(4.2.3) the coefficients a_{j_{1},...,j_{k}} are infinitely differentiable as real-valued functions on U

for all j_{1}, …, j_{k}. The space

C^{∞}(U, M_{k}(R^{n})) = C^{∞}(U, R^{n^{k}}) (4.2.4)

of all infinitely-differentiable tensor fields of type (0, k) on U is a linear subspace of (4.2.2) for each r.

Let a, b be tensor fields of types (0, k_{1}), (0, k_{2}) on U, respectively, for some nonnegative integers k_{1}, k_{2}, so that their product a⊗b is defined as a tensor field of type (0, k_{1} + k_{2}) on U, as in Subsection 4.1.1. If a, b are r-times continuously differentiable on U, then one can verify that

(4.2.5) a ⊗ b is r-times continuously differentiable

as a tensor field on U. If a, b are infinitely differentiable on U, then we get that

(4.2.6) a ⊗ b is infinitely differentiable

as a tensor field on U.

If a real-valued function f is differentiable on U, then the differential df of f defines a tensor of type (0, 1) on U, as in Section 3.14. Note that

dx_{j_{1}} ⊗ ⋯ ⊗ dx_{j_{k}} (4.2.7)

is the same as the tensor field of type (0, k) equal to (4.1.2) at every point. Thus a tensor field a of type (0, k) on U may be expressed as

a = ∑_{{1,...,n}^{k}} a_{j_{1},...,j_{k}} dx_{j_{1}} ⊗ ⋯ ⊗ dx_{j_{k}}, (4.2.8)

as in (4.1.3).

4.3 Vector fields

Let n be a positive integer, and let E be a nonempty subset of R^{n}. An R^{n}- valued function on E may be called a vector field on E. This may be considered as a tensor field of type (1, 0) on E, as in Definition 2.15 on p63 of [184].

Let k be a positive integer, and let a be a tensor field of type (0, k) on E. Suppose that a is as in (4.1.3), so that the value a_{x} of a at x ∈ E is given by

a_{x} = ∑_{{1,...,n}^{k}} a_{j_{1},...,j_{k}}(x) θ_{j_{1}} ⊗ ⋯ ⊗ θ_{j_{k}}, (4.3.1)

as a k-linear form on R^{n}. Let ξ_{1}, …, ξ_{k} be k vector fields on E, and let ξ_{l,j} be the jth component of ξ_{l} for j = 1, …, n and l = 1, …, k. We can evaluate a at ξ_{1}, …, ξ_{k} at each point in E to get a real-valued function

a(ξ_{1}, …, ξ_{k}) (4.3.2)

on E. More precisely, the value of this function at x ∈ E is

a_{x}(ξ_{1}(x), …, ξ_{k}(x)) = ∑_{{1,...,n}^{k}} a_{j_{1},...,j_{k}}(x) ∏_{l=1}^{k} ξ_{l,j_{l}}(x). (4.3.3)

Suppose for the moment that a is continuous as a tensor field of type (0, k) on E, as in Subsection 4.1.1. Suppose also that ξ_{1}, …, ξ_{k} are continuous vector fields on E, which is to say that they are continuous as R^{n}-valued functions on E. Under these conditions,

(4.3.4) a(ξ_{1}, …, ξ_{k}) is continuous as a real-valued function on E.

4.3.1 Associated differential operators

Now let U be a nonempty open subset of R^{n}. If ξ is a vector field on U, then

X_{ξ} = ∑_{j=1}^{n} ξ_{j} \frac{∂}{∂x_{j}} (4.3.5)

defines a first-order linear differential operator on U, where ξ_{j} is the jth component of ξ for each j = 1, …, n. If f is a differentiable real-valued function on U, then

X_{ξ}(f) = ∑_{j=1}^{n} ξ_{j} \frac{∂f}{∂x_{j}} (4.3.6)

is a real-valued function on U, whose value

(X_{ξ}(f))(x) = ∑_{j=1}^{n} ξ_{j}(x) \frac{∂f}{∂x_{j}} (x) (4.3.7)

4.4. PULLING SOME TENSORS BACK 75 at x ∈ U is the same as the directional derivative of f at x in the direction ξ(x). Equivalently, df(ξ) = X_{ξ}(f), (4.3.8) where df is considered as a tensor field of type (0, 1) on U. This means that df_{x}(ξ(x)) = (X_{ξ}(f))(x) (4.3.9) for every x ∈ U.

If f is r-times continuously differentiable on U for some positive integer r, and ξ is (r − 1)-times continuously differentiable on U, then (4.3.10) X_{ξ}(f) is (r − 1)-times continuously differentiable on U. If ξ and f are infinitely differentiable on U, then (4.3.11) X_{ξ}(f) is infinitely differentiable on U. Let a be a tensor field of type (0, k) on U, and let ξ_{1}, …, ξ_{k} be k vector fields on U. If a and ξ_{1}, …, ξ_{k} are r-times continuously differentiable on U, then (4.3.12) a(ξ_{1}, …, ξ_{k}) is r-times continuously differentiable on U. If a and ξ_{1}, …, ξ_{k} are infinitely differentiable on U, then (4.3.13) a(ξ_{1}, …, ξ_{k}) is infinitely differentiable on U.

4.4 Pulling some tensors back

Let n be a positive integer, let U be a nonempty open subset of R^{n}, and let k be a nonnegative integer. Also let n_{1} be a positive integer, let U_{1} be a nonempty open subset of R^{n_{1}}, and let ψ be a mapping from U_{1} into R^{n} such that ψ(U_{1}) ⊆ U. (4.4.1) Suppose that ψ is differentiable on U_{1}, in the sense that ψ is differentiable at every point in U_{1}, as in Section 3.8. If a is a tensor field of type (0, k) on U, then we would like to pull a back to get a tensor field ψ^{∗}(a) (4.4.2) of type (0, k) on U_{1}. This may also be denoted δψ(a), as in Definition 2.22 on p68 of [184].

If k = 0, then a is simply a real-valued function on U. In this case, we put ψ^{∗}(a) = a ∘ ψ, (4.4.3) which is a real-valued function on U_{1}.

Suppose that k ≥ 1, and let a_{x} be the value of a at x ∈ U, which is a k-linear form on R^{n}. Let w ∈ U_{1} be given, so that the differential dψ_{w} of ψ at w is a linear mapping from R^{n_{1}} into R^{n}. The value of ψ^{∗}(a) at w is defined to be the k-linear form on R^{n_{1}} given by

(ψ^{∗}(a))_{w} = (dψ_{w})^{∗}(a_{ψ(w)}), (4.4.4)

where the right side is defined as in Section 2.3. Note that ψ(w) ∈ U by hypothesis, so that a_{ψ(w)} is defined as a k-linear form on R^{n}. If v_{1}, …, v_{k} ∈ R^{n_{1}}, then

(4.4.5)  (ψ^{∗}(a))_{w}(v_{1}, …, v_{k}) = ((dψ_{w})^{∗}(a_{ψ(w)}))(v_{1}, …, v_{k})
= a_{ψ(w)}(dψ_{w}(v_{1}), …, dψ_{w}(v_{k})),

as in Section 2.3.

4.4.1 Some properties of pull-backs

Note that ψ^{∗}(a) is linear in a, as in Section 2.3. If a is a differential k-form on U, then a_{x} is an alternating k-linear form on R^{n} for each x ∈ U, as in Section 3.12. This implies that (4.4.4) is an alternating k-linear form on R^{n_{1}} for every w ∈ U_{1}, as in Section 2.4. This means that

(4.4.6) ψ^{∗}(a) is a differential k-form on U_{1}

in this case. Similarly, if a_{x} is a symmetric k-linear form on R^{n} for every x ∈ U, then (4.4.7) (ψ^{∗}(a))_{w} is a symmetric k-linear form on R^{n_{1}}

for every w ∈ U_{1}.

Suppose that a, b are tensor fields of types (0, k_{1}), (0, k_{2}) on U for some nonnegative integers k_{1}, k_{2}, respectively. Observe that

ψ^{∗}(a ⊗ b) = ψ^{∗}(a) ⊗ ψ^{∗}(b) (4.4.8)

as tensor fields of type (0, k_{1} +k_{2}) on U_{1}. This follows from the analogous statement for multilinear forms in Subsection 2.8.1. Similarly, if a, b are differential k_{1}, k_{2}-forms on U, then

ψ^{∗}(a ∧ b) = ψ^{∗}(a) ∧ ψ^{∗}(b), (4.4.9)

as differential (k_{1} + k_{2})-forms on U_{1}. This follows frm the analogous statement for alternating multilinear forms in Subsection 2.12.1.

4.5 More on pull-backs

Let us continue with the same hypotheses and notation as in the previous section. Remember that dx_{j} is a differential 1-form on R^{n} for each j = 1, …, n, whose value at each point is the jth standard coordinate function θ_{j}, as in Section 3.14. Equivalently, this may be considered as a tensor field of type (0, 1) on R^{n}, as in Sections 4.1 and 4.2.

Let ψ_{j} be the jth component of ψ for each j = 1, …, n, which is a differentiable real-valued function on U_{1}. Of course, the differential of ψ_{j} corresponds to the jth component of the differential of ψ in a simple way for each j = 1, …, n. One can use this to get that

ψ^{∗}(dx_{j}) = dψ_{j} (4.5.1)

on U_{1} for each j = 1, …, n. More precisely, the right side is a differential 1-form on U_{1}, as in Section 3.14, and thus a tensor field of type (0, 1). To get (4.5.1), one can take a = dx_{j} in (4.4.5), so that k = 1.

Let a be a tensor field of type (0, k) on U, expressed as in (4.2.8). Observe that

(4.5.2) ψ^{∗}(a) = ∑_{{1,...,n}^{k}} ψ^{∗}(a_{j_{1},...,j_{k}}) ψ^{∗}(dx_{j_{1}}) ⊗ ⋯ ⊗ ψ^{∗}(dx_{j_{k}})
= ∑_{{1,...,n}^{k}}(a_{j_{1},...,j_{k}} ∘ ψ) dψ_{j_{1}} ⊗ ⋯ ⊗ dψ_{j_{k}}

on U_{1}.

Similarly, a differential k-form α on U may be expressed as

α = ∑_{1≤j_{1}<···<j_{k}≤n} α_{j_{1},...,j_{k}} dx_{j_{1}} ∧ ⋯ ∧ dx_{j_{k}}, (4.5.3)

as in Section 3.14. More precisely, the sum on the right is taken over all k-tuples (j_{1}, …, j_{k}) of integers such that 1 ≤ j_{1} < ⋯ < j_{k} ≤ n. In this case, we get that

(4.5.4) ψ^{∗}(α) = ∑_{1≤j_{1}<···<j_{k}≤n} ψ^{∗}(α_{j_{1},...,j_{k}}) ψ^{∗}(dx_{j_{1}}) ∧ ⋯ ∧ ψ^{∗}(dx_{j_{k}})
= ∑_{1≤j_{1}<···<j_{k}≤n}(α_{j_{1},...,j_{k}} ∘ ψ) dψ_{j_{1}} ∧ ⋯ ∧ dψ_{j_{k}}

on U_{1}.

4.5.1 Pull-backs and composiitons

Let n_{0} be a positive integer, let U_{0} be a nonempty open subset of R^{n_{0}}, and let ϕ be a mapping from U_{0} into R^{n_{1}} such that

ϕ(U_{0}) ⊆ U_{1}. (4.5.5)

Suppose that ϕ is differentiable on U_{0} too. Thus ψ ∘ ϕ is a mapping from U_{0} into R^{n} with (ψ ∘ ϕ)(U_{0}) = ψ(ϕ(U_{0})) ⊆ ψ(U_{1}) ⊆ U. (4.5.6)

Remember that (4.5.7) ψ ∘ ϕ is differentiable on U_{0},

as in Section 3.9.

If a is a tensor field of type (0, k) on U, then one can check that

(ψ ∘ ϕ)^{∗}(a) = ϕ^{∗}(ψ^{∗}(a)) (4.5.8)

on U_{0}. This is very simple when k = 0. If k ≥ 1, then this follows from the chain rule, and the analogous statement for multilinear forms on Section 2.3.

4.5.2 Regularity of pull-backs

Let a be a tensor field of type (0, k) on U again, expressed as in (4.2.8). If a is continuous on U, and ψ is continuously differentiable on U_{1}, then

(4.5.9) ψ^{∗}(a) is continuous on U_{1}.

More precisely, if k = 0, then this holds when ψ is continuous on U_{1}. If k ≥ 1, then this can be obtained using (4.5.2).

Similarly, if a is r-times continuously differentiable on U for some positive integer r, and ψ is (r + 1)-times continuously differentiable on U_{1}, then

(4.5.10) ψ^{∗}(a) is r-times continuously differentiable on U_{1}.

If k = 0, then it is enough to ask that ψ be r-times continuously differentiable on U_{1}, as in Section 3.9. If a is infinitely differentiable on U, and ψ is infinitely differentiable on U_{1}, then it follows that

(4.5.11) ψ^{∗}(a) is infinitely differentiable on U_{1}.

4.6 Exterior differentiation

Let n be a positive integer, and let U be a nonempty open subset of R^{n}. If α is a continuously-differentiable differential form on U, then we would like to define the exterior derivative dα (4.6.1)

of α as a continuous differential form on U. More precisely, if α is a continuouslydifferentiable differential k-form on U for some nonnegative integer k, then

(4.6.2) dα is a continuous differential (k + 1)-form on U.

In particular, dα = 0 when k ≥ n. (4.6.3)

If α is a differential 0-form on U, then α is a real-valued function on U. If α is continuously differentiable on U, then we take

(4.6.4) dα to be the differential of α, considered as a differential 1-form on U, as in Section 3.14.

Suppose that α is a differential k-form on U, with 1 ≤ k ≤ n, which may be expressed as

α = ∑_{#I=k} α_{I} dx^{I}, (4.6.5)

as in Section 3.14. If α is continuously differentiable on U, then

(4.6.6) α_{I} is continuously differentiable

as a real-valued function on U for each subset I of {1, …, n} with exactly k elements, as in Section 3.13. In this case, we put

dα = ∑_{#I=k} dα_{I} ∧ dx^{I}. (4.6.7)

One can check that this defines a continuous differential (k+1)-form on U. Note that this is equal to 0 when k = n, as before.

If α is any continuously-differentiable differential form on U, then α may be expressed as a sum of continuously-differentiable differential k-forms on U, 0 ≤ k ≤ n. Under these conditions, dα is defined to be the sum of the exterior derivatives of these differential k-forms. This defines the exterior derivative as a linear mapping from C^{1}(U, AM(R^{n})) into C(U, AM(R^{n})).

If α is r-times continuously differentiable on U for some positive integer r, then one can verify that

(4.6.8) dα is (r − 1)-times continuously differentiable on U,

so that d(C^{r}(U, AM(R^{n}))) ⊆ C^{r−1}(U, AM(R^{n})). (4.6.9)

If α is infinitely differentiable on U, then it follows that

(4.6.10) dα is infinitely differentiable on U,

so that d(C^{∞}(U, AM(R^{n}))) ⊆ C^{∞}(U, AM(R^{n})). (4.6.11)

4.7 The anti-derivation property

Let us continue with the same notation and hypotheses as in the previous section. Let α be a continuously-differentiable differential k-form on U for some 0 ≤ k ≤ n, and let β be another continuously-differentiable differential form on U. We would like to show that

d(α ∧ β) = (dα) ∧ β + (−1)^{k} α ∧ (dβ) (4.7.1)

on U. This is the anti-derivation property of the exterior derivative, as in Section 2.11 on p61 of [184], and Theorem 2.20 on p65 of [184]. Note that it suffices to verify this when β is a differential l-form on U for some 0 ≤ l ≤ n.

Suppose for the moment that k = 0. If β is also a differential 0-form on U, then (4.7.1) is the same as saying that

d(α β) = (dα) β + α (dβ) (4.7.2)

on U. Of course, this holds by the usual product rule for partial derivatives. If β is a differential l-form on U for some l ≥ 1, then (4.7.1) is the same as saying that d(α β) = (dα) ∧ β + α (dβ). (4.7.3)

This can be verified directly from the definition of the exterior derivative using the product rule for partial derivatives again, or by reducing to the l = 0 case.

Thus we suppose from now on in this section that k ≥ 1. If β is a differential 0-form on U, then (4.7.1) is the same as saying that

d(α β) = (dα) β + (−1)^{k} α ∧ (dβ). (4.7.4)

This can be verified directly, as in the preceding paragraph, using basic properties of the wedge product, as in Section 2.12. One could also reduce to the previous case, using the fact that

α ∧ (dβ) = (−1)^{k} (dβ) ∧ α, (4.7.5)

because dβ is a differential 1-form on U.

4.7.1 The cases where k, l ≥ 1

This permits us to take β to be a differential l-form on U with l ≥ 1. In fact, we may as well suppose that α = α_{I} dx^{I} (4.7.6)

and β = β_{L} dx^{L}, (4.7.7)

where I, L are subsets of {1, …, n} with exactly k, l ≥ 1 elements, respectively, and α_{I}, β_{L} are continuously-differentiable real-valued functions on U. If

I ∩ L ≠ ∅, (4.7.8)

then one can check directly that both sides of (4.7.1) are equal to 0. Suppose now that I ∩ L = ∅, and put

M = I ∪ L. (4.7.9)

In this case, we have that

dx^{I} ∧ dx^{L} = sgn(τ) dx^{M} (4.7.10)

for a certain τ ∈ Sym(k + l), as in Section 3.11. Thus

α ∧ β = α_{I} β_{L} dx^{I} ∧ dx^{L} = α_{I} β_{L} sgn(τ) dx^{M}. (4.7.11)

4.8. THE SECOND EXTERIOR DERIVATIVE 81

This implies that d(α ∧ β) = sgn(τ) d(α_{I} β_{L}) ∧ dx^{M}, (4.7.12) by the definition of the exterior derivative. This means that d(α ∧ β) = sgn(τ) β_{L} dα_{I} ∧ dx^{M} + sgn(τ) α_{I} dβ_{L} ∧ dx^{M}, (4.7.13) as in (4.7.2).

It follows that d(α ∧ β) = β_{L} dα_{I} ∧ dx^{I} ∧ dx^{L} + α_{I} dβ_{L} ∧ dx^{I} ∧ dx^{L}, (4.7.14) because of (4.7.10). The second term on the right is the same as (−1)^{k} α_{I} dx^{I} ∧ dβ_{L} ∧ dx^{L}, (4.7.15) because dβ_{L} is a differential 1-form on U, as in Section 2.12. This shows that (4.7.1) holds in this case, as desired.

4.8 The second exterior derivative

Let n be a positive integer, and let U be a nonempty open subset of R^{n}. Also let (4.8.1) α be a twice continuously-differentiable differential form on U, so that dα is a continuously-differentiable differential form on U. We would like to show that d(dα) = 0 (4.8.2) on U.

If α is a differential 0-form on U, then

dα = ∑_{l=1}^{n}(∂_{l}α) dx_{l}, (4.8.3)

as in Section 3.14. This implies that

d(dα) = ∑_{l=1}^{n} d(∂_{l}α) ∧ dx_{l} = ∑_{j=1}^{n} ∑_{l=1}^{n}(∂_{j}∂_{l}α) dx_{j} ∧ dx_{l}. (4.8.4)

Remember that dx_{j} ∧ dx_{l} = −dx_{l} ∧ dx_{j} (4.8.5) for all j, l = 1, …, n, as in Section 2.12. We also have that ∂_{j} ∂_{l} α = ∂_{l} ∂_{j} α (4.8.6) on U for all j, l = 1, …, n, as in Subsection 3.5.2. One can use this to get that (4.8.2) holds in this case.

Suppose now that α is as in (4.7.6) for some nonempty subset I of {1, …, n}, where α_{I} is a twice continuously-differentiable real-valued function on U. Thus

dα = dα_{I} ∧ dx^{I}, (4.8.7)

and d(dα) = d(dα_{I}) ∧ dx^{I}, (4.8.8)

as in the previous section. More precisely, this uses the fact that

d(dx^{I}) = 0, (4.8.9)

by the definition of the exterior derivative. Of course, the right side of (4.8.8) is equal to 0, as in the preceding paragraph. It follows that (4.8.2) holds in this case, and for all twice continuously-differentiable differential forms on U.

If β is a continuously-differentiable differential form on U and

dβ = 0 (4.8.10)

on U, then β is said to be closed as a differential form on U. If γ is a continuously-differentiable differential form on U, then

dγ (4.8.11)

is said to be exact as a differential form on U. If γ is twice continuously differentiable on U, then (4.8.11) is closed on U, as before.

4.9 Differentiating pull-backs

Let n, n_{1} be positive integers, let U, U_{1} be nonempty open subsets of R^{n}, R^{n_{1}}, respectively, and let

(4.9.1) ψ be a twice continuously-differentiable mapping from U_{1} into R^{n}

such that ψ(U_{1}) ⊆ U. (4.9.2)

If α is a continuously-differentiable differential form on U, then we would like to show that d(ψ^{∗}(α)) = ψ^{∗}(dα), (4.9.3)

as differential forms on U_{1}. Remember that

(4.9.4) ψ^{∗}(α) is continuously differentiable on U_{1}

under these conditions, as in Subsection 4.5.2.

Suppose first that α is a differential 0-form on U, so that ψ^{∗}(α) = α ∘ ψ. Remember that dα, d(α∘ψ) are the same as the differentials of α, α∘ψ considered as differential 1-forms on U, U_{1}, respectively, as in Section 3.14. If w ∈ U_{1}, then

d(α ∘ ψ)_{w} = dα_{ψ(w)} ∘ dψ_{w}, (4.9.5) as linear functionals on R^{n_{1}}, by the chain rule, as in Section 3.9. This means that d(α ∘ ψ)_{w} = (dψ_{w})^{∗}(dα_{ψ(w)}), (4.9.6)

where the right side is as in Section 2.3. This is the same as saying that (4.9.3) holds at w, as in Section 4.4.

More precisely, if α is a differential 0-form on U, then it suffices to ask that

(4.9.7) ψ be a continuously-differentiable mapping from U_{1} into R^{n}

with ψ(U_{1}) ⊆ U, instead of (4.9.1). Note that ψ^{∗}(α) = α ∘ ψ is continuously differentiable on U_{1} in this case, as in Section 3.9.

4.9.1 Some basic cases

Suppose next that

(4.9.8) f is a twice continuously-differentiable real-valued function on U,

and that α = df. (4.9.9)

Thus ψ^{∗}(α) = ψ^{∗}(df) = d(f ∘ ψ) (4.9.10)

on U_{1}, as in the preceding paragraph. We also have that

dα = d(df) = 0 (4.9.11)

on U and d(ψ^{∗}(α)) = d(d(f ∘ α)) = 0 (4.9.12)

on U_{1}, as in the previous section. More precisely, this uses the fact that f ∘ ψ is twice continuously differentiable on U_{1}, because ψ and f are twice continuously differentiable by hypothesis, as in Subsection 3.9.1. It follows that both sides of (4.9.3) are equal to 0 under these conditions.

Let k be a positive integer, and suppose that

α = dx_{j_{1}} ∧ ⋯ ∧ dx_{j_{k}} (4.9.13)

for some positive integers j_{1} < ⋯ < j_{k} ≤ n. Observe that

ψ^{∗}(α) = ψ^{∗}(dx_{j_{1}}) ∧ ⋯ ∧ ψ^{∗}(dx_{j_{k}}), (4.9.14)

as in Subsection 4.4.1. This implies that

ψ^{∗}(α) = dψ_{j_{1}} ∧ ⋯ ∧ dψ_{j_{k}}, (4.9.15)

where ψ_{1}, …, ψ_{n} are the components of ψ, as in the preceding paragraph. Of course, d(dψ_{j}) = 0 (4.9.16) on U_{1} for each j = 1, …, n, as in the previous section, because ψ_{j} is twice continuously differentiable on U_{1}, by hypothesis. It follows that

d(ψ^{∗}(α)) = 0 (4.9.17)

on U_{1}, as in Section 4.7. Clearly dα = 0 on U, by definition of the exterior derivative, in Section 4.6. This means that both sides of (4.9.3) are equal to 0 in this case too.

4.9.2 The rest of the proof

Let I be a nonempty subset of {1, …, n}, and suppose that

α = α_{I} dx^{I}, (4.9.18)

where α_{I} is a twice-continuously-differentiable real-valued function on U. Of course, dα = dα_{I} ∧ dx^{I}, (4.9.19)

by the definition of the exterior derivative. Observe that

ψ^{∗}(α) = (α_{I} ∘ ψ) ψ^{∗}(dx^{I}). (4.9.20)

This implies that

(4.9.21) d(ψ^{∗}(α)) = d(α_{I} ∘ ψ) ∧ ψ^{∗}(dx^{I}) + (α_{I} ∘ ψ) d(ψ^{∗}(dx^{I}))
= ψ^{∗}(dα_{I}) ∧ ψ^{∗}(dx^{I}) = ψ^{∗}(dα),

where the first step is as in Section 4.7. The second term on the right side of the first line is equal to 0, as in (4.9.17). The second step also uses the earlier remarks about differential 0-forms. The third step uses a remark in Subsection 4.4.1. This implies that (4.9.3) holds for differential k-forms on U when k ≥ 1.

4.10 Connected sets

This section and the next deal with connectedness, path connectedness, and the relationship between them. Connectedness can be used to prove the intermediate value theorem, and it can also be helpful in calculus, as in Subsection 4.10.2. Another aspect of this will be mentioned in Section 6.14.

Let n be a positive integer, and let E be a subset of R^{n}. The property of connectedness of E can be defined in a standard way, that we shall not discuss here. In fact, one can define connectedness for subsets of arbitrary metric spaces.

It is well known that a subset of a metric space is connected if and only if it is connected as a subset of itself, with respect to the restriction of the metric to that set. In particular,

(4.10.1) E is connected as a subset of R^{n}, with respect to the standard Euclidean metric, if and only if

(4.10.2) E is connected as a subset of itself,

with respect to the restriction of the standard Euclidean metric on R^{n} to E.

Suppose for the moment that n = 1. It is well known that E is connected if and only if for every x, y ∈ E with x < y, we have that

(x, y) ⊆ E. (4.10.3)

Let m be a positive integer, and let f be a continuous mapping from E into R^{m}. If E is connected as a subset of R^{n}, then it is well known that

(4.10.4) f(E) is connected in R^{m}.

This uses the equivalence of (4.10.1) and (4.10.2).

4.10.1 Convex sets

We say that E is convex if for every x, y ∈ E and t ∈ R with 0 ≤ t ≤ 1, we have that (1 − t) x + t y ∈ E. (4.10.5)

It is well known that

(4.10.6) convex subsets of R^{n} are connected.

This can be obtained from another result about connectedness that will be mentioned in the next section. Note that connected subsets of the real line are convex.

One can define convexity of a subset of any vector space V over the real numbers in the same way. If N is a norm on V, as in Section 3.2, then one can check that

(4.10.7) open balls in V with respect to the metric associated to N are convex.

In particular, open balls in R^{n} with respect to the standard Euclidean metric are convex.

4.10.2 Locally constant functions

It is well known that an open set U in R^{n} is connected if and only if U cannot be expressed as the union of two nonempty open subsets of R^{n}. More precisely, the analogous statement holds in any metric space.

Let f be a function defined on U with values in a set Y. Let us say that f is locally constant on U is for each x ∈ U we have that

f(w) = f(x) (4.10.8) for all w ∈ U that are sufficiently close to x with respect to the standard Euclidean metric on R^{n}, depending on x. This is equivalent to saying that f is continuous as a mapping from U into Y when Y is equipped with the discrete metric, as in Section 3.1.

Alternatively, f is locally constant on U if and only if for each y ∈ Y,

f^{−1}({y}) = {w ∈ U : f(w) = y} (4.10.9)

is an open set in R^{n}. This implies that

f^{−1}(Y \ {z}) = U \ f^{−1}({z}) = {w ∈ U : f(w) ≠ z} (4.10.10)

for every z ∈ Y, because it is the union of f^{−1}({y}) over y ∈ Y \ {z}. In this case, if U is connected, then one can check that

(4.10.11) f is constant on U.

If U is not connected, then U can be expressed as the union of two nonempty open sets, as before. One can use this to find locally constant functions on U that are not constant.

Suppose now that f is a real-valued function on U. If f is locally constant on U, then

\frac{∂f}{∂x_{j}} = 0 (4.10.12)

at every point in U. Conversely, if the partial derivatives of f exist at every point in U and satisfy (4.10.12), then one can check that f is locally constant on U.

4.11 Path-connected sets

Let n be a positive integer again, and let E be a subset of R^{n}. We say that E is path connected if every x, y ∈ E can be connected by a continuous path in E. More precisely, this means that there are real numbers a ≤ b and a continuous mapping p from the closed interval [a, b] in the real line into R^{n} such that p(a) = x, p(b) = y, and

p([a, b]) ⊆ E. (4.11.1)

In this case, one can always take a = 0 and b = 1, although it is sometimes convenient to use different intervals. Note that this definition can also be used for subsets of arbitrary metric spaces.

It is well known that

(4.11.2) path-connected sets are connected,

and indeed this works in arbitrary metric spaces. This uses the fact that the image of a continuous path is a connected set. In particular,

(4.11.3) convex sets in R^{n} are path connected,

and thus connected, as mentioned in Subsection 4.10.1.

4.11.1 Path-connected components

If x, y ∈ E, then put x ∼_{E} y (4.11.4)

when x can be connected to y by a continuous path in E, as before. This holds for every x, y ∈ E exactly when E is path connected. One can verify that

(4.11.5) ∼_{E} defines an equivalence relation on E.

This implies that E can be partitioned into the equivalence classes corresponding to ∼_{E}, where x, y ∈ E are elements of the same equivalence class if and only if (4.11.4) holds.

These equivalence classes may be called the path-connected components of E. They are path connected sets, and maximal with respect to inclusion among path-connected subsets of E. This works in arbitrary metric spaces too.

Let U be an open subset of R^{n}, and let x ∼_{U} y be defined for x, y ∈ U as before. If x ∈ U, then one can check that

(4.11.6) x ∼_{U} w for all w ∈ U that are sufficiently close to x,

with respect to the standard Euclidean metric on R^{n}. This implies that

(4.11.7) the path-connected components of U are open sets.

If U is connected, then one can use this to get that U has only one pathconnected component, so that

(4.11.8) U is path connected.

It is not too difficult to show that the path connected components of U are maximal with respect to inclusion among connected subsets of U, so that they may also be called the connected components of U.

4.12 Volumes and determinants

These next few sections are concerned with n-dimensional volumes of subsets of R^{n} and other n-dimensional vector spaces over the real numbers. In this section, we look at the effect on volumes of mappings from R^{n} into itself.

Let n be a positive integer, and let T be a linear mapping from R^{n} into itself. If E is a reasonably nice subset of R^{n}, then its n-dimensional volume Vol_{n}(E) can be defined in standard ways. In this case, it is well known that

(4.12.1) Vol_{n}(T (E)) = | det T | Vol_{n}(E).

If one is familiar with n-dimensional Lebesgue measure on R^{n}, then this holds for all Lebesgue measurable subsets E of R^{n}, but we shall not pursue this here.

Let N be a norm on R^{n}, as in Section 3.2. Suppose for the moment that T is an isometry with respect to N, so that

N(T (v)) = N(v) (4.12.2)

for every v ∈ R^{n}, as in Section 3.15. Of course, this implies that T is one-to-one on R^{n}, so that T (R^{n}) = R^{n}. One can use (4.12.1) to get that

| det T | = 1, (4.12.3)

by taking E to be the unit ball in R^{n} with respect to N. More precisely, this uses some additional well-known facts, such as that N and the standard Euclidean norm on R^{n} can each be bounded by a constant multiple of the other, and we shall not pursue this here.

Let d_{N}(·, ·) be the metric on R^{n} associated to N, as in Section 3.2. One can use this metric to define n-dimensional Hausdorff measure on R^{n}, which is equal to a positive multiple of n-dimensional Lebesgue measure on R^{n}. Hausdorff measures are automatically preserved by isometries. This gives another way to look at (4.12.3), which we shall not pursue in detail here.

4.12.1 Orthogonal transformations

Suppose now that T is an isometry with respect to the standard Euclidean norm on R^{n}. Let ⟨·, ·⟩ be the standard inner product on R^{n}, as in Section 3.3. The condition that T be an isometry with respect to the standard Euclidean norm on R^{n} is equivalent to saying that

⟨T (u), T (v)⟩ = ⟨u, v⟩ (4.12.4)

for every u, v ∈ R^{n}, as in Section 3.15. This is also equivalent to saying that T is a one-to-one linear mapping from R^{n} onto itself such that

⟨T (u), w⟩ = ⟨u, T^{−1}(w)⟩ (4.12.5)

for every u, w ∈ R^{n}. Under these conditions, T is said to be an orthogonal transformation on R^{n}.

Of course, linear mappings from R^{n} into itself correspond to n × n matrices of real numbers in a standard way. An invertible linear mapping T on R^{n} is an orthogonal transformation if and only if

(4.12.6) the matrix associated to T^{−1} is the same as the transpose of the matrix associated to T,

because this is equivalent to (4.12.5).

If T is any invertible linear mapping on R^{n}, then

det T^{−1} = (det T)^{−1}. (4.12.7)

If T is an orthogonal transformation, then

det T^{−1} = det T, (4.12.8)

because the determinant of a matrix is the same as the determinant of its transpose. This implies that (det T)^{2} = 1, (4.12.9)

which is the same as (4.12.3).

4.13 Admissible volumes

On an arbitrary finite-dimensional vector space over the real numbers, it is not necessarily clear how to measure the volume of a set, without additional information. However, there is a nice family of ways of doing this, that are discussed in this section. One can use linear mappings to reduce to measuring volumes in Euclidean spaces, and different linear mappings can lead to different ways of measuring volumes.

Let n be a positive integer, and let W be an n-dimensional vector over the real numbers. Also let L be a one-to-one linear mapping from R^{n} onto W. If E is a reasonably nice subset of W, in the sense that L^{−1}(E) is a reasonably nice subset of R^{n}, then Vol_{n}(L^{−1}(E)) (4.13.1)

may be considered as a type of n-dimensional volume of E.

Let L_{1} be another one-to-one linear mapping from R^{n} onto W, so that L_{1}^{−1} ∘ L is a one-to-one linear mapping from R^{n} onto itself. If E is a reasonably nice subset of W, then

(4.13.2)  Vol_{n}(L_{1}^{−1}(E)) = Vol_{n}((L_{1}^{−1} ∘ L)(L^{−1}(E)))
= | det(L_{1}^{−1} ∘ L)| Vol_{n}(L^{−1}(E)),

using (4.12.1) in the second step.

Let us say that Λ is an admissible volume on W if there is a nonnegative real number c_{Λ,L} such that

Λ(E) = c_{Λ,L} Vol_{n}(L^{−1}(E)) (4.13.3)

for all reasonably nice subsets E of W. If

c_{Λ,L} > 0, (4.13.4)

then we say that Λ is nondegenerate on W. If one uses a different one-to-one linear mapping from R^{n} onto W, then one gets the same familty of admissible volumes on W, because of (4.13.2).

4.13.1 Admissible volumes and linear mappings

Let R be a linear mapping from W into itself, so that L^{−1} ∘ R ∘ L is a linear mapping from R^{n} into itself. Remember that

det R = det(L^{−1} ∘ R ∘ L), (4.13.5)

basically by the definition of the determinant of a linear mapping from W into itself, as in Subsection 2.5.1. If E is a reasonably nice subset of W, then

(4.13.6) Vol_{n}(L^{−1}(R(E))) = Vol_{n}((L^{−1} ∘ R ∘ L)(L^{−1}(E)))
= | det(L^{−1} ∘ R ∘ L)| Vol_{n}(L^{−1}(E))
= | det R| Vol_{n}(L^{−1}(E)),

using (4.9.3) in the second step. If Λ is an admissible volume on W, then it follows that Λ(R(E)) = | det R| Λ(E). (4.13.7)

4.14 Parallelepipeds

Let V be a vector space over the real numbers, let k be a positive integer, and let v_{1}, …, v_{k} be k elements of V. The parallelepiped in V associated to v_{1}, …, v_{k} is the subset of V defined by

P (v_{1}, …, v_{k}) = P_{V} (v_{1}, …, v_{k})
(4.14.1)  = {∑_{l=1}^{k} t_{l} v_{l} : t_{l} ∈ R, 0 ≤ t_{l} ≤ 1 for each l = 1, …, k}.

Let us say that this parallelepiped is nondegenerate when v_{1}, …, v_{k} are linearly independent in V.

Let W be another vector space over the real numbers, and let T be a linear mapping from V into W. Observe that

T (P_{V} (v_{1}, …, v_{k})) = P_{W} (T (v_{1}), …, T (v_{k})). (4.14.2)

Let n be a positive integer, and let e_{1}, …, e_{n} be the standard basis vectors in R^{n}. The corresponding parallelepiped in R^{n} is the same as the closed unit cube in R^{n}, P_{R^{n}}(e_{1}, …, e_{n}) = [0, 1]^{n}. (4.14.3)

Of course, the n-dimensional volume of this is equal to 1.

If B is any linear mapping from R^{n} into itself, then

P_{R^{n}}(B(e_{1}), …, B(e_{n})) = B(P_{R^{n}}(e_{1}, …, e_{n})) = B([0, 1]^{n}), (4.14.4)

where the first step is as in (4.14.2). Note that B(e_{1}), …, B(e_{n}) may be any n elements of R^{n}. The relation between volumes and determinants mentioned in Section 4.12 says that

Vol_{n}(P_{R^{n}}(B(e_{1}), …, B(e_{n}))) = | det B|. (4.14.5)

More precisely, this can be shown first, as in Theorem 8 on p329 of [20]. One can use this to obtain the previous relation for reasonably nice subsets of R^{n}, as in Theorem 9 and its corollary on p329f of [20].

4.14.1 Parallelepipeds and admissible volumes

Suppose that W has dimension equal to n, and let u_{1}, …, u_{n} be a basis for W. If L is a one-to-one linear mapping from R^{n} onto W, then

L^{−1}(u_{1}), …, L^{−1}(u_{n}) (4.14.6)

is a basis for R^{n}. In fact, we can choose L so that this is the standard basis for R^{n}. In particular,

Vol_{n}(L^{−1}(P_{W} (u_{1}, …, u_{n}))) = Vol_{n}(P_{R^{n}}(L^{−1}(u_{1}), …, L^{−1}(u_{n}))) (4.14.7)

is strictly positive.

Let Λ be an admissible volume on W, as in (4.13.3). Thus

Λ(P_{W} (u_{1}, …, u_{n})) = c_{Λ,L} Vol_{n}(P_{R^{n}}(L^{−1}(u_{1}), …, L^{−1}(u_{n}))). (4.14.8)

by (4.14.7). We may choose c_{Λ,L} so that this is any given nonnegative real number, because (4.14.7) is positive. More precisely, this determines c_{Λ,L} uniquely, so that Λ is uniquely determined in this way as well.

4.14.2 Admissible volumes and alternating forms

If ν is an alternating n-linear form on W, then it follows that there is a unique admissible volume Λ_{ν} on W such that

Λ_{ν}(P_{W} (u_{1}, …, u_{n})) = |ν(u_{1}, …, u_{n})|. (4.14.9)

If w_{1}, …, w_{n} are any n elements of W, then

Λ_{ν}(P_{W} (w_{1}, …, w_{n})) = |ν(w_{1}, …, w_{n})|. (4.14.10)

To see this, let R be the linear mapping from W into itself such that R(u_{j}) = w_{j} for each j. The left side of (4.14.10) is equal to

(4.14.11) Λ_{ν}(P_{W} (R(u_{1}), …, R(u_{n}))) = Λ_{ν}(R(P_{W} (u_{1}, …, u_{n})))
= | det R| Λ(P_{W} (u_{1}, …, u_{n})),

using (4.14.2) in the first step, and (4.13.7) in the second step. The right side of (4.14.10) is equal to

(4.14.12)  |ν(R(u_{1}), …, R(u_{n}))| = |(R^{∗}(ν))(u_{1}, …, u_{n})|
= | det R| |ν(u_{1}, …, u_{n})|,

using the definition of R^{∗}(ν) in Section 2.3 in the first step, and a remark from Subsection 2.5.1 in the second step.

4.15 Volumes and inner products

Let n be a positive integer, and let W be an n-dimensional vector space over the real numbers again. Also let ⟨·, ·⟩_{W} be an inner product on W, and let ∥·∥_{W} be the associated norm on W. Similarly, let ⟨·, ·⟩_{R^{n}} be the standard inner product on R^{n}, as in Section 3.3, whose associated norm ∥ · ∥_{R^{n}} is the standard Euclidean norm on R^{n}.

One can use an orthonormal basis for W to get an isometric linear mapping L from R^{n} onto W, as in Section A.8. This means that ⟨L(x), L(y)⟩_{W} = ⟨x, y⟩_{R^{n}} (4.15.1) for all x, y ∈ R^{n}, as in Section 3.15. Using L, we get an admissible volume Λ_{W} (E) = Vol_{n}(L^{−1}(E)) (4.15.2) on W, as in Section 4.13.

If L_{1} is another isometric linear mapping from R^{n} onto W, then L_{1}^{−1} ∘ L is an isometric linear mapping from R^{n} onto itself. Equivalently, (4.15.3) L_{1}^{−1} ∘ L is an orthogonal transformation on R^{n}, and in particular L_{1}^{−1} ∘L preserves the n-dimensional volume of reasonably nice subsets of R^{n}, as in Section 4.12. This implies that Λ_{W} (E) = Vol_{n}(L_{1}^{−1}(E)) (4.15.4) for all reasonable nice subsets E of W, as in Section 4.13.

Let e_{1}, …, e_{n} be the standard basis vectors in R^{n}, as usual. Any sequence of n vectors in R^{n} may be expressed as B(e_{1}), …, B(e_{n}) (4.15.5) for some linear mapping B from R^{n} into itself. This is an orthonormal basis for R^{n} with respect to the standard inner product if and only if B is an orthogonal transformation on R^{n}, as in Section A.8. In this case, we get that Vol_{n}(P_{R^{n}}(B(e_{1}), …, B(e_{n}))) = | det B| = 1, (4.15.6) where the first step is as in (4.14.5), and the second step is as in Section 4.12.

If w_{1}, …, w_{n} are n elements of W, then Λ_{W} (P_{W} (w_{1}, …, w_{n})) = Vol_{n}(P_{R^{n}}(L^{−1}(w_{1}), …, L^{−1}(w_{n}))), (4.15.7) by (4.14.2) and (4.15.2). If (4.15.8) w_{1}, …, w_{n} is an orthonormal basis for W, then (4.15.9) L^{−1}(w_{1}), …, L^{−1}(w_{n}) is an orthonormal basis for R^{n}, as in Section A.8. This implies that Λ_{W} (P_{W} ((w_{1}, …, w_{n}))) = 1, (4.15.10) by (4.15.6).

Diffeomorphisms and cells

5.1 Matrix-valued functions

Let m_{1} and m_{2} be positive integers, and let

M_{m_{1},m_{2}}(R) (5.1.1)

be the space of m_{1} × m_{2} matrices of real numbers. This is a vector space over the real numbers with respect to entrywise addition and scalar multiplication of matrices. It is sometimes convenient to identify M_{m_{1},m_{2}}(R) with R^{m_{1} m_{2}}, by listing the entries of an m_{1} × m_{2} matrix by a sequence of length m_{1} m_{2}.

The order in which the entries of an m_{1} × m_{2} matrix are listed will normally not really matter, as long as we use the same listing for all such matrices. This leads to a metric on M_{m_{1},m_{2}}(R), corresponding to the standard Euclidean metric on R^{m_{1} m_{2}}. In particular, this metric does not depend on the order in which the entries of an m_{1} × m_{2} matrix are listed, as long as we use the same listing for all such matrices, because the standard Euclidean metric on R^{k} is invariant under permutations of the coordinates for any positive integer k.

Let n be a positive integer, and let E be a nonempty subset of R^{n}. Also let f be a function on E with values in M_{m_{1},m_{2}}(R). Let us say that f is continuous on E if it corresponds to a continuous mapping from E into R^{m_{1} m_{2}} as in the previous two paragraphs. This does not depend on the order in which the entries of an m_{1} × m_{2} matrix are listed, as long as we use the same listing for all such matrices, as before. Equivalently, f is continuous on E if and only if the m_{1} m_{2} entries of f are continuous as real-valued functions on E, as in Section 3.4.

Of course, the space of all M_{m_{1},m_{2}}(R)-valued functions on E is a vector space over the real numbers, with respect to pointwise addition and scalar multiplication of these functions. The space

C(E, M_{m_{1},m_{2}}(R)) = C(E, R^{m_{1} m_{2}}) (5.1.2)

of all continuous M_{m_{1},m_{2}}(R)-valued functions on E is a linear subspace of the space of all M_{m_{1},m_{2}}(R)-valued functions on E.

5.1.1 Products of matrix-valued functions

If m_{3} is another positive integer, then the product of an m_{1} × m_{2} matrix of real numbers with an m_{2} ×m_{3} matrix of real numbers may be defined as an m_{1} ×m_{3} matrix of real numbers in the usual way. This defines a bilinear mapping from M_{m_{1},m_{2}}(R) × M_{m_{2},m_{3}}(R) into M_{m_{1},m_{3}}(R).

If f and g are functions on E with values in M_{m_{1},m_{2}}(R) and M_{m_{2},m_{3}}(R), respectively, then their product f g may be defined as a function on E with values in M_{m_{1},m_{3}}(R) using matrix multiplication at each point. If f, g are continuous on E, then one check that

(5.1.3) f g is continuous on E.

This is because the entries of f g are given by sums of products of the entries of f and g.

5.1.2 C^{r} Matrix-valued functions

Let U be a nonempty open subset of R^{n}, and let r be a positive integer. We say that a function f on U with values in M_{m_{1},m_{2}}(R) is r-times continuously differentiable on U if it corresponds to an r-times continuously-differentiable mapping from U into R^{m_{1} m_{2}} as before. This does not depend on the order in which the entries of an m_{1} × m_{2} matrix are listed, as long as we use the same listing for all such matrices, as usual. Equivalently, this means that the m_{1} m_{2} entries of f are r-times continuously differentiable as real-valued functions on U. If r = 0, then we may interpret this as meaning that f is continuous on U, as before.

Let us say that f is infinitely differentiable or smooth on U if f corresponds to an infinitely differentiable mapping from U into R^{m_{1} m_{2}}. This does not depend on the order in which the entries of an m_{1} × m_{2} matrix are listed, as long as we use the same listing for all such matrices, and indeed the smoothness of f is equivalent to the smoothness of the m_{1} m_{2} entries of f as real-valued functions on U.

The space C^{r}(U, M_{m_{1},m_{2}}(R)) = C^{r}(U, R^{m_{1} m_{2}}) (5.1.4)

of r-times continuously-differentiable functions on U with values in M_{m_{1},m_{2}}(R) is a linear subspace of the space of continuous functions on U with values in M_{m_{1},m_{2}}(R) for each r. Similarly, the space

C^{∞}(U, M_{m_{1},m_{2}}(R)) = C^{∞}(U, R^{m_{1} m_{2}}) (5.1.5)

of infinitely-differentiable functions on U with values in M_{m_{1},m_{2}}(R) is a linear subspace of (5.1.4) for each r.

Let f and g be functions on U with values in M_{m_{1},m_{2}}(R) and M_{m_{2},m_{3}}(R), so that f g is defined as a function on U with values in M_{m_{1},m_{3}}(R), as before. If f, g are r-times continuously differentiable on U for some r, then one can verify that (5.1.6) f g is r-times continuously differentiable on U.

If f, g are infinitely differentiable on U, then it follows that

(5.1.7) f g is infinitely differentiable on U.

5.2 General linear groups

Let m be a positive integer, and consider the space M_{m,m}(R) of m×m matrices of real numbers, as in the previous section. As before, it is sometimes convenient to identify M_{m,m}(R) with R^{m^{2}}, by listing the entries os an m × m matrix by a sequence of length m^{2}. The order in which the entries are listed will normally not really matter, as long as we use the same listing for all of these matrices, as usual.

Note that M_{m,m}(R) is an associative algebra over the real numbers with respect to matrix multiplication. The identity matrix in M_{m,m}(R) has diagonal entries equal to 1, and all other entries equal to 0. This is the multiplicative identity element in M_{m,m}(R).

An element of M_{m,m}(R) is said to be invertible if it has a multiplicative inverse in M_{m,m}(R). Let GL(m, R) (5.2.1) be the set of invertible elements of M_{m,m}(R). This is another example of a group, with respect to matrix multiplication in this case. This is known as the general linear group of m × m matrices with entries in R.

5.2.1 GL(m, R) And the determinant

Of course, the determinant defines a real-valued function on M_{m,m}(R). More precisely, the determinant corresponds to a homogeneous polynomial on R^{m^{2}} of degree m. In particular, this defines a continuous function on R^{m^{2}}.

It is well known that an element of M_{m,m}(R) is invertible if and only if its determinant is not zero, because of Cramer’s rule. One can use this and the continuity of the determinant to get that

(5.2.2) GL(m, R) is an open set in M_{m,m}(R).

This means that GL(m, R) corresponds to an open set in R^{m^{2}}, with respect to the standard Euclidean metric.

Equivalently, the complement of GL(m, R) in M_{m,m}(R) consists of matrices whose determinant is equal to 0. This corresponds to a closed set in R^{m^{2}}, because the determinant is continuous.

Cramer’s rule implies that the inverse of an element of GL(m, R) can be expressed in terms of the determinant of the matrix, and the determinants of its minors. In particular, the entries of the inverse are rational functions of the entries of the matrix, wth the determinant of the matrix as the denominator. This implies that

(5.2.3) the mapping from GL(m, R) into itself that sends an element of GL(m, R) to its multiplicative inverse in M_{m,m}(R) is smooth.

Another way to look at this mapping will be mentioned in Section 5.4.

5.3 Some spaces of linear mappings

Let m_{1} and m_{2} be positive integers, and let

L(R^{m_{2}}, R^{m_{1}}) (5.3.1)

be the space of linear mappings from R^{m_{2}} into R^{m_{1}}. This is a linear subspace of the space of all functions on R^{m_{2}} with values in R^{m_{1}}, as a vector space over the real numbers with respect to pointwise addition and scalar multiplication. It is sometimes convenient to identify L(R^{m_{2}}, R^{m_{1}}) with M_{m_{1},m_{2}}(R), using the usual correspondence between linear mappings from R^{m_{2}} into R^{m_{1}} with m_{1} × m_{2} matrices of real numbers. Thus we may also identify L(R^{m_{2}}, R^{m_{1}}) with R^{m_{1} m_{2}}, as in Section 5.1.

Let n be a positive integer, let E be a nonempty subset of R^{n}, and let f be a function on E with values in L(R^{m_{2}}, R^{m_{1}}). Let us say that f is continuous if f corresponds to a continuous function on E with values in M_{m_{1},m_{2}}(R), or equivalently R^{m_{1},m_{2}}, as in Section 5.1. Note that the space of functions on E with values in L(R^{m_{2}}, R^{m_{1}}) is a vector space over the real numbers, with respect to pointwise addition and scalar multiplication of these functions. The space C(E, L(R^{m_{2}}, R^{m_{1}})) = C(E, M_{m_{1},m_{2}}(R)) = C(E, R^{m_{1} m_{2}}) (5.3.2)

of all continuous functions on E with values in L(R^{m_{2}}, R^{m_{1}}) is a linear subspace of the space of all functions on E with values in L(R^{m_{2}}, R^{m_{1}}).

Let m_{3} be another positive integer, so that we may also consider linear mappings from R^{m_{3}} into R^{m_{1}} or R^{m_{2}}. Of course, if T_{2} is a linear mapping from R^{m_{3}} into R^{m_{2}}, and T_{1} is a linear mapping from R^{m_{2}} into R^{m_{1}}, then their composition T_{1} ∘ T_{2} is a linear mapping from R^{m_{3}} into R^{m_{1}}. It is easy to see that (T_{1}, T_{2}) ↦ T_{1} ∘ T_{2} (5.3.3)

defines a bilinear mapping from

L(R^{m_{2}}, R^{m_{1}}) × L(R^{m_{3}}, R^{m_{2}}) (5.3.4)

into L(R^{m_{3}}, R^{m_{1}}). This corresponds to multiplication of the matrices associated to T_{1} and T_{2}, as usual.

Let f and g be functions on E with values in L(R^{m_{2}}, R^{m_{1}}), L(R^{m_{3}}, R^{m_{2}}), respectively. Their product f g may be defined as a function on E with values in L(R^{m_{3}}, R^{m_{1}}), using composition of linear mappings as multiplication at each point. This means that the value of f g at x ∈ E is equal to

(f g)(x) = f(x) ∘ g(x), (5.3.5)

where the right side is the linear mapping from R^{m_{3}} into R^{m_{1}} obtained by composing g(x) with f(x). If f and g are continuus on E, then

(5.3.6) f g is continuous on E, as in Subsection 5.1.1.

5.3.1 C^{r} L(R^{m_{2}}, R^{m_{1}})-Valued functions

Let U be a nonempty open subset of R^{n}, let r be a positive integer, and let f be a function on U with values in L(R^{m_{2}}, R^{m_{1}}). We say that f is r-times continuously differentiable on U if it corresponds to an r-times continuously differentiable function on U with values in M_{m_{1},m_{2}}(R), or equivalently in R^{m_{1} m_{2}}, as in Subsection 5.1.2. If r = 0, then we may interpret this as meaning that f is continuous on U, as usual. Similarly, we say that f is infinitely differentiabile if f corresponds to an infinitely differentiable function on U with values in M_{m_{1},m_{2}}(R), or equivalently in R^{m_{1} m_{2}}, as before.

The space

C^{r}(U, L(R^{m_{2}}, R^{m_{1}})) = C^{r}(U, M_{m_{1},m_{2}}(R)) = C^{r}(U, R^{m_{1} m_{2}}) (5.3.7)

of r-times continuously-differentiable L(R^{m_{2}}, R^{m_{1}})-valued functions on U is a linear subspace of the space of continuous functions on U with values in L(R^{m_{2}}, R^{m_{1}}) for each r. Similarly, the space

C^{∞}(U, L(R^{m_{2}}, R^{m_{1}})) = C^{∞}(U, M_{m_{1},m_{2}}(R)) = C^{∞}(U, R^{m_{1} m_{2}}) (5.3.8)

of infinitely-differentiable functions on U with values in L(R^{m_{2}}, R^{m_{1}}) is a linear subspace of (5.3.7) for each r.

Let g be a function on U with values in L(R^{m_{3}}, R^{m_{2}}), so that f g is defined as a function on U with values in L(R^{m_{3}}, R^{m_{1}}), as in (5.3.5). If f and g are r-times continuously differentiable on U, then

(5.3.9) f g is r-times continuously differentiable on U,

as in Subsection 5.1.2. If f and g are infinitely differentiable on U, then we get that (5.3.10) f g is infinitely differentiable on U.

5.4 More on linear mappings

Let m_{1} and m_{2} be positive integers, and let T be a linear mapping from R^{m_{2}} into R^{m_{1}}. It is well known that there is a nonnegative real number C(T), depending on T, such that

∥T (v)∥_{2,R^{m_{1}}} ≤ C(T) ∥v∥_{2,R^{m_{2}}} (5.4.1)

for every v ∈ R^{m_{2}}. Here ∥ · ∥_{2,R^{m}} is the standard Euclidean norm on R^{m} for each m, as in Section 3.2, although one could consider other norms as well. In fact, one can take C(T) to be the standard Euclidean norm of the m_{1} × m_{2} matrix of real numbers associated to T, considered as an element of R^{m_{1} m_{2}}, as on p211 of [155].

Now let m be a positive integer, and let T be a one-to-one linear mapping from R^{m} onto itself. There is a positive real number C(T^{−1}) such that

∥T^{−1}(u)∥_{2,R^{m}} ≤ C(T^{−1}) ∥u∥_{2,R^{m}} (5.4.2)

for every u ∈ R^{m}, as before. This is the same as saying that

C(T^{−1})^{−1} ∥v∥_{2,R^{m}} ≤ ∥T (v)∥_{2,R^{m}} (5.4.3)

for every v ∈ R^{m}, by taking v = T^{−1}(u).

Let R be another linear mapping from R^{m} into itself, and suppose that R is fairly close to T, in the sense that

∥R(v) − T (v)∥_{2,R^{m}} ≤ (1/2) C(T^{−1})^{−1} ∥v∥_{2,R^{m}} (5.4.4)

for every v ∈ R^{m}. If v ∈ R^{m}, then we get that

C(T^{−1})^{−1}∥v∥_{2,R^{m}} ≤ ∥T (v)∥_{2,R^{m}} ≤ ∥R(v)∥_{2,R^{m}} + ∥R(v) − T (v)∥_{2,R^{m}}
(5.4.5)  ≤ ∥R(v)∥_{2,R^{m}} + (1/2) C(T^{−1})^{−1} ∥v∥_{2,R^{m}}.

This implies that

(1/2) C(T^{−1})^{−1} ∥v∥_{2,R^{m}} ≤ ∥R(v)∥_{2,R^{m}}. (5.4.6)

In particular, (5.4.6) implies that the kernel of R is trivial. This means that R is a one-to-one mapping from R^{m} onto itself, by well-known results in linear algebra. We also get that

∥R^{−1}(u)∥_{2,R^{m}} ≤ 2 C(T^{−1}) ∥u∥_{2,R^{m}} (5.4.7)

for every u ∈ R^{m}, by taking v = R^{−1}(u).

5.4.1 Linear mappings on R^{m}

Let L(R^{m}) = L(R^{m}, R^{m}) (5.4.8)

be the space of linear mappings from R^{m} into itself. This is an associative algebra over the real numbers with respect to composition of mappings. The identity mapping on R^{m} is the multiplicative identity element in this algebra. It is sometimes convenient to identify this algebra with M_{m,m}(R), as in the previous section.

Let GL(R^{m}) (5.4.9)

be the set of one-to-one linear mappings from R^{m} onto itself, which are the same as the invertible elements of L(R^{m}). This is a group, with respect to composition of mappings, which is known as the general linear group of R^{m}. This corresponds exactly to GL(m, R), using the usual correspondence between linear mappings from R^{m} into itself and m × m matrices of real numbers.

The fact that GL(m, R) corresponds to an open set in R^{m^{2}}, as in Subsection 5.2.1, can also be obtained from the remarks in this section. If R.T ∈ GL(R^{m}), then it is easy to see that

R^{−1} − T^{−1} = R^{−1} ∘ (T − R) ∘ T^{−1}, (5.4.10)

as linear mappings on R^{m}. One can use this to get continuity and other regularity properties of R ↦ R^{−1} on GL(R^{m}).

5.5 Continuity of the differential

Let m and n be positive integers, let W be a nonempty open subset of R^{n}, and let f be a differentiable mapping from W into R^{m}. This means that f is differentiable in the sense of Section 3.8 at every point in W, so that the differential f′(x) = df_{x} (5.5.1)

of f at x ∈ W defines a function on W with values in L(R^{n}, R^{m}).

Suppose from now on in this section that f is continuously differentiable on W, which implies that f is differentiable on W, as mentioned in Subsection 3.8.2. In this case,

(5.5.2) the differential of f is continuous as a function on W with values in L(R^{n}, R^{m}),

because the entries of the corresponding matrices are given by the partial derivatives of the components of f. In fact, continuous differentiability may be defined equivalently in terms of the existence and continuity of the differential in this way, as in Definition 9.20 and Theorem 9.21 on p219 of [155].

Let us also suppose from now on in this section that m = n. One can check that {x ∈ W : f′(x) = df_{x} ∈ GL(R^{n})} (5.5.3)

is an open set in R^{n}. This uses the continuity of the differential of f on W, the hypothesis that W be an open set in R^{n}, and the fact that GL(n, R) corresponds to an open set in R^{n^{2}}, as in Subsection 5.2.1 and the previous section. Although one could use the definition of continuity directly here, it is convenient to use a well-known characterization of continuity in terms of inverse images of open sets, which will be reviewed in Section 5.9.

Alternatively, (5.5.3) is the same as

{x ∈ W : det f′(x) = det df_{x} ≠ 0}. (5.5.4)

Observe that

(5.5.5) det f′(x) = det df_{x} is continuous as a real-valued function on W.

because of (5.5.2), the continuity of the determinant as a real-valued function on L(R^{n}), and the fact that compositions of continuous mappings are continuous, as in Section 3.4. As a slightly different approach, one can use the definition of the determinant to express

det f′(x) = det df_{x} (5.5.6)

in terms of a sum of products of partial derivatives of the components of f, and use the fact that sums and products of continuous real-valued functions are continuous too. One can verify that (5.5.4) is an open set in R^{n}, using (5.5.5) and the hypothesis that W be an open set in R^{n}, as before. Equivalently,

{x ∈ W : det f′(x) = det df_{x} = 0} (5.5.7)

is relatively closed in W, in the sense that is reviewed in Section 5.7. We also have that

(5.5.8) f′(x)^{−1} = (df_{x})^{−1} is continuous on (5.5.3).

More precisely, this means that the inverse of the differential of f at x, as a linear mapping from R^{n} onto itself, is continuous as a function of x on (5.5.3) with values in L(R^{n}). This follows from the continuity of the differential of f, as a function on W with values in L(R^{n}), and the continuity of the mapping that sends an element of GL(R^{n}) to its inverse, as in Subsections 5.2.1 and 5.4.1. This uses the fact that compositions of continuous mappings are continuous again too. Alternatively, the entries of the matrix associated to

f′(x)^{−1} = (df_{x})^{−1} (5.5.9)

can be expressed in terms of the partial derivatives of the components of f and (5.5.6) using Cramer’s rule, and one can use this to get that the entries of the matrix are continuous as real-valued functions of x on W.

5.6 The inverse function theorem

Let n be a positive integer, let W be a nonempty open subset of R^{n}, and let f be a continuously-differentiable mapping from W into R^{n}. Also let w ∈ W be given, and suppose that

f′(w) = df_{w} ∈ GL(R^{n}). (5.6.1)

Under these conditions, the inverse function theorem states that there are open subsets U(w) and V (w) of R^{n} with the following properties, that we shall describe in two parts.

In the first part, we have that

w ∈ U(w), U(w) ⊆ W, and f(w) ∈ V (w). (5.6.2)

We are also able to choose U(w) and V (w) so that

(5.6.3) the restriction of f to U(w) is a one-to-one mapping onto V (w) and (5.6.4) f′(x) = df_{x} ∈ GL(R^{n}) for each x ∈ U(w).

Let g = g_{w} be the inverse of the restriction of f to U(w), so that

(5.6.5) g is a one-to-one mapping from V (w) onto U(w),

with g(f(x)) = x (5.6.6)

for every x ∈ U(w). The second part of the inverse function theorem states that

(5.6.7) g is continuously differentiable on V (w),

and that g′(y) = f′(g(y))^{−1} (5.6.8)

for each y ∈ V (w). Equivalently, this means that

dg_{y} = (df_{g(y)})^{−1}, (5.6.9)

where the right side is the inverse of the differential of f at g(y), as a linear mapping from R^{n} onto itself.

Note that f(g(y)) = y (5.6.10)

for each y ∈ V (w), by construction. The differentiability of g at y implies that

(5.6.11) f′(g(y)) ∘ g′(y) = df_{g(y)} ∘ dg_{y} is the identity mapping on R^{n},

by the chain rule, as in Section 3.9. Of course, (5.6.8) or equivalently (5.6.9) follows from this. One can use this to get the continuity of the differential of g from the continuity of g and (5.5.8).

In this case, the restriction of f to U(w) is a C^{1} diffeomorphism from U(w) onto V (w), in the sense discussed in Section 5.11. Note that the mean value theorem and intermediate value theorem are very helpful for some properties like these when n = 1, as in Section 5.11. If f is r-times continuously differentiable for some positive integer r, or infinitely differentiable, then g has the same property, as we shall see in Section 5.12. This means that the restriction of f to U(w) is a C^{r} or C^{∞} diffeomorphism from U(w) onto V (w), as appropriate.

5.7 Closed sets and limit points

Let n be a positive integer, and let E be a subset of R^{n}. A point x ∈ R^{n} is said to be a limit point of E with respect to the standard Euclidean metric if for every positive real number r there is a w ∈ E such that w ≠ x and

∥x − w∥_{2} < r, (5.7.1)

where ∥ · ∥_{2} = ∥ · ∥_{2,R^{n}} is the standard Euclidean norm on R^{n}, as in Section 3.2.

102 CHAPTER 5. DIFFEOMORPHISMS AND CELLS

We say that E is a closed set in R^{n} if for every x ∈ R^{n} such that x is a limit point of E with respect to the standard Euclidean metric, we have that x ∈ E. (5.7.2) It is well known and not difficult to show that this happens if and only if the complement R^{n} \ E = {z ∈ R^{n} : z ∉ E} (5.7.3) of E in R^{n} is an open set with respect to the standard Euclidean metric.

Note that R^{n} \ (R^{n} \ E) = E. (5.7.4) The previous statement is equivalent to saying that (5.7.5) U ⊆ R^{n} is an open set if and only if R^{n} \ U is a closed set.

The closure of E in R^{n} is defined to be the set (5.7.6) \overline{E} = \{x ∈ R^{n} : x ∈ E or x is a limit point of E\}. It is easy to see that E is a closed set in R^{n} if and only if E = \overline{E}. (5.7.7) It is well known and not too difficult to show that for any subset E of R^{n}, (5.7.8) \overline{E} is a closed set in R^{n}.

Let us say that a subset A of E ⊆ R^{n} is relatively closed in E if A contains all of its limit points in E. Equivalently, this means that A = \overline{A} ∩ E. (5.7.9) Note that E is automatically relatively closed as a subset of itself. If E is a closed set in R^{n}, then A ⊆ E is relatively closed in E if and only if A is a closed set in R^{n}. This uses the fact that a limit point of A in R^{n} is also a limit point of E when A ⊆ E. If A_{1} is a closed set in R^{n}, then one can check that A_{1} ∩ E (5.7.10) is relatively closed in E. Every relatively closed subset of E is of this form, as in (5.7.9).

Let us say that a subset U of E ⊆ R^{n} is relatively open if for every x ∈ U there is an r > 0 such that B(x, r) ∩ E ⊆ U. (5.7.11) It is well known and not too difficult to show that this happens if and only if there is an open subset W of R^{n} such that U = W ∩ E. (5.7.12)

Clearly E is automatically relatively open as a subset of itself. If E is an open set in R^{n}, then U ⊆ E is relatively open in E if and only if U is an open set in R^{n}.

It is well known and not too difficult to show that A ⊆ E is relatively closed in E if and only if E \ A = {x ∈ E : x ∉ A} (5.7.13)

is relatively open in E. This means that U ⊆ E is relatively open in E if and only if E \ U is relatively closed in E. The relatively open and closed subsets of E are the same as the subsets of E that are open or closed when E is considered as a metric space, with respect to the restriction of the standard Euclidean metric on R^{n} to E.

Let us say that E is bounded in R^{n} if E is contained in a ball. If E is bounded, then one can check that

(5.7.14) \overline{E} is bounded in R^{n}

too.

The boundary of E in R^{n} is defined by

∂E = \overline{E} ∩ \overline{(R^{n} \ E)}. (5.7.15)

Note that this is automatically a closed set in R^{n}.

5.8 Compact sets

Let n be a positive integer, and let K be a subset of R^{n}. We say that K is compact if every open covering of K in R^{n} can be reduced to a finite subcovering. It is well known that

(5.8.1) K is compact in R^{n} if and only if K is closed and bounded.

More precisely, a compact subset of any metric space is closed and bounded, and the converse uses additional properties of R^{n}.

Let E be a subset of R^{n}, and suppose for the moment that K ⊆ E. It is well known and not too difficult to show that K is compact if and only if every covering of K by relatively open subsets of E can be reduced to a finite subcovering. This means that K is compact as a subset of R^{n} if and only if K is compact as a subset of E, with respect to the restriction of the standard Euclidean metric on R^{n} to E.

If K is compact and A is a closed subset of R^{n}, then it is well known that

(5.8.2) K ∩ A is compact.

Although this follows from (5.8.1), this can be verified more directly, with an argument that works in any metric space.

If every infinite subset of K has a limit point in K, then K is said to have the limit point property. If every sequence of elements of K has a subsequence that converges to an element of K, then K is said to be sequentially compact. It is well known that compactness, the limit point property, and sequential compactness are equivalent in any metric space.

Suppose that U ⊆ R^{n} is an open set, and that

K ⊆ U. (5.8.3)

This implies that for every x ∈ K there is a positive real number r(x) such that

B(x, r(x)) ⊆ U. (5.8.4)

Observe that the open balls B(x, r(x)/2), x ∈ K, form an open covering of E in R^{n}. If K is nonempty and compact, then there are finitely many elements x_{1}, …, x_{l} of K such that

K ⊆ ⋃_{j=1}^{l} B(x_{j}, r(x_{j})/2). (5.8.5)

If we put

r_{0} = min_{1≤j≤l}(r(x_{j})/2), (5.8.6)

then r_{0} is a positive real number, and one can check that

B(x, r_{0}) ⊆ U (5.8.7)

for every x ∈ K.

5.8.1 Continuity and compactness

Let m be a positive integer, and let f be a continuous mapping from E into R^{m}. If K is compact in R^{n}, and K ⊆ E, then it is well known that

f(K) = {f(x) : x ∈ K} (5.8.8)

is compact in R^{m}.

Let f be a continuous real-valued function on E. If K is compact in R^{n}, K ⊆ E, and K ≠ ∅, then the extreme value theorem states that

(5.8.9) f attains its maximum and minimum on K.

This can be obtained from the fact that f(K) is closed and bounded in the real line, because f(K) is compact.

A mapping f from E into R^{m} is said to be uniformly continuous if for every ϵ > 0 there is a δ > 0 such that for every x, w ∈ E with

∥x − w∥_{2,R^{n}} < δ, (5.8.10)

we have that ∥f(x) − f(w)∥_{2,R^{m}} < ϵ. (5.8.11)

Here ∥ · ∥_{2,R^{n}} and ∥ · ∥_{2,R^{m}} are the standard Euclidean norms on R^{n} and R^{m}, respectively, as in Section 3.2. It is easy to see that uniform continuity on E implies ordinary continuity on E. If f is continuous on E, and if E is compact in R^{n}, then it is well known that

(5.8.12) f is uniformly continuous on E.

5.9 More on continuous mappings

Let n and m be positive integers, and let W be an open subset of R^{n}. It is well known that a mapping f from W into R^{m} is continuous if and only if for every open set V ⊆ R^{m}, f^{−1}(V) = {x ∈ W : f(x) ∈ V } (5.9.1)

is an open set in R^{n}. If W is any subset of R^{n}, then f is continuous on W if and only if for every open set V ⊆ R^{m},

(5.9.2) f^{−1}(V) is a relatively open set in W.

We also have that f is continuous on W if and only if for every relatively open set V_{0} ⊆ f(W), (5.9.3) f^{−1}(V_{0}) is relatively open in W.

Similarly, it is well known that a mapping f from a closed set E ⊆ R^{n} into R^{m} is continuous if and only if for every closed set A ⊆ R^{m},

f^{−1}(A) = {x ∈ E : f(x) ∈ A} (5.9.4)

is a closed set in R^{n}. If E is any subset of R^{n}, then f is continuous on E if and only if for every closed set A ⊆ R^{m},

(5.9.5) f^{−1}(A) is a relatively closed set in E.

Observe that f^{−1}(R^{m} \ A) = E \ f^{−1}(A) (5.9.6)

for every A ⊆ R^{m}. Alternatively, f is continuous on E if and only if for every relatively closed set A_{0} ⊆ f(E),

(5.9.7) f^{−1}(A_{0}) is relatively closed in E.

5.9.1 Homeomorphisms

A mapping f from a subset E of R^{n} into R^{m} is said to be a homeomorphism onto its image f(E) = {f(x) : x ∈ E} if f is one-to-one and continuous, and if

(5.9.8) the inverse f^{−1} of f is continuous

as a mapping from f(E) onto E.

106 CHAPTER 5. DIFFEOMORPHISMS AND CELLS

If E is compact, and f is a one-to-one continuous mapping from E into R^{m}, then it is well known that (5.9.9) f is a homeomorphism of E onto f(E). Note that (5.9.10) f(E) is a compact subset of R^{m} in this case, so that (5.9.11) f(E) is a closed set in R^{m} in particular, as in the previous section. In order to show that f^{−1} is continuous on f(E), it suffices to get that (5.9.12) (f^{−1})^{−1}(A) is a closed set in R^{m} for every closed set A ⊆ R^{n}, as before.

In this case, one can check that (f^{−1})^{−1}(A) = f(A ∩ E). (5.9.13) We also have that (5.9.14) A ∩ E is compact in R^{n}, because E is compact and A is a closed set, as in the previous section. This implies that (5.9.15) f(A ∩ E) is compact in R^{m}, because f is continuous on E. It follows that (5.9.16) f(A ∩ E) is a closed set in R^{m}, as mentioned in the previous section.

5.10 Homeomorphisms and open mappings

Let m and n be positive integers, let E be a subset of R^{n}, and let f be a mapping from E into R^{m}. Suppose for the moment that f is a homeomorphism from E onto f(E), which means that (5.10.1) f^{−1} is a homeomorphism from f(E) onto E. Let k be another positive integer, and let g be a mapping from f(E) into R^{k}. If g is a homeomorphism from f(E) onto g(f(E)), then (5.10.2) g ∘ f is a homeomorphism from E onto g(f(E)).

A mapping f from an open subset W of R^{n} into R^{m} is said to be an open mapping if for every open subset U of R^{n} with U ⊆ W, we have that (5.10.3) f(U) is an open set in R^{m}.

5.10. HOMEOMORPHISMS AND OPEN MAPPINGS 107

In particular, this implies that (5.10.4) f(W) is an open set in R^{m}. If W is any subset of R^{n}, then we may say that f is an open mapping as a mapping from W onto its image f(W) in R^{m} if for every relatively open subset U of W, (5.10.5) f(U) is relatively open in f(W). Of course, (5.10.5) implies (5.10.3) when (5.10.4) holds. Similarly, if W is an open set in R^{n}, then every relatively open subset U of W is an open set in R^{n}, so that (5.10.3) implies (5.10.5).

If f is one-to-one on W, then (5.10.6) f is an open mapping as a mapping from W onto f(W) if and only if (5.10.7) f^{−1} is a continuous mapping from f(W) onto W. This follows from one of the characterizations of continuity mentioned in the previous section. In this case, f is a homeomorphism from W onto f(W) if and only if f is continuous and an open mapping as a mapping from W onto f(W). If W and f(W) are open sets, then (5.10.6) is the same as saying that (5.10.8) f is an open mapping from W into R^{m}, as in the preceding paragraph.

5.10.1 A sufficient condition

Suppose that for every w ∈ W there is an open subset U(w) in R^{n} such that w ∈ U(w), U(w) ⊆ W, (5.10.9) and (5.10.10) the restriction of f to U(w)

is an open mapping from U(w) into R^{m}.

Under these conditions, one can check that (5.10.8) holds. More precisely, if U is an open subset of R^{n} with U ⊆ W and w ∈ U, then U ∩ U(w) is an open set in R^{n} that is contained in U(w), so that (5.10.11) f(U ∩ U(w)) is an open set in R^{m}, by hypothesis. One can use this to verify that f(U) is an open set in R^{m}, directly from the definition of an open set. Indeed, every element of f(U) is of the form f(w) for some w ∈ U, and in this case f(U ∩ U(w)) is an open set in R^{m} that contains f(w) and is contained in f(U).

Alternatively, observe that

U = ⋃_{w∈U}(U ∩ U(w)). (5.10.12)

This implies that

f(U) = f( ⋃_{w∈U}(U ∩ U(w))) = ⋃_{w∈U} f(U ∩ U(w)). (5.10.13)

We also have that

(5.10.14)  ⋃_{w∈U} f(U ∩ U(w)) is an open set in R^{m},

because the union of any family of open sets is an open set as well. This means that f(U) is an open set in R^{m}, as desired.

5.11 C^{1} Diffeomorphisms

Let n be a positive integer, let W be a nonempty open subset of R^{n}, and let f be a mapping from W into R^{n}.

Suppose for the moment that f is one-to-one on W, and that f(W) is an open set in R^{n} too. If f is differentiable at x ∈ W, and f^{−1} is differentiable at f(x), then the chain rule implies that

(5.11.1) d(f^{−1})_{f(x)} ∘ df_{x} = df_{x} ∘ d(f^{−1})_{f(x)} = the identity mapping on R^{n},

as in Section 3.9. This means that df_{x} is invertible as a linear mapping on R^{n}, with (df_{x})^{−1} = d(f^{−1})_{f(x)}. (5.11.2)

Suppose for the moment again that f is continuously differentiable on W, and that df_{x} ∈ GL(R^{n}) (5.11.3)

for every x ∈ W. Under these conditions, one can check that

(5.11.4) f is an open mapping from W into R^{n},

using the inverse function theorem, as in Section 5.6. This also uses the criterion for f to be an open mapping mentioned in Subsection 5.10.1. More precisely in this case, (5.10.10) corresponds to the fact that V (w) is an open set in R^{n}, and that g_{w} is a continuous mapping from V (w) onto U(w), in the notation of Section 5.6. In particular, this means that f(W) is an open set in R^{n}.

5.11. C^{1} DIFFEOMORPHISMS 109

5.11.1 The definition of a diffeomorphism

Suppose that f is one-to-one on W again, and that f is continuously differentiable on W. If f(W) is an open set in R^{n}, and

(5.11.5) f^{−1} is continuously differentiable on f(W),

then we say that

(5.11.6) f is a C^{1} diffeomorphism from W onto f(W).

This implies that (5.11.3) holds for every x ∈ W, as before. Of course, if (5.11.6) holds, then (5.11.7) f is a homeomorphism from W onto f(W).

Alternatively, if (5.11.3) holds for every x ∈ W, then f(W) is an open set in R^{n}, as before. If f is also one-to-one on W, then one can check that (5.11.5) holds, using the inverse function theorem. More precisely, if w ∈ U, then the restriction of f^{−1} to the open set V (w) in R^{n} mentioned in Section 5.6 is the same as the function g_{w} mentioned there. This means that f^{−1} is continuously differentiable on V (w). It follows that (5.11.5) holds, because w ∈ U is arbitrary, and V (w) is an open set that contains f(w).

Suppose that f is a C^{1} diffeomorphism from W onto f(W), which implies that (5.11.8) f^{−1} is a C^{1} diffeomorphism from f(W) onto W.

If g is a mapping from f(W) into R^{n} that is a C^{1} diffeomorphism from f(W) onto g(f(W)), then

(5.11.9) g ∘ f is a C^{1} diffeomorphism from W onto g(f(W)).

5.11.2 The n = 1 case

Suppose now that n = 1, and that f is a differentiable real-valued function on W. If x ∈ W, then we let f′(x) be the usual derivative of f at x, which is a real number. Thus the differential of f at x, as a linear mapping from R into itself, corresponds to multiplication by f′(x). In this case, (5.11.3) is the same as saying that f′(x) ≠ 0. (5.11.10)

Suppose also that W is connected, so that W is basically an open interval in R, which may be unbounded, like an open half-line, or the real line. If (5.11.10) holds for every x ∈ W, then either

f′(x) > 0 (5.11.11)

for every x ∈ W, or f′(x) < 0 (5.11.12)

for every x ∈ W. This follows from the intermediate value theorem when f′ is continuous on W. There is a version of the intermediate value theorem for the derivative of a differentiable real-valued function on an interval, even if the derivative is not continuous, as in Theorem 5.12 on p108 of [155]. One can use this to get the same conclusion without asking that f′ be continuous on W.

If (5.11.11) holds on W, then

(5.11.13) f is strictly increasing on W,

by the mean-value theorem. Similarly, if (5.11.12) holds on W, then

(5.11.14) f is strictly decreasing on W.

In both cases, we get that f is one-to-one on W. It is easy to see that f is an open mapping on W in both cases too, using the intermediate value theorem. This uses the fact that f is continuous on W, because f is differentiable.

5.12 Diffeomorphisms with more regularity

Let n and r be positive integers, let W be a nonempty open subset of R^{n}, and let f be an r-times continuously differentiable mapping from W into R^{n}. Thus the differential df_{x} of f at x ∈ W defines a function on W with values in the space L(R^{n}) of linear mappings from R^{n} into itself. Observe that

df ∈ C^{r−1}(W, L(R^{n})), (5.12.1)

in the notation of Subsections 5.3.1 and 5.4.1, because the entries of the n × n matrix corresponding to df_{x} are given by the partial derivatives of the components of f at x ∈ W.

Using (5.12.1), we get that

det df ∈ C^{r−1}(W, R), (5.12.2)

because the determinant is a smooth function on L(R^{n}). Alternatively,

det df (5.12.3)

can be expressed in terms of sums of products of partial derivatives of the cmponents of f, each of which is (r − 1)-times continuously differentiable on W, by hypothesis.

Suppose also now that df_{x} ∈ GL(R^{n}) for every x ∈ W. In this case, we get that (df)^{−1} ∈ C^{r−1}(W, L(R^{n})), (5.12.4)

because the mapping that sends an element of GL(R^{n}) to its inverse is smooth, as in Subsections 5.2.1 and 5.4.1. This could be obtained from Cramer’s rule too, using (5.12.2).

5.12. DIFFEOMORPHISMS WITH MORE REGULARITY 111

5.12.1 Regularity of the inverse mapping

Suppose that f is one-to-one on W as well, and remember that f^{−1} is continuously differentiable on W, as in Subsection 5.11.1, and that the differential of f^{−1} is as in (5.11.2). Equivalently, this means that d(f^{−1})_{y} = (df_{f^{−1}(y)})^{−1} (5.12.5) for every y ∈ f(W). Let us use this to get that d(f^{−1}) ∈ C^{r−1}(f(W), L(R^{n})), (5.12.6) which is the same as saying that f^{−1} ∈ C^{r}(f(W), R^{n}). (5.12.7)

More precisely, suppose for the moment that f^{−1} ∈ C^{l}(f(W), R^{n}), (5.12.8) where 0 ≤ l ≤ r − 1. Observe that (5.12.9) df_{f^{−1}(y)} is l-times continuously differentiable as a function of y ∈ f(W) with values in L(R^{n}), because of (5.12.1). This uses the fact that compositions of l-times continuously differentiable functions are l-times continuously differentiable too, as in Section 3.9. It follows that (5.12.10) (df_{f^{−1}(y)})^{−1} is l-times continuously differentiable as a function of y ∈ f(W) with values in L(R^{n}), as before.

This means that d(f^{−1}) ∈ C^{l}(f(W), L(R^{n})), (5.12.11) because of (5.12.5). This is the same as saying that f^{−1} ∈ C^{l+1}(f(W), R^{n}), (5.12.12) as before. One can repeat the process as needed to get (5.12.6), and thus (5.12.7).

5.12.2 C^{r} And C^{∞} diffeomorphisms

Let us say that (5.12.13) f is a C^{r} diffeomorphism from W onto f(W) under these conditions. This implies that (5.12.14) f^{−1} is a C^{r} diffeomorphism from f(W) onto W.

112 CHAPTER 5. DIFFEOMORPHISMS AND CELLS If g is a mapping from f(W) into R^{n} that is a C^{r} diffeomorphism from f(W) onto g(f(W)), then (5.12.15) g ∘ f is a C^{r} diffeomorphism from W onto g(f(W)).

Similarly, if f is infinitely differentiable on W, then df ∈ C^{∞}(W, L(R^{n})), (5.12.16) and det df ∈ C^{∞}(W, R). (5.12.17) If df_{x} ∈ GL(R^{n}) for every x ∈ W, then it follows that (df)^{−1} ∈ C^{∞}(W, L(R^{n})), (5.12.18) as before. If f is also one-to-one on W, then we get that f^{−1} ∈ C^{∞}(W, R^{n}), (5.12.19) as in (5.12.7). In this case, we say that (5.12.20) f is a C^{∞} diffeomorphism from W onto f(W), and we have that (5.12.21) f^{−1} is a C^{∞} diffeomorphism from f(W) onto W. If g is a mapping from f(W) into R^{n} that is a C^{∞} diffeomorphism from f(W) onto g(f(W)) as well, then (5.12.22) g ∘ f is a C^{∞} diffeomorphism from W onto g(f(W)).

5.13 Cells and supports of functions

Let n be a positive integer, and let a_{1}, …, a_{n} and b_{1}, …, b_{n} be real numbers with a_{j} ≤ b_{j} for each j = 1, …, n. The Cartesian product

C = ∏_{j=1}^{n}[a_{j}, b_{j}] (5.13.1)

of the closed intervals [a_{j}, b_{j}], 1 ≤ j ≤ n, is called a cell in R^{n}.

It is well known that (5.13.2) cells in R^{n} are compact, with respect to the standard Euclidean metric. In order to show that subsets of R^{n} that are closed and bounded are also compact, one often starts by showing that cells are compact. Note that a continuous mapping from a cell in R^{n} into R^{m} for some positive integer m is uniformly continuous, as in Subsection 5.8.1.

Let e_{1}, …, e_{n} be the standard basis vectors in R^{n}, and let α_{1}, …, α_{n} be real numbers. Consider the corresponding parallelepiped

P_{R^{n}}(α_{1} e_{1}, …, α_{n} e_{n}) (5.13.3)

in R^{n}, as in Section 4.14. This is the same as the cell

∏_{j=1}^{n} I_{j}, (5.13.4)

where I_{j} = [0, α_{j}] when α_{j} ≥ 0, and I_{j} = [α_{j}, 0] when α_{j} ≤ 0. Every cell in R^{n} is a translate of a cell of this type. Note that the n-dimensional volume of the cell C is given by

Vol_{n}(C) = ∏_{j=1}^{n}(b_{j} − a_{j}). (5.13.5)

5.13.1 Integrals over cells

Let f be a continuous real-valued function on C, which is in fact uniformly continuous on C, as mentioned earlier. One can check that for each l = 1, …, n,

∫_{a_{l}}^{b_{l}} f(x_{1}, …, x_{l}, …, x_{n}) dx_{l} (5.13.6)

is uniformly continuous as a function of the n − 1 variables

x_{1}, …, x_{l−1}, x_{l+1}, …, x_{n}. (5.13.7)

More precisely, (5.13.6) is uniformly continuous as a real-valued function on the cell in R^{n−1} corresponding to the Cartesian product of the closed intervals [a_{j}, b_{j}] with j ≠ l.

Of course,

∫_{C} f(x) dx (5.13.8)

may be defined as an n-dimensional Riemann integral over C in a standard way. It is well known that this is the same as the iterated integral

∫_{a_{1}}^{b_{1}} ⋯ ( ∫_{a_{n}}^{b_{n}} f(x_{1}, …, x_{n}) dx_{n}) ⋯ dx_{1}. (5.13.9)

In fact, it is well known that (5.13.8) is the same as iterated integrals like (5.13.9), where x_{1}, …, x_{n} are integrated in any order. This may be shown intially for n = 2, which can be used to get the analogous statement for any n.

114 CHAPTER 5. DIFFEOMORPHISMS AND CELLS

5.13.2 Supports of functions

Let m be another positive integer, and let f be a function on R^{n} with values in R^{m}. The support of f is defined the closure of the set of x ∈ R^{n} such that f(x) ≠ 0, which may be denoted supp f = \overline{\{x ∈ R^{n} : f(x) ≠ 0\}}. (5.13.10) Note that the support of f is automatically a closed set in R^{n}. If the set of x ∈ R^{n} such that f(x) ≠ 0 is bounded, then the support of f is bounded too, and thus compact.

Let f be a continuous real-valued function on R^{n} with bounded support. This implies that there is a cell C in R^{n} such that supp f ⊆ C. (5.13.11) In this case, the integral of f over R^{n} may be defined by ∫_{R^{n}} f(x) dx = ∫_{C} f(x) dx, (5.13.12) as in (3) on p247 of [155]. One can check that this does not depend on the particular cell C in R^{n} that satisfies (5.13.11).

5.14 Some smooth functions on R

Let ϕ be an infinitely-differentiable real-valued function on the real line such that

(5.14.1)  ϕ(t) > 0 when t > 0
= 0 when t ≤ 0,

as in Subsection 3.7.2. If a, b are real numbers with a < b, then put ϕ_{a,b}(t) = ϕ(t − a) ϕ(b − t). (5.14.2) This is an infinitely-differentiable real-valued function on the real line such that

(5.14.3)  ϕ_{a,b}(t) > 0 when a < t < b
= 0 when t ≤ a and when b ≤ t.

If r ∈ R, then ϕ_{a,b}(((a + b)/2) + r) = ϕ(((b − a)/2) + r) ϕ(((b − a)/2) − r). (5.14.4) This implies that ϕ_{a,b}(((a + b)/2) + r) = ϕ_{a,b}(((a + b)/2) − r). (5.14.5)

Note that

∫_{a}^{b} ϕ_{a,b}(u) du > 0. (5.14.6)

Put

ψ_{a,b}(t) = ( ∫_{0}^{1} ϕ_{a,b}(u) du)^{−1} ϕ_{a,b}(t) (5.14.7)

for each t ∈ R. This is an infinitely-differentiable real-valued function on the real line such that

(5.14.8)  ψ_{a,b}(t) > 0 when a < t < b
= 0 when t ≤ a and when b ≤ t

and

∫_{a}^{b} ψ_{a,b}(t) dt = 1. (5.14.9)

We can integrate ψ_{a,b} on R to get an infinitely-differentiable real-valued function η_{a,b} on R such that η′_{a,b} = ψ_{a,b} (5.14.10)

on R, and η_{a,b}(t) = 0 when t ≤ a. (5.14.11)

This means that

η_{a,b}(t) = ∫_{a}^{t} ψ_{a,b}(u) du (5.14.12)

when t ≥ a, so that 0 < η_{a,b}(t) ≤ 1 (5.14.13)

when t > a, and η_{a,b}(t) = 1 (5.14.14)

when t ≥ b. Note that η_{a,b} is strictly increasing on [a, b], by construction.

Note that 1 − η_{a,b} is an infinitely-differentiable real-valued function on R such that 1 − η_{a,b}(t) = 1 (5.14.15)

when t ≤ a, 0 < 1 − η_{a,b}(t) < 1 (5.14.16)

when a < t < b, and 1 − η_{a,b}(t) = 0 (5.14.17)

when t ≥ b. We also have that 1 − η_{a,b} is strictly decreasing on [a, b]. More precisely,

1 − η_{a,b}(t) = ∫_{t}^{b} ψ_{a,b}(u) du (5.14.18)

when t ≤ b, by (5.14.9) and (5.14.12). One can check that

1 − η_{a,b}(((a + b)/2) + r) = η_{a,b}(((a + b)/2) − r) (5.14.19) for every r ∈ R. This uses (5.14.5), which implies that ψ_{a,b} has the same property.

If c, d are real numbers with b ≤ c < d, then put

ξ(t) = ξ_{a,b,c,d}(t) = η_{a,b}(t) (1 − η_{c,d}(t)) (5.14.20)

for each t ∈ R. This is an infinitely-differentiable real-valued function on R with ξ(t) = 0 (5.14.21)

when t ≤ a and when d ≤ t. Similarly,

0 < ξ(t) < 1 (5.14.22)

when a < t < b and when c < t < d, and

ξ(t) = 1 (5.14.23)

when b ≤ t ≤ c. In fact, ξ is strictly increasing on [a, b], and strictly decreasing on [c, d].

5.15 Some smooth functions on R^{n}

Let n be a positive integer, let a_{1}, …, a_{n} and b_{1}, …, b_{n} be real numbers with a_{j} ≤ b_{j} for each j = 1, …, n, and let C = ∏_{j=1}^{n}[a_{j}, b_{j}] be the corresponding cell in R^{n}, as in Section 5.13. Also let a′_{1}, …, a′_{n} and b′_{1}, …, b′_{n} be real numbers with a′_{j} < a_{j} and b_{j} < b′_{j} (5.15.1)

for each j. Note that

C ⊆ ∏_{j=1}^{n}(a′_{j}, b′_{j}), (5.15.2)

and that the right side is an open set in R^{n}.

As in the previous section, there is an infinitely-differentiable real-valued function ξ_{j} on R for each j such that

0 ≤ ξ_{j} ≤ 1 (5.15.3)

on R, (5.15.4) ξ_{j}(t) > 0 if and only if a′_{j} < t < b′_{j},

and (5.15.5) ξ_{j}(t) = 1 if and only if a_{j} ≤ t ≤ b_{j}.

Put

ζ(x) = ∏_{j=1}^{n} ξ_{j}(x_{j}) (5.15.6)

5.15. SOME SMOOTH FUNCTIONS ON R^{N} 117 for each x ∈ R^{n}, which defines an infinitely-differentiable real-valued function on R^{n} such that 0 ≤ ζ(x) ≤ 1 (5.15.7) for every x ∈ R^{n}. Observe that

(5.15.8) ζ(x) > 0 if and only if x ∈ ∏_{j=1}^{n}(a′_{j}, b′_{j}),

because of (5.15.4). Similarly, (5.15.9) ζ(1) = 1 if and only if x ∈ C, because of (5.15.5). Using (5.15.8), we get that the support of ζ is given by

supp ζ = ∏_{j=1}^{n}[a′_{j}, b′_{j}]. (5.15.10)

Now let a, b be positive real numbers with a < b, so that a^{2} < b^{2}, and let η_{a^{2},b^{2}} (5.15.11) be the infinitely-differentiable real-valued function on R corresponding to a^{2} and b^{2} as in the previous section. If x ∈ R^{n}, then put

ρ_{a,b}(x) = 1 − η_{a^{2},b^{2}}(∥x∥_{2}^{2}) = 1 − η_{a^{2},b^{2}}( ∑_{j=1}^{n} x_{j}^{2}), (5.15.12)

where ∥x∥_{2} is the standard Euclidean norm on R^{n}, as in Section 3.2. This is an infinitely-differentiable real-valued function on R^{n}, with 0 ≤ ρ_{a,b}(x) ≤ 1 (5.15.13) for every x ∈ R^{n}. In fact, we have that ρ_{a,b}(x) = 1 (5.15.14) when ∥x∥_{2} ≤ a, 0 < ρ_{a,b}(x) < 1 (5.15.15) when a < ∥x∥_{2} < b, and ρ_{a,b}(x) = 0 (5.15.16) when ∥x∥_{2} ≥ b. It follows that supp ρ_{a,b} = {x ∈ R^{n} : ∥x∥_{2} ≤ b}. (5.15.17)

Integration and n-surfaces

6.1 Some partitions of unity

Let n be a positive integer, and let ϕ_{1}, …, ϕ_{l} be finitely many real-valued functions on R^{n}, with 0 ≤ ϕ_{k} ≤ 1 (6.1.1)

on R^{n} for each k = 1, …, n. Put ψ_{1} = ϕ_{1}, and

ψ_{k} = ( ∏_{p=1}^{k−1}(1 − ϕ_{p})) ϕ_{k} (6.1.2)

for each k = 2, …, l, as in (28) on p251 of [155]. Observe that 0 ≤ 1 − ϕ_{p} ≤ 1 for each p = 1, …, l, so that 0 ≤ ψ_{k} ≤ 1 (6.1.3)

on R^{n} for each k = 1, …, l. We also have that

ψ_{k}(x) = 0 when ϕ_{k}(x) = 0, (6.1.4)

so that (6.1.5) supp ψ_{k} ⊆ supp ϕ_{k}

for every k = 1, …, l.

Let us check that

∑_{p=1}^{k} ψ_{p} = 1 − ∏_{p=1}^{k}(1 − ϕ_{p}) (6.1.6)

for each k = 1, …, l, as in (29) on p251 of [155]. This follows from the definition of ψ_{1} when k = 1, and we can use induction to go from k to k + 1 when k < l. This uses the fact that

ψ_{k+1} = ( ∏_{p=1}^{k}(1 − ϕ_{p})) ϕ_{k+1} = ∏_{p=1}^{k}(1 − ϕ_{p}) − ∏_{p=1}^{k+1}(1 − ϕ_{p}) (6.1.7)

6.2. THE STANDARD SIMPLEX IN R^{N} 119 when k < l. It follows that

∑_{p=1}^{l} ψ_{p}(x) = 1 (6.1.8)

on the set

⋃_{q=1}^{l}{x ∈ R^{n} : ϕ_{q}(x) = 1}, (6.1.9)

as on p251 of [155]. This is a type of partition of unity for the set (6.1.9), as on p251 of [155].

If the ϕ_{k}’s are continuous on R^{n}, then (6.1.10) the ψ_{k}’s are continuous on R^{n} too. Similarly, if the ϕ_{k}’s are r-times continuously differentiable on R^{n} for some positive integer r, then (6.1.11) the ψ_{k}’s are r-times continuously differentiable on R^{n} as well. If the ϕ_{k}’s are infinitely differentiable on R^{n}, then it follows that (6.1.12) the ψ_{k}’s are infinitely differentiable on R^{n}. This is related to Exercise 6 on p289 of [155].

Let m be another positive integer, let E be a subset of R^{n}, and let f be a function on E with values in R^{m}. If E is contained in (6.1.9), then we get that

f(x) = ∑_{p=1}^{l} ψ_{p}(x) f(x) (6.1.13)

for every x ∈ E, by (6.1.8). Similarly, if f is defined on all of R^{n}, and the support of f is contained in (6.1.9), then (6.1.13) holds for every x ∈ R^{n}.

6.2 The standard simplex in R^{n}

Let n be a positive integer, and consider

(6.2.1) Q^{n} = {x ∈ R^{n} : x_{j} ≥ 0 for each j = 1, …, n, and ∑_{j=1}^{n} x_{j} ≤ 1}.

This is known as the standard simplex or standard n-simplex in R^{n}, as in Example 10.4 on p247 of [155], and as mentioned on p141 of [184]. Note that (6.2.2) Q^{n} is closed and bounded in R^{n}, and thus compact.

120 CHAPTER 6. INTEGRATION AND N-SURFACES

In fact, Q^{n} ⊆ [0, 1]^{n}, (6.2.3)

where the right side is the unit cube in R^{n}, consisting of x ∈ R^{n} such that 0 ≤ x_{j} ≤ 1 for each j = 1, …, n. This is the same as the Cartesian product of n copies of the closed unit interval [0, 1] in the real line, which is a cell in R^{n} in particular. If n = 1, then we have equality in (6.2.3).

Let f be a continuous real-valued function on Q^{n}, so that f is bounded and uniformly continuous on Q^{n}, because Q^{n} is compact. The integral

∫_{Q^{n}} f(x) dx (6.2.4)

of f over Q^{n} may be defined as an n-dimensional Riemann integral in a standard way. Another way to look at this is discussed in Example 10.4 on p247 of [155]. Of course, if n = 1, then this is the same as an ordinary Riemann integral on [0, 1], and so we suppose that n ≥ 2.

Let Q^{n−1} be the standard simplex in R^{n−1}, which is defined in the same way as before. If x ∈ Q^{n}, then

(x_{1}, …, x_{n−1}) ∈ Q^{n−1}. (6.2.5)

More precisely, if (6.2.5) holds, then

(x_{1}, …, x_{n−1}, x_{n}) ∈ Q^{n} (6.2.6)

if and only if

0 ≤ x_{n} ≤ 1 − ∑_{j=1}^{n−1} x_{j}, (6.2.7)

and every element of Q^{n} occurs in this way.

If (6.2.5) holds, then put

f_{n−1}(x_{1}, …, x_{n−1}) = ∫_{0}^{1−∑_{j=1}^{n−1} x_{j}} f(x_{1}, …, x_{n−1}, x_{n}) dx_{n}. (6.2.8)

One can check that f_{n−1} is bounded and uniformly continuous on Q^{n−1}, because f is bounded and uniformly continuous on Q^{n}. We also have that

f_{n−1}(x_{1}, …, x_{n−1}) = 0 when ∑_{j=1}^{n−1} x_{j} = 1, (6.2.9)

by construction.

Let us extend f_{n−1} to a real-valued function on [0, 1]^{n−1} by putting

f_{n−1}(x_{1}, …, x_{n−1}) = 0 (6.2.10)

6.2. THE STANDARD SIMPLEX IN R^{N} 121

when (x_{1}, …, x_{n−1}) ∈ [0, 1]^{n−1} \ Q^{n−1}, which means that

∑_{j=1}^{n−1} x_{j} > 1. (6.2.11)

This defines a continuous function on [0, 1]^{n−1}, because of (6.2.9).

Thus the integral of f_{n−1} over [0, 1]^{n−1} may be defined in the usual way, which is the same as the integral of f_{n−1} over Q^{n−1}. This is one way to look at the integral of f over Q^{n}.

6.2.1 More on integration over Q^{n}

We can also extend f to a real-valued function on [0, 1]^{n} by putting

f(x) = 0 (6.2.12)

when x ∈ [0, 1]^{n} \ Q^{n}, which means that

∑_{j=1}^{n} x_{j} > 1. (6.2.13)

However, this extension will not be continuous at x ∈ [0, 1]^{n} such that

∑_{j=1}^{n} x_{j} = 1, (6.2.14)

unless (6.2.12) holds. Note that the previous extension of f_{n−1} to [0, 1]^{n−1} may be expressed as

f_{n−1}(x_{1}, …, x_{n−1}) = ∫_{0}^{1} f(x_{1}, …, x_{n−1}, x_{n}) dx_{n}, (6.2.15)

using this extension of f to [0, 1]^{n}.

To deal with this, a continuous real-valued function F = F_{δ} on [0, 1]^{n} may be defined on [0, 1]^{n} for 0 < δ < 1, with

(6.2.16)  F (x) = f(x) when ∑_{j=1}^{n} x_{j} ≤ 1 − δ

= 0  when ∑_{j=1}^{n} x_{j} ≥ 1,

and

|F (x)| ≤ |f(x)| when 1 − δ < ∑_{j=1}^{n} x_{j} < 1, (6.2.17)

as on p247 of [155]. More precisely, we also have that

|f(x) − F (x)| ≤ |f(x)| when 1 − δ < ∑_{j=1}^{n} x_{j} < 1, (6.2.18)

because F (x) is defined so that it has the same sign as f(x) in this case. Let C be a nonnegative real number such that

|f(x)| ≤ C (6.2.19)

for every x ∈ Q^{n}. If F_{n−1} = F_{δ.n−1} is defined on [0, 1]^{n−1} as in (6.2.15), then one can check that

|f_{n−1}(x_{1}, …, x_{n−1}) − F_{n−1}(x_{1}, …, x_{n−1})| ≤ C δ (6.2.20)

on [0, 1]^{n−1}, as in (7) on p247 of [155]. In particular, this implies that the integral of f_{n−1} over [0, 1]^{n−1} is approximated by the integral of F over [0, 1]^{n−1}.

Of course, we could start by integrating f in any of the variables x_{1}, …, x_{n}, with the same properties as before. The result would be uniformly approximated by the integral of F in the same variable.

6.3 Jacobians and changes of variables

Let n be a positive integer, let W be a nonempty open subset of R^{n}, and let ϕ be a mapping from W into R^{n}. If ϕ is differentiable at a point x ∈ W, then the determinant det ϕ′(x) = det dϕ_{x} (6.3.1)

of the differential ϕ′(x) = dϕ_{x} of ϕ at x is known as the Jacobian of ϕ at x. Note that if n = 1, then the Jacobian of ϕ at x is the same as the ordinary derivative of ϕ at x.

Suppose that ϕ is continously differentiable on W, so that the differential ϕ′(x) of ϕ at x is continuous as a function of x ∈ W with values in the space L(R^{n}) of linear mappings from R^{n} into itself, as in Section 5.5. This implies that (6.3.2) det ϕ′(x) is continuous

as a real-valued function of x ∈ W, as before.

Suppose now that ϕ(W) is an open set in R^{n}, and that

(6.3.3) ϕ is a C^{1} diffeomorphism from W onto ϕ(W),

as in Subsection 5.11.1. Let K be a compact subset of R^{n} that is contained in W, and let f be a continuous real-valued function on ϕ(K). Note that ϕ(K) is a compact set in R^{n} too, as in Subsection 5.8.1. It is well known that

∫_{ϕ(K)} f(y) dy = ∫_{K} f(ϕ(x)) | det ϕ′(x)| dx, (6.3.4) at least if K is nice enough for the Riemann integral to be defined. There are versions of this using Lebesgue measure and integration with milder hypotheses, but we shall not pursue this here.

Remember that if T is a linear mapping from R^{n} into itself and E is a reasonably nice subset of R^{n}, then the n-dimensional volume of T (E) is equal to | det T | times the n-dimensional volume of E, as mentioned in Section 4.12. Basically (6.3.4) behaves approximately like this on sufficiently small sets. Of course, if ϕ is a linear mapping on R^{n}, then its Jacobian is constant, and (6.3.4) is somewhat simpler. This is also somewhat simpler when n = 1, in which case it may normally be stated with milder hypotheses.

This version of the change of variables formula is essentially the same as the one mentioned in (1) in Section 4.4 on p141 of [184]. Another version is given in Theorem 10.9 on p252 of [155], which will be discussed on the next section.

6.4 Another version of changing variables

Let us continue with the same notation and hypotheses as in the previous section. Let f be a continuous real-valued function R^{n} with compact support and supp f ⊆ ϕ(W). (6.4.1)

This implies that

(6.4.2) ϕ^{−1}(supp f) is a compact set in R^{n},

as in Subsection 5.8.1, because ϕ^{−1} is continuous as a mapping from ϕ(W) into R^{n}. Of course, ϕ^{−1}(supp f) ⊆ W, (6.4.3)

because ϕ^{−1}(ϕ(W)) = W.

Let f_{ϕ} be the real-valued function defined on R^{n} by

(6.4.4)  f_{ϕ}(x) = f(ϕ(x)) when x ∈ W
= 0  when x ∈ R^{n} \ W.

One can check that (6.4.5) f_{ϕ} is continuous on R^{n},

with (6.4.6) supp f_{ϕ} = ϕ^{−1}(supp f).

Let F_{ϕ} be the real-valued function on R^{n} defined by

(6.4.7)  F_{ϕ}(x) = f(ϕ(x)) | det ϕ′(x)| when x ∈ W
= 0  when x ∈ R^{n} \ W.

One can verify that (6.4.8) F_{ϕ} is continuous on R^{n}, with (6.4.9) supp F_{ϕ} = supp f_{ϕ} = ϕ^{−1}(supp f).

Under these conditions, Theorem 10.9 on p252 of [155] states that

∫_{R^{n}} f(y) dy = ∫_{R^{n}} F_{ϕ}(x) dx, (6.4.10)

where both sides of the equation are defined as in Subsection 5.13.2.

Part of the proof of (6.4.10) is to use partitions of unity to reduce to the case of (6.4.11) functions f supported in sufficiently small balls in R^{n}.

In that case, one can use another result on the local behavior of C^{1} diffeomorphisms between open subsets of R^{n}, in Theorem 10.7 on p249 of [155].

Of course, the change of variables formula is simpler when n = 1. One can use this to obtain the change of variables formula for any n when the diffeomorphism is “primitive” in the sense of Definition 10.5 on p248 of [155], as mentioned on p252 of [155]. A linear mapping from R^{n} onto itself is called a “flip” when it interchanges two coordinates of a point in R^{n} and does not change the other coordinates, as in Definition 10.6 on p249 of [155]. It is easy to see that the change of variables formula holds for these mappings, as mentioned on p252 of [155].

Theorem 10.7 on p249 of [155] says that locally,

(6.4.12) a C^{1} diffeomorphism can be expressed as a composition of primitive mappings, flips, and a translation.

This is used to get that the change of variables formula holds for functions supported in sufficiently small balls, as before. Thus one would like to express f as a sum of finitely many continuous functions supported in sufficiently small balls of this type. This can obtained using a partition of unity, as in (6.1.13). This also uses the compactness of the support of f, to cover f by finitely many balls that are contained in balls that are a bit larger and sufficiently small too.

6.5 Some remarks about linear independence

Let n and m be positive integers with n < m, and let u_{1}, …, u_{m} be the standard basis vectors in R^{m}. Suppose that

(6.5.1) v_{1}, …, v_{n} are n linearly independent vectors in R^{m}.

Let L be the linear span of v_{1}, …, v_{n} in R^{m}, so that

dim L = n, (6.5.2)

as a vector space over the real numbers.

6.5. SOME REMARKS ABOUT LINEAR INDEPENDENCE 125

Under these conditions, it is well known that there is a set I_{2} ⊆ {1, …, m} with exactly m − n elements such that (6.5.3) v_{1}, …, v_{n} together with u_{k}, k ∈ I_{2}, is a basis for R^{m}. One way to see this is to take I_{2} to be a maximal subset of {1, …, m} so that this collection of vectors is linearly independent. Alternatively, one can take I_{2} to be a minimal subset of {1, …, m} so that the linear span of this collection of vectors is equal to R^{m}.

Put I_{1} = {1, …, m} \ I_{2}, (6.5.4) so that I_{1} is a subset of {1, …, m} with exactly n elements. Thus I_{1} = {l_{1}, …, l_{n}}, (6.5.5) where 1 ≤ l_{1} < ⋯ < l_{n} ≤ m. (6.5.6) Let P_{1} be the linear mapping from R^{m} onto R^{n} such that for each y ∈ R^{m}, (6.5.7) the jth coordinate of P_{1}(y) is equal to the l_{j}-th coordinate of y for each j = 1, …, n. Note that P_{1}(u_{k}) = 0 when k ∈ I_{2}, (6.5.8) by construction. More precisely, the kernel of P_{1} is spanned by the u_{k}’s, k ∈ I_{2}.

Let e_{1}, …, e_{n} be the standard basis vectors in R^{n}. Remember that there is a unique linear mapping T from R^{n} into R^{m} such that T (e_{j}) = v_{j} (6.5.9) for each j = 1, …, n. Of course, L = T (R^{n}), (6.5.10) by construction. The condition that the v_{j}’s be linearly independent in R^{m} says exactly that (6.5.11) T is one-to-one on R^{n}.

Observe that P_{1}(T (R^{n})) = P_{1}(L) = P_{1}(R^{m}) = R^{n}, (6.5.12) using (6.5.3) and (6.5.8) in the second step. This implies that (6.5.13) P_{1} ∘ T is one-to-one on R^{n}, because P_{1} ∘ T is a linear mapping from R^{n} into itself.

126 CHAPTER 6. INTEGRATION AND N-SURFACES

6.5.1 Considering L as a graph

Equivalently, (6.5.14) P_{1} is one-to-one on L. This means that if v ∈ L, then (6.5.15) the coordinates of v corresponding to elements of I_{2}

are uniquely determined by

the coordinates of v corresponding to elements of I_{1}.

It follows that (6.5.16) L corresponds to the graph of a linear mapping

from R^{n} into R^{m−n},

by arranging the coordinates appropriately.

More precisely, put R = T ∘ (P_{1} ∘ T)^{−1}, (6.5.17) where (P_{1} ∘ T)^{−1} is the inverse of P_{1} ∘ T, as a one-to-one linear mapping from R^{n} onto itself. Thus R is a one-to-one linear mapping from R^{n} into R^{m}, and R(R^{n}) = T (R^{n}) = L. (6.5.18) By construction, we also have that P_{1} ∘ R = P_{1} ∘ T ∘ (P_{1} ∘ T)^{−1} = I_{R^{n}}, (6.5.19) where I_{R^{n}} is the identity mapping on R^{n}. This basically means that L corresponds to the graph of a mapping from R^{n} into R^{m−n} obtained from the components of R corresponding to elements of I_{2}, with the coordinates arranged appropriately.

6.6 Some nice local graphs

Let us continue with the same notation and hypotheses as in the previous section. Also let W be an open set in R^{n}, and let F be a continuously-differentiable mapping from W into R^{m}. Suppose that w ∈ W and (6.6.1) dF_{w} is one-to-one on R^{n}, as a linear mapping from R^{n} into R^{m}. If we put v_{j} = dF_{w}(e_{j}) (6.6.2) for each j = 1, …, n, then (6.5.1) holds, because of (6.6.1).

6.6. SOME NICE LOCAL GRAPHS 127

This leads to subsets I_{1}, I_{2} of {1, …, m} as before, as well as the linear mapping P_{1} from R^{m} onto R^{n}. We also have that (6.6.3) P_{1} ∘ dF_{w} is a one-to-one mapping from R^{n} onto itself, as in (6.5.12) and (6.5.13). Under these conditions, we would like to use the inverse function theorem to say more about F near w.

Put f = P_{1} ∘ F, (6.6.4) which is a continuously-differentiable mapping from W into R^{n}. Note that df_{w} = P_{1} ∘ dF_{w}, (6.6.5) which is an easy instance of the chain rule that can be seen more directly. It follows that df_{w} ∈ GL(R^{n}), (6.6.6) by (6.6.3). The inverse function theorem implies that (6.6.7) f has a local inverse near w that is continuously differentiable, as in Section 5.6.

6.6.1 Using the local inverse

We would like to use this to get that near w, (6.6.8) F takes values in the graph of a continuously-differentiable function

defined on a neighborhood of f(w) in R^{n} with values in R^{m−n}, with the coordinates arranged appropriately. More precisely, the inverse function theorem implies that there are open subsets U(w) and V (w) of R^{n} such that w ∈ U(w), U(w) ⊆ W, f(w) ∈ V (w), and (6.6.9) the restriction of f to U(w) is a one-to-one mapping onto V (w). In particular, the injectivity of f on U(w) implies that (6.6.10) F is one-to-one on U(w), by the definition (6.6.4) of f.

The inverse g = g_{w} of the restriction of f to U(w) is continuously differentiable on V (w), as before. It follows that (6.6.11) F ∘ g is continuously differentiable as a mapping from V (w) into R^{m}. Observe that (6.6.12) P_{1} ∘ F ∘ g = f ∘ g is the identity mapping on V (w), 128 CHAPTER 6. INTEGRATION AND N-SURFACES

by construction. This implies that F (U(w)) = (F ∘ g)(V (w)) (6.6.13) corresponds to the graph of a continuously-differentiable function on V (w) with values in R^{m−n}, with the coordinates arranged appropriately.

If F is r-times continuously differentiable on W for some positive integer r, then f is r-times continuously differentiable on W, and (6.6.14) g is r-times continuously differentiable on V (w) too, as in Subsection 5.12.1. This implies that (6.6.15) F ∘ g is r-times continuously differentiable on V (w) as well, as in Subsection 3.9.1.

Similarly if F is infinitely differentiable on W, then f is infinitely differentiable on W, and (6.6.16) g is infinitely differentiable on V (w). This means that (6.6.17) F ∘ g is infinitely differentiable on V (w) too.

6.7 Injective linear mappings

Let n and m be positive integers with n ≤ m, and let e_{1}, …, e_{n} be the standard basis vectors in R^{n}, as usual. Remember that a linear mapping T from R^{n} into R^{m} is one-to-one on R^{n} if and only if (6.7.1) T (e_{1}), …, T (e_{n}) are linearly independent in R^{m}. In this case, there is a linear mapping P_{1} from R^{m} onto R^{n} such that P_{1} ∘ T ∈ GL(R^{n}), (6.7.2) as in Section 6.5. More precisely, if m = n, then one can simply take P_{1} to the the identity mapping on R^{n}. Conversely, if (6.7.2) holds for any linear mapping P_{1} from R^{m} into R^{n}, then T is one-to-one on R^{n}.

If P_{1} is any linear mapping from R^{m} into R^{n}, then T ↦ P_{1} ∘ T (6.7.3) defines a linear mapping from the space L(R^{n}, R^{m}) of linear mappings from R^{n} into R^{m} into the space L(R^{n}) of linear mappings from L(R^{n}) of linear mappings from R^{n} into itself. This corresponds to a linear mapping from the space M_{m,n}(R) of m × n matrices of real numbers into M_{n,n}(R), as in Section 5.3. This may be identified with a linear mapping from R^{m n} into R^{n^{2}}, as in Section 5.1. In particular, this mapping is continuous with respect to the standard Euclidean metrics on R^{m n} and R^{n^{2}}.

One can use this to get that

{T ∈ L(R^{n}, R^{m}) : P_{1} ∘ T ∈ GL(R^{n})} (6.7.4)

is an open set in L(R^{n}, R^{m}), in the sense that it corresponds to an open set in R^{m n}. This uses the fact that GL(R^{n}) corresponds to an open set in R^{n^{2}}, as in Subsection 5.2.1 and Section 5.4. Note that (6.7.2) implies that

P_{1}(R^{m}) = R^{n}. (6.7.5)

This means that (6.7.4) is the empty set unless (6.7.5) holds.

Of course, (6.7.4) is the same as

{T ∈ L(R^{n}, R^{m}) : det(P_{1} ∘ T) ≠ 0}. (6.7.6)

It is easy to see that det(P_{1} ∘ T) (6.7.7)

is given by a polynomial in the entries of the matrix associated to T, with P_{1} fixed. In particular, this corresponds to a continuous real-valued function on R^{m n}, so that (6.7.6) corresponds to an open set in R^{m n}, as before.

It follows that

(6.7.8) {T ∈ L(R^{n}, R^{m}) : T is one-to-one on R^{n}}

is an open set in L(R^{n}, R^{m}) too. More precisely, (6.7.8) is the same as the union of (6.7.4) over all linear mappings P_{1} from R^{m} into R^{n}. In fact, one can restrict one’s attention to linear mappings P_{1} that satisfy (6.7.5), because otherwise (6.7.4) is the empty set, as before. The fact that (6.7.8) is an open set in L(R^{n}, R^{n}) could also be obtained from an argument analogous to one in Section 5.4. Of course, if m = n, then (6.7.8) is the same as GL(R^{n}).

6.8 Embeddings and immersions

Let n and m be positive integers with n ≤ m again, and let W be a nonempty open subset of R^{n}. Also let F be a continuously-differentiable mapping from W into R^{m}, so that the differential of F is continuous as a function on W with values in L(R^{n}, R^{m}), as in Section 5.5. One can check that

(6.8.1) {x ∈ W : F′(x) = dF_{x} is one-to-one on R^{n}}

is an open set in R^{n}, using the fact that (6.7.8) is an open set in L(R^{n}, R^{m}). This is the same as a remark in Section 5.5 when m = n.

Let us say that F is an immersion of W into R^{m} if

(6.8.2) F′(x) = dF_{x} is one-to-one on R^{n} for every x ∈ W.

This corresponds to part (a) of Definition 1.27 on p22 of [184]. This implies that (6.8.3) F is one-to-one on a neighborhood of every point in W,

as in Subsection 6.6.1. If F is r-times continuously differentiable on W for some positive integer r, or infinitely differentiable on W, then one may say that F is a C^{r} or C^{∞} immersion, as appropriate.

Suppose that F is an immersion of W into R^{m}. If we also have that

(6.8.4) F is one-to-one on W,

then (W, F) may be called a submanifold of R^{m}, as in part (b) of Definition 1.27 on p22 of [184]. In this case, if x ∈ W, then the tangent space of this submanifold at x ∈ W is the linear subspace

dF_{x}(R^{n}) (6.8.5)

of R^{m}. Note that dim dF_{x}(R^{n}) = n, (6.8.6)

because of (6.8.2). If F is r-times continuously differentiable on W, or infinitely differentiable on W, then (W, F) may be called a C^{r} or C^{∞} submanifold of R^{m}, as appropriate.

If (6.8.7) F is a homeomorphism from W onto F (W),

as in Subsection 5.9.1, then F may be called an imbedding of W into R^{m}, as in part (c) of Definition 1.27 on p22 of [184]. However, some colleagues have mentioned that the term embedding is sometimes used for one-to-one immersions. In order to be precise, we may say that F is an embedding in the strong sense when (6.8.7) holds. As before, if F is r-times continuously differentiable on W, or infinitely differentiable on W, then one may say that F is a C^{r} or C^{∞} embedding of W into R^{m}, as appropriate.

6.8.1 Some remarks about manifolds

One can consider broader notions of manifolds, but we shall not pursue this in detail here. One can start with n-dimensional topological manifolds, which are topological spaces that are locally homeomorphic to open sets in R^{n}, and which are often asked to satisfy some additional conditions. If r is a positive integer, then one can consider n-dimensional C^{r} manifolds, which are locally C^{r} diffeomorphic to open sets in R^{n} in a suitable sense. Similarly, one can consider n-dimensional C^{∞} or smooth manifolds, which are locally C^{∞} diffeomorphic to open sets in R^{n} in an analogous sense. These topics are discussed beginning on p5 of [184], for instance.

The notions of immersions and submanifolds can be extended to mappings between C^{r} and C^{∞} manifolds, as on p22 of [184]. In particular, broader notions of manifolds are already helpful for considering submanifolds of Euclidean spaces.

Basically, a (reasonably nicely embedded) C^{r} or C^{∞} n-dimensional submanifold of R^{m} should look locally like a C^{r} or C^{∞} submanifold in the previous sense at each point, as appropriate. Equivalently, this means that it looks locally like the graph of a C^{r} or C^{∞} mapping from an open set in R^{n} into R^{m−n}, with the coordinates of R^{m} arranged in a suitable way, as in Subsection 6.6.1.

The unit sphere in R^{n+1} with respect to the standard Euclidean metric is the same as

S^{n} = {y ∈ R^{n+1} : ∑_{j=1}^{n+1} y_{j}^{2} = 1}. (6.8.8)

This is a nice example of a smooth n-dimensional manifold, which is a submanifold of R^{n+1}.

6.9 Surjective linear mappings

Let n and m be positive integers with n < m, and let e_{1}, …, e_{n} be the standard basis for R^{n}, as usual. Also let T be a linear mapping from R^{m} into R^{n}. If

T (R^{m}) = R^{n}, (6.9.1)

then for each j = 1, …, n there is a v_{j} ∈ R^{m} such that

T (v_{j}) = e_{j}. (6.9.2)

Conversely, this condition implies (6.9.1).

If v_{1}, …, v_{n} are any elements of R^{m}, then there is a unique linear mapping R_{0} from R^{n} into R^{m} such that

R_{0}(e_{j}) = v_{j} (6.9.3)

for each j = 1, …, n. In this case, (6.9.2) is the same as saying that

T (R_{0}(e_{j})) = e_{j}. (6.9.4)

It is easy to see that this holds if and only if

(6.9.5) T ∘ R_{0} is the identity mapping on R^{n}.

Let R_{1} be a linear mapping from R^{n} into R^{m}. If

T (R_{1}(R^{n})) = R^{n}, (6.9.6)

then it is easy to see that (6.9.1) holds. Note that (6.9.6) holds if and only if

T ∘ R_{1} ∈ GL(R^{n}), (6.9.7) because T ∘ R_{1} is a linear mapping from R^{n} into itself. This means that T ∘ R_{1} is one-to-one on R^{n}, which implies that

(6.9.8) R_{1} is one-to-one on R^{n}.

Of course, T ↦ T ∘ R_{1} (6.9.9)

defines a linear mapping from L(R^{m}, R^{n}) into L(R^{n}), which corresponds to a linear mapping from M_{n,m}(R) into M_{n,n}(R), as in Section 5.3. This may be identified with a linear mapping from R^{n m} into R^{n^{2}}, as in Section 5.1, which is continuous in particular, with respect to the standard Euclidean metrics on R^{n m} and R^{n^{2}}.

It follows that

{T ∈ L(R^{m}, R^{n}) : T ∘ R_{1} ∈ GL(R^{n})} (6.9.10)

is an open set in L(R^{m}, R^{n}), in the sense that it corresponds to an open set in R^{n m}, because GL(R^{n}) corresponds to an open set in R^{n^{2}}, as in Subsection 5.2.1 and Section 5.4. This is the empty set unless (6.9.8) holds, as before.

Note that (6.9.10) is the same as

{T ∈ L(R^{m}, R^{n}) : det(T ∘ R_{1}) ≠ 0}. (6.9.11)

One can check that det(T ∘ R_{1}) (6.9.12)

is a polynomial in the entries of the matrix associated to T, with R_{1} fixed. This can be used as another way to get that (6.9.11) corresponds to an open set in R^{n m}.

Let us check that

{T ∈ L(R^{m}, R^{n}) : T (R^{m}) = R^{n}} (6.9.13)

is an open set in L(R^{m}, R^{n}) as well. In fact, (6.9.13) is the same as the union of (6.9.10) over all linear mappings R_{1} from R^{n} into R^{m}. More precisely, we may restrict our attention to linear mappings R_{1} that satisfy (6.9.8), because (6.9.10) is the empty set otherwise, as before.

6.10 The implicit function theorem

Let n and m be positive integers with n < m again, and let W be a nonempty open subset of R^{m}. Also let ϕ be a continuously-differentiable mapping from W into R^{n}, so that the differential of ϕ is continuous as a mapping from W into L(R^{m}, R^{n}), as in Section 5.5. Observe that

{x ∈ W : dϕ_{x}(R^{m}) = R^{n}} (6.10.1)

is an open set in R^{m}, because (6.9.13) is an open set in L(R^{m}, R^{n}).

6.10. THE IMPLICIT FUNCTION THEOREM 133

Suppose that w ∈ W and dϕ_{w}(R^{m}) = R^{n}. (6.10.2) We would like to add some components to ϕ to get a mapping Φ from W into R^{m} with dΦ_{w} ∈ GL(R^{m}), (6.10.3) so that we can use the inverse function theorem. Note that dim(ker dϕ_{w}) = m − n, (6.10.4) because of (6.10.2) and standard arguments from linear algebra. This implies that there is a linear mapping P_{0} from R^{m} onto R^{m−n} such that (6.10.5) P_{0} is one-to-one on ker dϕ_{w}, as in Section 6.5. More precisely, we can take P_{0} so that its components are the same as m − n of the coordinates of an element of R^{m}, as before.

We would like to define Φ so that its first n components are the same as the components of ϕ, and the other m−n components are equal to the components of P_{0}. In this case, the first n components of dΦ_{w} are the same as the components of dϕ_{w}, and the other m − n components of dΦ_{w} are equal to the components of P_{0}, so that (6.10.6) ker dΦ_{w} = (ker dϕ_{w}) ∩ (ker P_{0}) = {0}, using (6.10.5) in the second step. Thus (6.10.3) holds, so that we can use the inverse function theorem, as in Section 5.6. This means that there are open sets U(w), V (w) in R^{m} such that w ∈ U(w), U(w) ⊆ W, Φ(w) ∈ V (w), (6.10.7) and (6.10.8) the restriction of Φ to U(w) is a C^{1} diffeomorphism onto V (w), as before. Let Ψ = Ψ_{w} be the inverse of the restriction of Φ to U(w), which is a C^{1} diffeomorphism from V (w) onto U(w).

6.10.1 The zero set of ϕ

Suppose that ϕ(w) = 0, (6.10.9) and consider U_{0}(w) = {x ∈ U(w) : ϕ(x) = 0}. (6.10.10) Put V_{0}(w) = Φ(U_{0}(w)), (6.10.11) which is the same as the set of points in V (w) whose first n coordinates are equal to 0. Of course, Φ(w) ∈ V_{0}(w), (6.10.12)

because w ∈ U_{0}(w), by (6.10.9). We can identify V_{0}(w) with an open set in R^{m−n}, using the last m − n coordinates of the elements of V_{0}(w). Note that

Ψ(V_{0}(w)) = U_{0}(w), (6.10.13)

by construction.

This shows that U_{0}(w) is an (m − n)-dimensional C^{1} submanifold of R^{m}. This is a version of the implicit function theorem. If x ∈ U_{0}(w), then one can check that the tangent space to U_{0}(w) at x is equal to

ker dϕ_{x}. (6.10.14)

This uses the fact that dΨ_{Φ(x)} = (dΦ_{x})^{−1}, (6.10.15)

as in Section 5.6.

If ϕ is r-times continuously differentiable on W for some positive integer r, then

(6.10.16) the restriction of Φ to U(w) is a C^{r} diffeomorphism onto V (w),

Ψ is a C^{r} diffeomorphism from V (w) onto U(w), and U_{0}(w) is a C^{r} submanifold of R^{m}. Similarly, if ϕ is infinitely differentiable on W, then

(6.10.17) the restriction of Φ to U(w) is a C^{∞} diffeomorphism onto V (w),

Ψ is a C^{∞} diffeomorphism from V (w) onto U(W), and U_{0}(w) is a C^{∞} submanifold of R^{m}.

Put W_{0} = {x ∈ W : ϕ(x) = 0}, (6.10.18)

and suppose that dϕ_{x}(R^{m}) = R^{n} (6.10.19)

for every x ∈ W_{0}. In this case, the previous remarks imply that W_{0} is an (m−n)- dimensional C^{1} submanifold of R^{m}. If ϕ is r-times continuously differentiable on W, or infinitely differentiable on W, then W_{0} is a C^{r} or C^{∞} submanifold of R^{m}, as appropriate. This corresponds to Theorem 1.38 on p31 of [184].

6.11 Injectivity and alternating forms

Let W be a vector space over the real numbers, and let W_{0} be a linear subspace of W. If μ is a k-linear form on W for some positive integer k, then let μ_{0} be the k-linear form on W_{0} obtained by restricting μ to elements of W_{0}. More precisely, (6.11.1) μ_{0} is the restriction of μ to the set (W_{0})^{k} of k-tuples of elements of W_{0}, considered as a subset of the set W^{k} of k-tuples of elements of W. Of course, if μ is alternating or symmetric on W, then μ_{0} has the same property on W_{0}.

Suppose for the moment that

dim W_{0} < k. (6.11.2)

If μ is an alternating k-linear form on W, then μ_{0} is an alternating k-linear form on W_{0}, and thus μ_{0} = 0, (6.11.3)

as in Subsections 1.11.1 and 2.5.1.

Let ι_{0} be the obvious inclusion mapping of W_{0} into W, which sends every element of W_{0} to itself, and is considered as a linear mapping from W_{0} into W. Note that ι_{0}^{∗}(μ) may be defined as a k-linear form on W_{0}, as in Section 2.3. In this case, ι_{0}^{∗}(μ) = μ_{0}. (6.11.4)

Let V be another vector space over the real numbers, and let T be a linear mapping from V into W. Suppose that

T (V) ⊆ W_{0}, (6.11.5)

and let T_{0} be the same as T, considered as a linear mapping from V into W_{0}. Observe that T = ι_{0} ∘ T_{0}, (6.11.6)

as a linear mapping from V into W. If T^{∗}(μ) is the k-linear form on V as in Section 2.3, then we get that

T^{∗}(μ) = (ι_{0} ∘ T_{0})^{∗}(μ) = T_{0}^{∗}(ι_{0}^{∗}(μ)), (6.11.7)

where the second step is as in Section 2.3. This means that

T^{∗}(μ) = T_{0}^{∗}(μ_{0}), (6.11.8)

because of (6.11.4).

Of course, we can simply take W_{0} = T (V). Suppose for the moment that

dim V = k. (6.11.9)

If (6.11.10) T is not one-to-one on V,

then dim T (V) < k. (6.11.11)

If μ is an alternating k-linear form on W, then we get that

T^{∗}(μ) = 0, (6.11.12)

by (6.11.3) and (6.11.8).

6.11.1 Injectivity of the differential

Let n and m be positive integers, and let U, V be nonempty open subsets of R^{n}, R^{m}, respectively. Also let ψ be a mapping from U into R^{m} with

ψ(U) ⊆ V, (6.11.13)

and let α be a differential n-form on V. Suppose that ψ is differentiable on U, so that ψ^{∗}(α) may be defined as a differential n-form on U, as in Section 4.4.

Let x ∈ U be given, and let α_{ψ(x)} be the value of α at ψ(x) ∈ V, which is an alternating k-linear form on R^{m}. Remember that the value of ψ^{∗}(α) at x is the alternating k-linear form on R^{n} given by

(ψ^{∗}(α))_{x} = (dψ_{x})^{∗}(α_{ψ(x)}), (6.11.14)

as in Section 4.4. Suppose that

(6.11.15) dψ_{x} is not one-to-one

as a linear mapping from R^{n} into R^{m}. Under these conditions, we get that

(ψ^{∗}(α))_{x} = 0 (6.11.16)

as in (6.11.12).

6.12 Integrating n-forms on R^{n}

Let n be a positive integer, let E be a nonempty subset of R^{n}, and let α be a differential n-form on E. Remember that α may be expressed as

α = α_{0} dx_{1} ∧ ⋯ ∧ dx_{n}, (6.12.1)

where α_{0} is a real-valued function on E, as in Section 3.14.

Suppose that E is compact in R^{n}, and that α is continuous on E. This means that α_{0} is continuous as a real-valued function on E, as in Subsection 3.12.1. If E is reasonably nice, then the integral of α over E may be defined by

∫_{E} α = ∫_{E} α_{0}(x) dx, (6.12.2)

where the right side is an ordinary Riemann integral, as in (1) in Section 4.5 on p141 of [184]. In particular, we may do this when E is a cell in R^{n}, as in Subsection 5.13.1, or the standard simplex in R^{n}, as in Section 6.2. If one uses the Lebesgue integral, then the right side of (6.12.2) may be defned more broadly.

If β is a differential form on R^{n}, then the support

supp β (6.12.3)

6.12. INTEGRATING N-FORMS ON R^{N} 137

of β may be defined as in Subsection 5.13.2. This is the closure of the set of x ∈ R^{n} at which the value of β is not zero, as an element of AM(R^{n}). Equivalentlyf, if we identify AM(R^{n}) with R^{2^{n}}, as in Subsection 3.12.1, then the support of β is the same as the support of the corresponding function on R^{n} with values in R^{2^{n}}.

Let α be a differential n-form on R^{n} as in (6.12.1), and note that the support of α as a differential form on R^{n} is the same as the support of α_{0} as a real-valued function on R^{n}. If α is continuous, and the support of α is compact, then the integral of α over R^{n} may be defined by

∫_{R^{n}} α = ∫_{R^{n}} α_{0}(x) dx, (6.12.4)

where the right side is as in Subsection 5.13.2. If C is a cell in R^{n} that contains the support of α, then

∫_{R^{n}} α = ∫_{C} α = ∫_{C} α_{0}(x) dx, (6.12.5)

as before.

6.12.1 A family of integrals

Let β be a differential (n − 1)-form on R^{n}. We may express β as

β = ∑_{j=1}^{n} β_{j} dx_{1} ∧ ⋯ ∧ dx_{j−1} ∧ dx_{j+1} ∧ ⋯ ∧ dx_{n}, (6.12.6)

as in Section 3.14, where β_{j} is a real-valued function on R^{n}. Suppose that

(6.12.7) β is continuously differentiable

as a differential form on R^{n}, which means that β_{j} is continuously differentiable as a real-valued function on R^{n} for each j = 1, …, n, as in Section 3.13. Remember that the exterior derivative dβ of β is a continuous differential n-form on R^{n}, as in Section 4.6.

Observe that

dβ = ∑_{j=1}^{n} dβ_{j} ∧ dx_{1} ∧ ⋯ ∧ dx_{j−1} ∧ dx_{j+1} ∧ ⋯ ∧ dx_{n}, (6.12.8)

by the definition of the exterior derivative. Remember that

dβ_{j} = ∑_{l=1}^{n}(∂_{l}β_{j}) dx_{l} (6.12.9)

for each j, as in Section 3.14. Using this, we get that

dβ = ∑_{j=1}^{n}(∂_{j}β_{j}) dx_{j} ∧ dx_{1} ∧ ⋯ ∧ dx_{j−1} ∧ dx_{j+1} ∧ ⋯ ∧ dx_{n}, (6.12.10)

because dx_{l} ∧ dx_{l} = 0 for every l, as in Section 3.11. It follows that

dβ = ∑_{j=1}^{n}(−1)^{j−1} (∂_{j}β_{j}) dx_{1} ∧ ⋯ ∧ dx_{n}, (6.12.11)

because dx_{j} ∧ dx_{l} = −dx_{l} ∧ dx_{j} for each j, l, as before.

Suppose that (6.12.12) β also has compact support in R^{n},

which means that β_{j} has compact support for each j. This implies that ∂_{j}β_{j} has compact support for each j, so that dβ has compact support too. Observe that

(6.12.13)  ∫_{R^{n}} dβ = ∫_{R^{n}} ∑_{j=1}^{n}(−1)^{j−1}(∂_{j}β_{j})(x) dx

= ∑_{j=1}^{n}(−1)^{j−1} ∫_{R^{n}}(∂_{j}β_{j})(x) dx.

It is easy to see that

∫_{R^{n}}(∂_{j}β_{j})(x) dx = 0 (6.12.14)

for each j, by integrating in the jth variable first, and using the fundamental theorem of calculus. This means that

∫_{R^{n}} dβ = 0 (6.12.15)

under these conditions.

6.13 More on changes of variables

Let m and n be positive integers, let W be a nonempty open subset of R^{n}, and let ψ be a mapping from W into R^{m}. Suppose that

(6.13.1) ψ is differentiable on W,

in the sense that ψ is differentiable at every point in W. Let E be a nonempty subset of W, and let α be a differential k-form on a subset of R^{m} that contains ψ(E) for some positive integer k. Under these conditions, we can define the pull-back ψ^{∗}(α) (6.13.2)

of α with respect to ψ as a differential k-form on E, in essentially the same way as in Section 4.4. More precisely, if x ∈ E, and α_{ψ(x)} is the value of α at ψ(x), as an alternating k-linear form on R^{m}, then the value of ψ^{∗}(α) at x, as an alternating k-linear form on R^{n}, is defined by

(ψ^{∗}(α))_{x} = (dψ_{x})^{∗}(α_{ψ(x)}), (6.13.3)

6.13. MORE ON CHANGES OF VARIABLES 139 where the right side is as in Section 2.3.

Suppose now that k = m = n, and that α is as in (6.12.1) for some realvalued function α_{0} defined on a subset of R^{n} that contains ψ(E). In this case, ψ^{∗}(α) = (α_{0} ∘ ψ) ψ^{∗}(dx_{1} ∧ ⋯ ∧ dx_{n}) = (α_{0} ∘ ψ) dψ_{1} ∧ ⋯ ∧ dψ_{n} (6.13.4) on E, where the second step is as in Section 4.5. Remember that the components ψ_{1}, …, ψ_{n} of ψ are differentiable as real-valued functions on W.

The differential n-form dx_{1} ∧ ⋯ ∧ dx_{n} (6.13.5) is equal to θ_{1} ∧ ⋯ ∧ θ_{n} (6.13.6) at every point in R^{n}, as an alternating n-linear form on R^{n}, as in Section 3.14. Remember that μ_{det} is the alternating n-linear form on R^{n} corresponding to the determinant, as in Section 1.15, and that this is equal to (6.13.6), as in Section 2.11. It follows that ψ^{∗}(dx_{1} ∧ ⋯ ∧ dx_{n}) = (det dψ) dx_{1} ∧ ⋯ ∧ dx_{n} (6.13.7) on W, as in Section 2.5. This means that ψ^{∗}(α) = (α_{0} ∘ ψ) (det dψ) dx_{1} ∧ ⋯ ∧ dx_{n} (6.13.8) on E, by (6.13.4).

6.13.1 Integrals and diffeomorphisms

Suppose that ψ(W) is an open set in R^{n} too, and that (6.13.9) ψ is a C^{1} diffeomorphism from W onto ψ(W). Suppose also that E is compact as a subset of R^{n}, and that α is continuous on ψ(E), so that α_{0} is continuous as a real-valued function on ψ(E). This implies that α_{0} ∘ ψ is continuous on E, so that (6.13.10) ψ^{∗}(α) is continuous on E, because of (6.13.8). Remember that (6.13.11) det dψ is continuous as a real-valued function on W, because ψ is continuously differentiable on W, as in Section 6.3.

If we also have that det ψ′(x) = det dψ_{x} > 0 (6.13.12)

140 CHAPTER 6. INTEGRATION AND N-SURFACES for every x ∈ E, then ∫_{ψ(E)} α = ∫_{E} ψ^{∗}(α), (6.13.13) at least if E is nice enough for the Riemann integral to be defined. This follows from (6.13.8) and the change of variables formula in Section 6.3. Similarly, if det ψ′(x) = det dψ_{x} < 0 (6.13.14) for every x ∈ E, then ∫_{ψ(E)} α = − ∫_{E} ψ^{∗}(α), (6.13.15) if E is nice enought for the Riemann integral to be defined. This corresponds to (2) in Section 4.5 on p141 of [184]. As before, there are versions of this using Lebesgue measure and integration with milder hypotheses.

6.14 Diffeomorphisms and orientations

Let us continue with the same notation and hypotheses as in the previous section. Remember that (6.14.1) dψ_{x} is invertible as a linear mapping from R^{n} into itself for every x ∈ W, because ψ is a C^{1} diffeomorphism, as in Subsection 5.11.1. Equivalently, this means that det dψ_{x} ≠ 0 (6.14.2) for every x ∈ W. This implies that (6.14.3) sign(det dψ_{x}) is locally constant on W, as in Subsection 4.10.2, because of (6.13.11). If (6.14.4) W is connected, then it follows that (6.14.5) sign(det dψ_{x}) is constant on W, as before.

We say that (6.14.6) ψ is orientation-preserving on W if (6.13.12) holds for every x ∈ W. Similarly, (6.14.7) ψ is orientation-reversing on W if (6.13.14) holds for every x ∈ W.

6.14.1 Another version

The version of the change of variables formula in Section 6.4 may be restated in terms of differential forms as well. Let α be a differential n-form on R^{n} that is continuous and has compact support, with

supp α ⊆ ψ(W). (6.14.8)

Thus α may be expressed as in (6.12.1), where α_{0} is a continuous real-valued function on R^{n} with compact support, and

(6.14.9) supp α_{0} = supp α ⊆ ψ(W).

It follows that

(6.14.10) ψ^{−1}(supp α) = ψ^{−1}(supp α_{0}) is a compact set in R^{n},

because ψ^{−1} is continuous as a mapping from ψ(W) into R^{n}, as in Subsection 5.8.1. Note that (6.14.11) ψ^{−1}(supp α) = ψ^{−1}(supp α_{0}) ⊆ W,

because ψ^{−1}(ψ(W)) = W.

Let β_{0} be the real-valued function defined on R^{n} by

(6.14.12)  β_{0}(x) = α_{0}(ψ(x)) det dψ_{x} when x ∈ W
= 0  when x ∈ R^{n} \ W.

One can check that (6.14.13) β_{0} is continuous on R^{n},

as in Section 6.4. We also have that

(6.14.14) supp β_{0} = ψ^{−1}(supp α_{0}) = ψ^{−1}(supp α),

as before. Put β = β_{0} dx_{1} ∧ ⋯ ∧ dx_{n}, (6.14.15)

which is a differential n-form on R^{n} that is continuous, with

(6.14.16) supp β = supp β_{0}.

Note that β = ψ^{∗}(α) on W, (6.14.17)

by (6.13.8). It is easy to see that

∫_{R^{n}} α = ∫_{R^{n}} β (6.14.18)

when ψ is orientation-preserving on W, using the remarks in Section 6.4. Similarly,

∫_{R^{n}} α = − ∫_{R^{n}} β (6.14.19)

when ψ is orientation-reversing on W.

6.15 n-Surfaces in R^{m}

Let m and n be positive integers, let U be a nonempty open subset of R^{n}, and let ϕ be a continuously-differentiable mapping from U into R^{m}. Also let K be a nonempty compact subset of U, which is reasonably nice, so that one can take Riemann integrals of continuous real-valued functions on K. One may take K to be a cell in R^{n} that is contained in U, for instance, or the standard simplex in R^{n}, if that is contained in U. One could consider wider classes of sets K using Lebesgue measure and integration, as usual.

Let V be a nonempty open subset of R^{m}. If

ϕ(K) ⊆ V, (6.15.1)

then the restriction of ϕ to K is considered to be

(6.15.2) an n-surface in V,

as a mapping from K into V. This corresponds to Definition 10.10 on p254 of [155]. We may use Φ = Φ(K) to refer to this n-surface, which involves both K and the restriction of ϕ to K. The set K may be called the parameter domain of the n-surface, as in [155].

More precisely, we shall also use the differential of ϕ at points in K. If K is reasonably nice, then the differential of ϕ at points in K is uniquely determined by the restriction of ϕ to K. In particular, this holds when K is the closure of an open set in R^{n}.

If ϕ is r-times continuously differentiable on U for some positive integer r, then we may say that (6.15.3) Φ is a C^{r} n-surface in V.

Similarly, if ϕ is infinitely differentiable on U, then we may say that

(6.15.4) Φ is a C^{∞} n-surface in V.

6.15.1 Integration over n-surfaces

Let α be a differential n-form on ϕ(K) that is continuous on ϕ(K). The integral of α over the n-surface Φ is defined by

∫_{Φ} α = ∫_{K} ϕ^{∗}(α). (6.15.5)

More precisely, ϕ^{∗}(α) is a differential n-form on K that is continuous on K under these conditions, so that the right side may be defined as in Section 6.12. This basically corresponds to (35) on p254 of [155], although this version is closer to Theorem 10.24 on p264 of [155].

Remember that ϕ^{−1}(V) = {x ∈ U : ϕ(x) ∈ V } (6.15.6)

6.15. N-SURFACES IN R^{M} 143

is an open subset of R^{n}, because ϕ is continuous, as mentioned in Section 5.9. Note that K ⊆ ϕ^{−1}(V), (6.15.7)

by (6.15.1). Of course, ϕ(ϕ^{−1}(V)) ⊆ V (6.15.8)

automatically. If α is a differential form on V, then ϕ^{∗}(α) may be defined as a differential form on ϕ^{−1}(V) in the usual way.

6.15.2 Changing variables in R^{n}

Let U_{0} be another nonempty open subset of R^{n}, and let θ be a C^{1} diffeomorphism from U_{0} onto U. Put ϕ_{0} = ϕ ∘ θ, (6.15.9)

which is a continuously-differentiable mapping from U_{0} into R^{m}. Also put

K_{0} = θ^{−1}(K), (6.15.10)

so that K_{0} is a compact subset of R^{n} that is contained in U_{0}, and

ϕ_{0}(K_{0}) = ϕ(K) ⊆ V. (6.15.11)

Thus the restriction of ϕ_{0} to K_{0} is considered to be another n-surface in V, which may be expressed as Φ_{0} = Φ_{0}(K_{0}). More precisely, this also uses the differential of ϕ_{0} at points in K_{0}, as before.

Of course, ϕ_{0}^{∗}(α) = θ^{∗}(ϕ^{∗}(α)) (6.15.12)

on K_{0}, as in Subsection 4.5.1. This implies that

∫_{Φ_{0}} α = ∫_{K_{0}} ϕ_{0}^{∗}(α) = ∫_{K_{0}} θ^{∗}(ϕ^{∗}(α)). (6.15.13)

If θ is orientation-preserving on U_{0}, then it follows that

∫_{Φ_{0}} α = ∫_{K} ϕ^{∗}(α) = ∫_{Φ} α, (6.15.14)

where the first step is as in Subsection 6.13.1. This also uses the fact that

θ(K_{0}) = K, (6.15.15)

because θ(U_{0}) = U, by hypothesis. Similarly, if θ is orientation-reversing on U_{0}, then

∫_{Φ_{0}} α = − ∫_{K} ϕ^{∗}(α) = − ∫_{Φ} α. (6.15.16)

6.15.3 Mapping V into R^{p}

Let p be another positive integer, and let η be a continuously-differentiable mapping from V into R^{p}. Put

ψ = η ∘ ϕ, (6.15.17)

which is defined as a mapping from ϕ^{−1}(V) into R^{p}. In fact, ψ is continuously differentiable on ϕ^{−1}(V), as in Section 3.9.

Let W be an open subset of R^{p} such that

ψ(K) = η(ϕ(K)) ⊆ W. (6.15.18)

The restriction of ψ to K may be considered as an n-surface in W, as before. We may use Ψ = Ψ(K) to refer to this n-surface.

Let β be a differential n-form on ψ(K) that is continuous on ψ(K), so that ψ^{∗}(β) is a differential n-form on K that is continuous on K, as before. Similarly, η^{∗}(β) defines a differential n-form on ϕ(K) that is continuous on ϕ(K), and ϕ^{∗}(η^{∗}(β)) defines a differential n-form on K that is continuous on K. Note that

ψ^{∗}(β) = ϕ^{∗}(η^{∗}(β)) (6.15.19)

on K, as in Subsection 4.5.1. Under these conditions, we have that

∫_{Ψ} β = ∫_{K} ψ^{∗}(β) = ∫_{K} ϕ^{∗}(η^{∗}(β)) = ∫_{Φ} η^{∗}(β). (6.15.20)

This corresponds to Theorem 10.25 on p265 of [155].

Simplices and chains

7.1 Affine mappings

Let V and W be vector spaces over the real numbers. A mapping f from V into W is said to be affine if

(7.1.1) f − f(0) is a linear mapping from V into W,

as in p266 of [155]. Equivalently, this means that there is a linear mapping T from V into W such that f(v) = f(0) + T (v) (7.1.2)

for every v ∈ V. It is easy to see that the space of all affine mappings from V into W is a linear subspace of the space of all mappings from V into W, as a vector space over the real numbers with respect to pointwise addition and scalar multiplication of functions. The space L(V, W) of all linear mappings from V into W is a linear subspace of the space of all affine mappings from V into W.

Let Z be another vector space over the real numbers. If f is an affine mapping from V into W, and g is an affine mapping from W into Z, then one can check that

(7.1.3) g ∘ f is an affine mapping from V into Z.

Let f be an affine mapping from V into W, and let T be as in (7.1.2). Observe that (7.1.4) f is one-to-one on V

if and only if (7.1.5) T is one-to-one on V.

Similarly, f(V) = W (7.1.6)

if and only if T (V) = W. (7.1.7)

Let n be a positive integer, and let e_{1}, …, e_{n} be the standard basis vectors in R^{n}. If f is an affine mapping from R^{n} into W, then f is uniquely determined by f(0), f(e_{1}), …, f(e_{n}). (7.1.8)

More precisely, if T is the corresponding linear mapping from R^{n} into W as in (7.1.2), then T (e_{j}) = f(e_{j}) − f(0) (7.1.9)

for each j = 1, …, n, and this determines T uniquely. It is easy to see that (7.1.8) may be any n + 1 elements of W, because T (e_{1}), …, T (e_{n}) may be any n elements of W.

Let m be another positive integer, and let f be an affine mapping from R^{n} into R^{m}. If T is the linear mapping from R^{n} into R^{m} corresponding to f as in (7.1.2), then T is the same as the differential of f at every point in R^{n}.

7.2 Some remarks about determinants

Let n be a positive integer, and let σ ∈ Sym(n) be given. Also let T_{σ} be the linear mapping from R^{n} into itself defined by

(T_{σ}(x))_{j} = x_{σ(j)} (7.2.1)

for each x ∈ R^{n} and j = 1, …, n, as in Section 1.5. If e_{1}, …, e_{n} is the standard basis for R^{n}, then it is easy to see that

T_{σ}(e_{l}) = e_{σ^{−1}(l)} (7.2.2)

for each l = 1, …, n.

Remember that det T = μ_{det}(T (e_{1}), …, T (e_{n})) (7.2.3)

for any linear mapping T from R^{n} into itself, as in Section 2.5. Here μ_{det} is the alternating n-linear form on R^{n} associated to the determinant as in Section 1.15. Thus det T_{σ} = μ_{det}(e_{σ^{−1}(1)}, …, e_{σ^{−1}(n)}). (7.2.4)

One can use this to check that

det T_{σ} = sgn(σ). (7.2.5)

Let us put e_{0} = 0, (7.2.6)

for convenience. Let τ be a permutation on {0, 1, …, n}, and let R_{τ} be the linear mapping from R^{n} into itself such that

R_{τ} (e_{l}) = e_{τ(l)} − e_{τ(0)} (7.2.7)

7.2. SOME REMARKS ABOUT DETERMINANTS 147 for each l = 1, …, n. Note that sgn(τ) can be defined in the same way as in Section 1.4, which corresponds to identifying τ with an element of Sym(n + 1) in the obvious way.

Suppose for the moment that τ(0) = 0, (7.2.8) so that R_{τ} (e_{l}) = e_{τ(l)} (7.2.9) for each l = 1, …, n. Let τ_{0} be the restriction of τ to {1, …, n}, which is an element of Sym(n). Note that R_{τ} = T_{τ_{0}^{−1}} (7.2.10) in this case. Thus (7.2.11) det R_{τ} = det T_{τ_{0}^{−1}} = sgn(τ_{0}^{−1}) = sgn(τ_{0}), usng (7.2.5) in the second step. This implies that det R_{τ} = sgn(τ), (7.2.12) because sgn(τ) = sgn(τ_{0}), as in Section 2.7.

7.2.1 The case where τ(0) ≠ 0

Suppose now that τ(0) ≠ 0, (7.2.13) so that τ^{−1}(0) ≠ 0 too. Let ρ be the mapping from {1, …, n} into itself defined by

(7.2.14)  ρ(l) = τ(l) when l ≠ τ^{−1}(0)
= τ(0) when l = τ^{−1}(0).

It is easy to see that ρ ∈ Sym(n). Observe that

(7.2.15)  R_{τ} (e_{l}) = e_{ρ(l)} − e_{ρ(τ^{−1}(0))} when l ≠ τ^{−1}(0)
= −e_{τ(0)} = −e_{ρ(l)} when l = τ^{−1}(0).

Let L_{ρ} be the linear mapping from R^{n} into itself such that

(7.2.16)  L_{ρ}(e_{l}) = e_{ρ(l)} when l ≠ τ^{−1}(0)
= −e_{ρ(l)} when l = τ^{−1}(0).

One can check that det R_{τ} = det L_{ρ}. (7.2.17)

More precisely, one can verify that

μ_{det}(R_{τ} (e_{1}), …, R_{τ} (e_{n})) = μ_{det}(L_{ρ}(e_{1}), …, L_{ρ}(e_{n})). (7.2.18)

Let T_{ρ^{−1}} be the linear mapping on R^{n} associated to ρ^{−1} as at the beginning of the section. It is easy to see that

det L_{ρ} = − det T_{ρ^{−1}}, (7.2.19)

using (7.2.3) and the definition of L_{ρ}. This means that

det L_{ρ} = − sgn(ρ), (7.2.20)

because of (7.2.5). It follows that

det R_{τ} = − sgn(ρ), (7.2.21)

by (7.2.17).

Let ρ_{1} be the permutation on {0, 1, …, n} that is equal to ρ on {1, …, n}, and satisfies ρ_{1}(0) = 0. Thus

sgn(ρ_{1}) = sgn(ρ), (7.2.22)

as in Section 2.7 again. One can check that τ is the same as the composition of ρ_{1} with the permutation on {0, 1, …, n} that interchanges 0 and τ(0), and leaves the other points fixed. This implies that

sgn(τ) = − sgn(ρ_{1}). (7.2.23)

It follows that (7.2.24) det R_{τ} = − sgn(ρ_{1}) = sgn(τ),

so that (7.2.12) holds in this case as well.

7.3 Affine simplices

Let n be a positive integer, and let e_{1}, …, e_{n} be the standard basis in R^{n}, as usual. Remember that Q^{n} denotes the standard simplex in R^{n}, as in Section 6.2. This may be described equivalently as the set of points in R^{n} of the form

∑_{j=1}^{n} t_{j} e_{j}, (7.3.1)

where t_{j} is a nonnegative real number for each j, and ∑_{j=1}^{n} t_{j} ≤ 1, as on p266 of [155].

Let m be another positive integer, and let p_{0}, p_{1}, …, p_{n} be elements of R^{m}. These elements of R^{m} are not required to be distinct. Consider the affine mapping ϕ = ϕ_{p_{0},p_{1},...,p_{n}} (7.3.2) from R^{n} into R^{m} defined by

ϕ_{p_{0},p_{1},...,p_{n}}(x) = p_{0} + ∑_{j=1}^{n} x_{j} (p_{j} − p_{0}). (7.3.3)

This is the unique affine mapping from R^{n} into R^{m} such that

(7.3.4) ϕ(0) = p_{0} and ϕ(e_{j}) = p_{j} for each j = 1, …, n,

as in Section 7.1. Note that every affine mapping from R^{n} into R^{m} corresponds to elements p_{0}, p_{1}, …, p_{n} of R^{m} in this way.

Let Φ = Φ(p_{0}, p_{1}, …, p_{n}) = [p_{0}, p_{1}, …, p_{n}] (7.3.5)

be the n-surface in R^{m} associated to ϕ_{p_{0},p_{1},...,p_{n}} with parameter domain Q^{n}, as in Section 6.15. Let us call this the

(7.3.6) parameterized affine n-simplex in R^{m} associated to p_{0}, p_{1}, …, p_{n}.

This may also be called the oriented affine n-simplex in R^{n} associated to p_{0}, p_{1}, …, p_{n}, as on p266 of [155]. Indeed, one is often primarily concerned with the way that the affine n-simplex is oriented, rather than the particular affine parameterization, as in Section 7.5. The points p_{0}, p_{1}, …, p_{n} may be called the vertices of this affine n-simplex.

Of course,

ϕ_{p_{0},p_{1},...,p_{n}}(Q^{n}) = {p_{0} + ∑_{j=1}^{n} x_{j} (p_{j} − p_{0}) : x ∈ Q^{n}}. (7.3.7)

Equivalently,

ϕ_{p_{0},p_{1},...,p_{n}}(Q^{n}) = {(1 − ∑_{j=1}^{n} x_{j}) p_{0} + ∑_{j=1}^{n} x_{j} p_{j} : x ∈ Q^{n}}. (7.3.8)

If m = n, p_{0} = 0, and p_{j} = e_{j} for each j = 1, …, n, then ϕ_{p_{0},p_{1},...,p_{n}} is the identity mapping on R^{n}, so that ϕ_{p_{0},p_{1},...,p_{n}}(Q^{n}) = Q^{n}.

Alternatively, ϕ_{p_{0},p_{1},...,p_{n}}(Q^{n}) consists of the points in R^{m} of the form

∑_{j=0}^{n} t_{j} p_{j}, (7.3.9)

where t_{0}, t_{1}, …, t_{n} are nonnegative real numbers such that ∑_{j=0}^{n} t_{j} = 1. This description shows that

(7.3.10) ϕ_{p_{0},p_{1},...,p_{n}}(Q^{n}) does not depend on the order in which the vertices p_{0}, p_{1}, …, p_{n} are listed.

This will be discussed further in the next section.

7.4 Some affine mappings on R^{n}

Let V and W be vector spaces over the real numbers, and let f be an affine mapping from V into W, as in Section 7.1. Suppose that v_{1}, …, v_{k} are finitely many elements of V, and thet t_{1}, …, t_{k} are real numbers such that

∑_{j=1}^{k} t_{j} = 1. (7.4.1)

Under these conditions, one can check that

f( ∑_{j=1}^{k} t_{j} v_{j}) = ∑_{j=1}^{k} t_{j} f(v_{j}). (7.4.2)

In fact, one can verify that this property characterizes affine mappings, with k = 2.

Let n be a positive integer, let e_{1}, …, e_{n} be the standard basis in R^{n}, and let us put e_{0} = 0, as in Section 7.2. Note that Q^{n} is the set of points in R^{n} of the form

∑_{j=0}^{n} t_{j} e_{j}, (7.4.3)

where t_{0}, t_{1}, …, t_{n} are nonnegative real numbers with ∑_{j=0}^{n} t_{j} = 1. Let m be another positive integer, let p_{0}, p_{1}, …, p_{n} be elements of R^{m}, and let ϕ_{p_{0},p_{1},...,p_{n}} be as in the previous section. One can use (7.4.2) to get the description of the elements of ϕ_{p_{0},p_{1},...,p_{n}}(Q^{n}) in (7.3.9).

7.4.1 Some affine mappings from permutations

Let τ be a permutation on {0, 1, …, n}, and let R_{τ} be the linear mapping from R^{n} into itself such that R_{τ} (e_{l}) = e_{τ(l)} − e_{τ(0)} (7.4.4)

for each l = 1, …, n, as in Section 7.2. If x ∈ R^{n}, then put

ξ_{τ} (x) = e_{τ(0)} + R_{τ} (x), (7.4.5)

This is the unique affine mapping from R^{n} into itself such that

ξ_{τ} (e_{l}) = e_{τ(l)} (7.4.6)

for each l = 0, 1, …, n. This is the same as the identity mapping on R^{n} when τ is the identity mapping on {0, 1, …, n}, and

ξ_{τ} = ϕ_{e_{τ(0)},e_{τ(1)},...,e_{τ(n)}}, (7.4.7)

with m = n, for any permutation τ on {0, 1, …, n}. In particular,

ξ_{τ} (Q^{n}) = Q^{n} (7.4.8)

7.5. INTEGRALS OVER AFFINE SIMPLICES 151

for each τ, as in the preceding paragraph.

The differential of ξ_{τ} at any point in R^{n} is equal to R_{τ}, as in Section 7.1. Remember that the determinant of R_{τ} is equal to sgn(τ), as in Section 7.2. This means that (7.4.9) ξ_{τ} is orientation-preserving on R^{n} when τ is an even permutation, and that (7.4.10) ξ_{τ} is orientation-reversing on R^{n} when τ is an odd permutation, as in Section 6.14.

Observe that ϕ_{p_{0},p_{1},...,p_{n}}(ξ_{τ} (e_{l})) = ϕ_{p_{0},p_{1},...,p_{n}}(e_{τ(l)}) = p_{τ(l)} (7.4.11) for each l = 0, 1, …, n. This implies that ϕ_{p_{0},p_{1},...,p_{n}} ∘ ξ_{τ} = ϕ_{p_{τ(0)},p_{τ(1)},...,p_{τ(n)}}. (7.4.12) In particular, ϕ_{p_{τ(0)},p_{τ(1)},...,p_{τ(n)}}(Q^{n}) = ϕ_{p_{0},p_{1},...,p_{n}}(ξ_{τ} (Q^{n})) = ϕ_{p_{0},p_{1},...,p_{n}}(Q^{n}), (7.4.13) as mentioned in the previous section.

7.5 Integrals over affine simplices

Let us continue with the same notation and hypotheses as in the previous section. Let β be a differential n-form on Q^{n} that is continuous on Q^{n}, so that ξ_{τ}^{∗}(β) is another differential n-form on Q^{n} that is continuous on Q^{n}. Observe that ∫_{Q^{n}} ξ_{τ}^{∗}(β) = sgn(τ) ∫_{Q^{n}} β, (7.5.1) as in Subsection 6.13.1. More precisely, this uses the fact that the differential of ξ_{τ} at any point is equal to R_{τ}, whose determinant is equal to sgn(τ), as before.

Similarly, let α be a differential n-form on ϕ_{p_{0},p_{1},...,p_{n}}(Q^{n}) that is continuous on this set. Note that ϕ_{p_{τ(0)},p_{τ(1)},...,p_{τ(n)}}^{∗}(α) = ξ_{τ}^{∗}(ϕ_{p_{0},p_{1},...,p_{n}}^{∗}(α)) (7.5.2) on Q^{n}, because of (7.4.12). It follows that

∫_{Φ(p_{τ(0)},p_{τ(1)},...,p_{τ(n)})} α = ∫_{Q^{n}} ϕ_{p_{τ(0)},p_{τ(1)},...,p_{τ(n)}}^{∗}(α)
(7.5.3)  = ∫_{Q^{n}} ξ_{τ}^{∗}(ϕ_{p_{0},p_{1},...,p_{n}}^{∗}(α))
= sgn(τ) ∫_{Q^{n}} ϕ_{p_{0},p_{1},...,p_{n}}^{∗}(α) = sgn(τ) ∫_{Φ(p_{0},p_{1},...,p_{n})} α.

152 CHAPTER 7. SIMPLICES AND CHAINS This uses (7.5.1) in the third step. Here Φ(p_{0}, p_{1}, …, p_{n}) and Φ(p_{τ(0)}, p_{τ(1)}, …, p_{τ(n)}) = [p_{τ(0)}, p_{τ(1)}, …, p_{τ(n)}] (7.5.4) are as in Section 7.3, and the integrals over them are as in Subsection 6.15.1. Of course, (7.5.3) may be considered as a version of the remarks in Subsection 6.15.2. This corresponds to Theorem 10.27 on p267 of [155].

7.5.1 Orientations of affine simplices

Let us put Φ(p_{τ(0)}, p_{τ(1)}, …, p_{τ(n)}) ≃ sgn(τ) Φ(p_{0}, p_{1}, …, p_{n}), (7.5.5) or equivalently [p_{τ(0)}, p_{τ(1)}, …, p_{τ(n)}] ≃ sgn(τ) [p_{0}, p_{1}, …, p_{n}]. (7.5.6) If τ is an even permutation, then this means that Φ(p_{0}, p_{1}, …, p_{n}) and Φ(p_{τ(0)}, p_{τ(1)}, …, p_{τ(n)}) (7.5.7)

have the same orientation,

even though they may be parameterized differently as n-surfaces in R^{m}. Similarly, if τ is an odd permutation, then this means that Φ(p_{0}, p_{1}, …, p_{n}) and Φ(p_{τ(0)}, p_{τ(1)}, …, p_{τ(n)}) (7.5.8)

have opposite orientations.

This corresponds to some remarks on p267 of [155]. Of course, this notation is compatible with (7.5.3).

Remember that ϕ_{p_{0},p_{1},...,p_{n}} is one-to-one on R^{n} if and only if the corresponding linear mapping

ϕ_{p_{0},p_{1},...,p_{n}}(x) − p_{0} = ∑_{j=1}^{n} x_{j} (p_{j} − p_{0}) (7.5.9)

is a one-to-one linear mapping from R^{n} into R^{m}, as in Section 7.1. This happens if and only if (7.5.10) p_{j} − p_{0}, 1 ≤ j ≤ n, are linearly indpendent in R^{m}. If m = n, then we may say that Φ(p_{0}, p_{1}, …, p_{n}) is positively oriented when the determinant of (7.5.9) is positive, and negatively oriented when the determinant is negative, as on p267 of [155].

7.5.2 0-Simplices

An oriented affine 0-simplex in R^{m} is defined to be a point p_{0} ∈ R^{m} with a sign attached to indicate the orientation, i.e., +p_{0} or −p_{0}, as on p267 of [155]. This may be denoted Φ = ϵ p_{0}, (7.5.11)

with ϵ = ±1. If f is a differential 0-form defined at p_{0}, which is to say a real-valued function, then we put

∫_{Φ} f = ϵ f(p_{0}) (7.5.12)

in this case, as in [155].

Alternatively, (7.5.13) the standard 0-simplex Q^{0}

may be defined as the set {0} with 0 as its only element, as on p141 of [184]. A parameterized 0-simplex in R^{m} may be considered as a mapping from Q^{0} into R^{m}, as on p142, 191 of [184]. This is essentially the same as a point in R^{m}, without including a sign, as in the preceding paragraph. If p ∈ R^{m}, then we may use Φ(p) (7.5.14)

for the corresponding parameterized 0-simplex in R^{m}. The corresponding integral may be defined as in (7.5.12), without the extra factor of ϵ on the right side.

However, signs may also be included in chains, as in the next section. Using this, the difference mentioned in the previous paragraphs does not really matter.

7.6 Affine chains

Let m and n be positive integers, and let V be a nonempty open subset of R^{m}. An affine n-chain Γ in V consists of finitely many parameterized affine n-simplices Φ_{1}, …, Φ_{k} (7.6.1)

in V with signs ϵ_{1}, …, ϵ_{k} (7.6.2)

attached. More precisely, for each l = 1, …, k, Φ_{l} corresponds to an affine mapping ϕ_{l} from R^{n} into R^{m} such that

ϕ_{l}(Q^{n}) ⊆ V, (7.6.3)

as in Section 7.3, where Q^{n} is the standard simplex in R^{n}, as in Section 6.2. This basically corresponds to the definition in Section 10.28 on p268 of [155].

In [155], the signs ϵ_{l} = ±1 are not included directly in this way, and instead they correspond to using odd permutations of the vertices of the simplices. As in [155], the n-simplices are not required to be distinct, so that they may occur with multiplicity. We can also allow n = 0 here, using parameterized affine 0-simplices in R^{m}, as in Subsection 7.5.2. This is equivalent to using oriented affine 0-simplices in R^{m}, by including the signs directly here.

We shall consider “simplices” that are parameterized by mappings that are r-times continuously differentiable for some nonnegative integer r, or infinitely differentiable, in Section 7.9. We shall also consider corresponding notions of chains there. We shall say a bit more about some aspects of chains in Section 7.13.

7.6.1 Integrals over affine chains

Let α be a differential n-form on V that is continuous on V. If Γ is as before, then we put

∫_{Γ} α = ∑_{l=1}^{k} ϵ_{l} ∫_{Φ_{l}} α, (7.6.4)

as in (82) on p268 of [155]. Remember that the integrals on the right are defined as in Subsection 6.15.1 when n ≥ 1, and as in Subsection 7.5.2 when n = 0.

An affine n-chain Γ as before is often expressed as a formal sum

Γ = ∑_{l=1}^{k} ϵ_{l} Φ_{l}, (7.6.5)

as in (83), (84) on p268 of [155]. The commutativity of addition reflects the fact that we are not really concerned with the order in which the terms are listed, as in the right side of (7.6.4). Similarly, if c_{1}, …, c_{k} are integers, then the formal sum

∑_{l=1}^{k} c_{l} Φ_{l} (7.6.6)

may be considered as an affine n-chain in V. If c_{l} ≠ 0, then this means that sign(c_{l}) Φ_{l} occurs with multiplicity |c_{l}| in the chain.

Note that this type of formal sum is different from taking sums of affine mappings from R^{n} into R^{m} when n ≥ 1, as mentioned on p268 of [155]. Similarly, if n = 0, then this type of formal sum is different from taking sums of elements of R^{m}.

7.6.2 Defining ≃ for affine chains

Let Φ̃_{1}, …, Φ̃_{k} be parameterized affine n-simplices in V with signs ϵ̃_{1}, …, ϵ̃_{k}, so that

Γ̃ = ∑_{l=1}^{k} ϵ̃_{l} Φ̃_{l} (7.6.7)

7.7. BOUNDARIES OF AFFINE SIMPLICES 155

is an affine n-chain in V. Suppose that ϵ_{l} Φ_{l} ≃ ϵ̃_{l} Φ̃_{l} (7.6.8) for each l = 1, …, k, in the notation of Subsection 7.5.1. Under these conditions, we extend this notation by putting Γ ≃ Γ̃. (7.6.9)

If α is as before, then (7.6.8) implies that

ϵ_{l} ∫_{Φ_{l}} α = ϵ̃_{l} ∫_{Φ_{l}} α (7.6.10)

for each l = 1, …, k, as in Section 7.5. It follows that

∫_{Γ} α = ∫_{Γ} α, (7.6.11)

by summing over l, as in (7.6.4).

7.7 Boundaries of affine simplices

Let m and n be positive integers, and let p_{0}, p_{1}, …, p_{n} be elements of R^{m}, so that Φ(p_{0}, p_{1}, …, p_{n}) defines a parameterized affine n-simplex in R^{m}, as in Section 7.3. The boundary of Φ(p_{0}, p_{1}, …, p_{n}) is defined as an affine (n − 1)- chain in R^{m} by

∂Φ(p_{0}, p_{1}, …, p_{n}) = ∑_{l=0}^{n}(−1)^{l} Φ(p_{0}, …, p_{l−1}, p_{l+1} …, p_{n}), (7.7.1)

as in (85) on p269 of [155]. Note that (7.7.2) the image of Φ(p_{0}, …, p_{j−1}, p_{j+1}, …, p_{n})

is contained in the image of Φ(p_{0}, p_{1}, …, p_{n})

for each j, by a remark in Section 7.3. If n = 1, then we get that ∂Φ(p_{0}, p_{1}) = Φ(p_{1}) − Φ(p_{0}), (7.7.3) where Φ(p) is the parameterized 0-simplex corresponding to p ∈ R^{m} as in Subsection 7.5.2.

If n = 2, then ∂Φ(p_{0}, p_{1}, p_{2}) = Φ(p_{1}, p_{2}) − Φ(p_{0}, p_{2}) + Φ(p_{1}, p_{2}), (7.7.4) as on p269 of [155]. Note that Φ(p_{0}, p_{2}) ≃ −Φ(p_{2}, p_{0}), (7.7.5) in the notation of Subsection 7.5.1. We can extend this notation to affine nchains, as in Subsection 7.6.2, to get that

∂Φ(p_{0}, p_{1}, p_{2}) ≃ Φ(p_{0}, p_{1}) + Φ(p_{1}, p_{2}) + Φ(p_{2}, p_{0}), (7.7.6)

as in [155].

If n is any positive integer and 1 ≤ l ≤ n, then

Φ(p_{0}, …, p_{l−1}, p_{l+1}, …, p_{n}) = [p_{0}, …, p_{l−1}, p_{l+1}, …, p_{n}] (7.7.7)

is the parameterized affine (n − 1)-simplex in R^{m} associated to these vertices as in Section 7.3. If n ≥ 2, then this corresponds to the affine mapping

ϕ_{l} = ϕ_{p_{0},...,p_{l−1},p_{l+1},...,p_{n}} (7.7.8)

from R^{n−1} into R^{m} defined by

ϕ_{l}(x) = p_{0} + ∑_{j=1}^{l−1} x_{j} (p_{j} − p_{0}) + ∑_{j=l}^{n−1} x_{j} (p_{j+1} − p_{0}), (7.7.9)

as on p269 of [155]. Remember that the parameter domain of (7.7.7) is the standard simplex Q^{n−1} in R^{n−1}.

The l = 0 term on the right side of (7.7.1) uses the parameterized affine (n − 1)-simplex Φ(p_{1}, …, p_{n}) = [p_{1}, …, p_{n}] (7.7.10)

in R^{m}. If n ≥ 2, then this corresponds to the affine mapping

ϕ_{0} = ϕ_{p_{1},...,p_{n}} (7.7.11)

from R^{n−1} into R^{m} defined by

ϕ_{0}(x) = p_{1} + ∑_{j=1}^{n−1} x_{j} (p_{j+1} − p_{1}), (7.7.12)

as on p269 of [155].

7.7.1 Boundaries of affine chains

Let V be a nonempty open subset of R^{m}, and let Γ be an affine n-chain in V, as in (7.6.5). The boundary of Γ is defined as an affine (n − 1)-chain in V by

∂Γ = ∑_{l=1}^{k} ϵ_{l} ∂Φ_{l}. (7.7.13)

This corresponds to (90) on p270 of [155], in the case of affine chains. The boundary of a parameterized affine 0-simplex may be interpreted as being equal to 0, so that the boundary of an affine 0-chain is interpreted as being equal to 0 as well.

7.8 Boundaries and permutations

Let n be a positive integer, and let l be an integer with 0 ≤ l ≤ n. Put

E_{n,l} = {0, …, l − 1, l + 1, n}, (7.8.1)

which is a set with n elements. Let ρ_{n,l} be the unique mapping from

{0, 1, …, n − 1} (7.8.2)

onto E_{n,l} that is strictly increasing, so that

(7.8.3)  ρ_{n,l}(j) = j  when j ≤ l − 1
= j + 1 when j ≥ l.

Equivalently, ρ_{n,l}^{−1} is the unique mapping from E_{n,l} onto (7.8.2) that is strictly increasing, with

(7.8.4)  ρ_{n,l}^{−1}(k) = k  when k ≤ l − 1
= k − 1 when k ≥ l + 1.

If τ is a permutation on {0, 1, …, n}, then the restriction of τ to E_{n,l} defines a one-to-one mapping onto E_{n,τ(l)}. Put

τ_{l} = ρ_{n,τ(l)}^{−1} ∘ τ ∘ ρ_{n,l}, (7.8.5)

which defines a permutation on (7.8.2). We can define sgn(τ) and sgn(τ_{l}) as in Section 1.4, by identifying these permutations with elements of Sym(n + 1) and Sym(n), respectively, in the obvious way.

More precisely,

sgn(τ) = ∏_{0≤j<k≤n} sign(τ(k) − τ(j)), (7.8.6)

as before. Similarly,

sgn(τ_{l}) = ∏_{0≤p<q≤n−1} sign(τ_{l}(q) − τ_{l}(p)). (7.8.7)

It is easy to see that this is the same as saying that

sgn(τ_{l}) = ∏_{{0≤j<k≤n \atop j,k≠l}} sign(τ(j) − τ(k)) (7.8.8)

in this case. It follows that

(7.8.9) sgn(τ) = sgn(τ_{l}) ( ∏_{j=0}^{l−1} sign(τ(l) − τ(j))) ( ∏_{k=l+1}^{n} sign(τ(k) − τ(l))).

158 CHAPTER 7. SIMPLICES AND CHAINS

We would like to check that

(7.8.10) ( ∏_{j=0}^{l−1} sign(τ(l) − τ(j))) ( ∏_{k=l+1}^{n} sign(τ(k) − τ(l))) = (−1)^{τ(l)−l}.

Observe that

(7.8.11)  τ(l) = #{j = 0, …, l − 1 : τ(j) < τ(l)}
+#{k = l + 1, …, n : τ(k) < τ(l)}.

Of course,

(7.8.12)  l = #{j = 0, …, l − 1 : τ(j) < τ(l)}
+#{j = 0, …, l − 1 : τ(j) > τ(l)}.

This implies that

(7.8.13)  τ(l) − l = #{k = l + 1, …, n : τ(k) < τ(l)}
−#{j = 0, …, l − 1 : τ(j) > τ(l)}.

It is easy to use this to get (7.8.10). This means that sgn(τ_{l}) = (−1)^{τ(l)−l} sgn(τ). (7.8.14)

7.8.1 Permuting vertices of affine simplices

Let m be another positive integer, let p_{0}, p_{1}, …, p_{n} be elements of R^{m}, and let Φ(p_{0}, p_{1}, …, p_{n}) be as in Section 7.3. Remember that Φ(p_{τ(0)}, p_{τ(1)}, …, p_{τ(n)}) ≃ sgn(τ) Φ(p_{0}, p_{1}, …, p_{n}), (7.8.15) in the notation of Subsection 7.5.1. Similarly, one can verify that Φ(p_{τ(0)}, …, p_{τ(l−1)}, p_{τ(l+1)}, …, p_{τ(n)}) (7.8.16)

≃ sgn(τ_{l}) Φ(p_{0}, …, p_{τ(l)−1}, p_{τ(l)+1}, …, p_{n}),

as parameterized affine (n − 1)-simplices in R^{m}. Basically, p_{0}, …, p_{τ(l)−1}, p_{τ(l)+1}, …, p_{n} (7.8.17) and p_{τ(0)}, …, p_{τ(l−1)}, p_{τ(l+1)}, …, p_{τ(n)} (7.8.18) list the same set of vertices in R^{m}, where the order in the second list corresponds to the restriction of τ to E_{n,l}.

This implies that Φ(p_{τ(0)}, …, p_{τ(l−1)}, p_{τ(l+1)}, …, p_{τ(n)}) (7.8.19)

≃ (−1)^{τ(l)−l} sgn(τ) Φ(p_{0}, …, p_{τ(l)−1}, p_{τ(l)+1}, …, p_{n}),

7.9. C^{R} SIMPLICES AND CHAINS 159

by (7.8.14). It follows that

∂Φ(p_{τ(0)}, p_{τ(1)}, …, p_{τ(n)}) (7.8.20)

= ∑_{l=0}^{n}(−1)^{l} Φ(p_{τ(0)}, …, p_{τ(l−1)}, p_{τ(l+1)}, …, p_{n})

≃ ∑_{l=0}^{n}(−1)^{τ(l)} sgn(τ) Φ(p_{0}, …, p_{τ(l)−1}, p_{τ(l)+1}, …, p_{n})
= sgn(τ) ∂Φ(p_{0}, p_{1}, …, p_{n}),

by the definition (7.7.1) of the boundary.

Let V be a nonempty open subset of R^{m}, and let Γ, Γ̃ be affine n-chains in V, as in (7.6.5) and (7.6.7), respectively. If (7.6.8) holds, then we get that

∂Γ ≃ ∂Γ̃, (7.8.21)

because of (7.8.20).

7.9 C^{r} Simplices and chains

Let m, n, and r be positive integers, and let U, V be nonempty open subsets of R^{n}, R^{m}, respectively. Remember that Q^{n} is the standard simplex in R^{n}, as in Section 6.2, and suppose that Q^{n} ⊆ U. (7.9.1)

Let ϕ be an r-times continuously-differentiable mapping from U into R^{m} with

ϕ(Q^{n}) ⊆ V. (7.9.2)

This defines a C^{r} n-surface Φ in V with parameter domain Q^{n}, as in Section 6.15.

Under these conditions,

(7.9.3) Φ is considered as a parameterized C^{r} n-simplex in V.

One may also call Φ an oriented C^{r} n-simplex in V, which basically corresponds to the definition at the beginning of Section 10.30 on p269 of [155] when r = 2.

Similarly, if ϕ is infinitely differentiable on U, then

(7.9.4) Φ is considered as a parameterized C^{∞} n-simplex in V.

This corresponds to a definition in Section 4.6 on p141 in [184]. As before, Φ may be called an oriented C^{∞} n-simplex in V in this case as well. Of course, a parameterized affine n-simplex in V is a parameterized C^{∞} n-simplex in V in particular.

If r = 0, then one can simply ask that

(7.9.5) ϕ be a continuous mapping from Q^{n} into V.

This is mentioned on p142, 191 of [184], and is used in algebraic topology.

We may interpret a parameterized C^{r} or C^{∞} 0-simplex in V to be the same as a parameterized affine 0-simplex in V, as in Subsection 7.5.2. Similarly, an oriented C^{r} or C^{∞} 0-simplex may be interpreted as being the same as an oriented affine 0-simplex in V.

7.9.1 C^{r} chains

A C^{r} or C^{∞} n-chain Γ in V consists of finitely many parameterized C^{r} or C^{∞} n-simplices

Φ_{1}, …, Φ_{k} (7.9.6)

in V, respectively, with signs

ϵ_{1}, …, ϵ_{k} (7.9.7)

attached. This basically corresponds to the definition on p270 of [155] when r = 2. As in Section 7.6, the n-simplices are not required to be distinct. Note that an affine n-chain in V is a C^{∞} n-chain in V in particular. Some aspects of these notions will be discussed further in Section 7.13.

As before, Γ is often expressed as a formal sum

Γ = ∑_{l=1}^{k} ϵ_{l} Φ_{l}. (7.9.8)

Similarly, if c_{1}, …, c_{k} are integers, then the formal sum

∑_{l=1}^{k} c_{l} Φ_{l} (7.9.9)

may be considered as a C^{r} or C^{∞} n-chain in V, as appropriate. In an analogous definition on p142 of [184], one considers formal sums of parameterized C^{∞} nsimplices in V with coefficients in R. A related definition on p192 of [184] uses formal sums of continuous n-simplices in V with integer coefficients. We shall say more about this in Section 7.13 as well.

Let Γ be a C^{1} n-chain in V as in (7.9.8), and let α be a differential n-form on V that is continuous on V. Under these conditions, we put

∫_{Γ} α = ∑_{l=1}^{k} ϵ_{l} ∫_{Φ_{l}} α, (7.9.10)

where the integrals on the right are defined as in Subsection 6.15.1 when n ≥ 1, and as in Subsection 7.5.2 when n = 0. This corresponds to (87) on p270 of [155], and to (10) on p143 of [184].

7.10 Simplices and compositions

Let us continue with the same notation and hypotheses as in the previous section. If ϕ and U are as before, then

ϕ^{−1}(V) = {x ∈ U : ϕ(x) ∈ V } (7.10.1)

is an open set in R^{n}, because ϕ is continuous on U, as in Section 5.9. We also have that Q^{n} ⊆ ϕ^{−1}(V), (7.10.2)

because of (7.9.1) and (7.9.2). Note that

ϕ(ϕ^{−1}(V)) ⊆ V. (7.10.3)

Let q be another positive integer, and let η be a mapping from V into R^{q} that is r-times continuously differentiable or infinitely differentiable on V, as appropriate. Observe that ψ = η ∘ ϕ (7.10.4)

defines a mapping from ϕ^{−1}(V) into R^{q} that is r-times continuously differentiable or infinitely differentiable, as appropriate. Let W be a nonempty open subset of R^{q} with η(V) ⊆ W, (7.10.5)

so that ψ(Q^{n}) = η(ϕ(Q^{n})) ⊆ η(V) ⊆ W. (7.10.6)

Using ψ, we get a C^{r} or C^{∞} n-surface Ψ in W, as appropriate, with parameter domain Q^{n}, as in Section 6.15. This may be considered as a parameterized C^{r} or C^{∞} n-simplex in W, as appropriate.

Under these conditions, we may use the notation

Ψ = η ∘ Φ. (7.10.7)

Of course, if ϕ and η are both affine mappings, then

ψ is affine (7.10.8)

too, as in Section 7.1. Sometimes we may be concerned with cases where ϕ is affine and η may not be, which is related to some of the remarks in Section 10.30 starting on p269 of [155].

7.10.1 Chains and compositions

Suppose that Γ is a C^{r} or C^{∞} n-chain in V as in (7.9.8). Thus

Ψ_{l} = η ∘ Φ_{l} (7.10.9) is a parameterized C^{r} or C^{∞} n-simplex in W for each l = 1, …, k, as appropriate. Put

η ∘ Γ = ∑_{l=1}^{k} ϵ_{l} Ψ_{l}, (7.10.10)

which is a C^{r} or C^{∞} n-chain in W. This is related to some remarks on p270 of [155].

Let β be a differential n-form on W that is continuous on W, so that η^{∗}(β) is a differential n-form on V that is continuous on V. If Γ is a C^{1} n-chain in V, then ∫_{η∘Γ} β = ∫_{Γ} η^{∗}(β). (7.10.11)

Of course, both sides of the equation are defined as in (7.9.10). This follows from the analogous statement for integrals over n-surfaces in Subsection 6.15.3.

7.11 Boundaries of C^{r} simplices

Let n be a positive integer, let e_{1}, …, e_{n} be the standard basis in R^{n}, and put e_{0} = 0, as before. Consider the parameterized affine n-simplex

Φ(e_{0}, e_{1}, …, e_{n}) = [e_{0}, e_{1}, …, e_{n}] (7.11.1)

in R^{n} associated to e_{0}, e_{1}, …, e_{n} as in Section 7.3. The corresponding affine mapping ϕ_{e_{0},e_{1},...,e_{n}} (7.11.2)

from R^{n} into itself is the identity mapping on R^{n}. In particular, this maps the standard simplex Q^{n} in R^{n} onto itself.

We may consider (7.11.1) as the standard parameterized affine n-simplex in R^{n}. Its boundary

∂Φ(e_{0}, e_{1}, …, e_{n}) = ∑_{l=0}^{n}(−1)^{l} Φ(e_{0}, …, e_{l−1}, e_{l+1}, …, e_{n}) (7.11.3)

is an affine (n − 1)-chain in R^{n}, as in Section 7.7. This may be called

(7.11.4) the positively oriented boundary of Q^{n},

as in Section 10.31 on p270 of [155].

7.11.1 C^{r} Simplices as compositions

Let m be another positive integer, and let U, V be nonempty open subsets of R^{n}, R^{m}, respectively, with Q^{n} ⊆ U. Also let ϕ be a mapping from U into R^{m} such that ϕ(Q^{n}) ⊆ V, and suppose that ϕ is r-times continuously differentiable on U for some nonnegative integer r, or infinitely differentiable on U. This defines 7.11. BOUNDARIES OF C^{R} SIMPLICES 163

a parameterized C^{r} or C^{∞} n-simplex Φ in V, as appropriate, as in Section 7.9. Observe that Φ = ϕ ∘ Φ(e_{0}, e_{1}, …, e_{n}), (7.11.5)

using the notation in the previous section on the right side. More precisely, (7.11.2) corresponds to ϕ in the previous section, and ϕ here corresponds to η in the previous section.

Under these conditions, the boundary of Φ may be defined as a C^{r} or C^{∞} (n − 1)-chain in V, as appropriate, by

(7.11.6)  ∂Φ = ϕ ∘ (∂Φ(e_{0}, e_{1}, …, e_{n}))
= ∑_{l=0}^{n}(−1)^{l} ϕ ∘ Φ(e_{0}, …, e_{l−1}, e_{l+1}, …, e_{n}),

where the right side is as in the previous section. This basically corresponds to the definition of the boundary in (89) on p270 of [155]. One can check that this also corresponds to the definition of the boundary in (4) on p142 of [184]. To see this, let us look a bit more closely at (7.11.3).

If 1 ≤ l ≤ n, then the lth term on the right side of (7.11.3) uses the parameterized affine (n − 1)-simplex

Φ(e_{0}, …, e_{l−1}, e_{l+1}, …, e_{n}) = [e_{0}, …, e_{l−1}, e_{l+1}, …, e_{n}] (7.11.7)

in R^{n}. If n ≥ 2, then this corresponds to the affine mapping

ϕ_{l} = ϕ_{e_{0},...,e_{l−1},e_{l+1},...,e_{n}} (7.11.8)

from R^{n−1} into R^{n} defined by

(7.11.9)  ϕ_{l}(x) = e_{0} + ∑_{j=1}^{l−1} x_{j} (e_{j} − e_{0}) + ∑_{j=l}^{n−1} x_{j} (e_{j+1} − e_{0})

= ∑_{j=1}^{l−1} x_{j} e_{j} + ∑_{j=l}^{n−1} x_{j} e_{j+1},

where the first step is as in (7.7.9). Similarly, if l = 0, then the right side of (7.11.3) uses the parameterized affine (n − 1)-simplex

Φ(e_{1}, …, e_{n}) = [e_{1}, …, e_{n}] (7.11.10)

in R^{n}. If n ≥ 2, then this corresponds to the affine mapping

ϕ_{0} = ϕ_{e_{1},...,e_{n}} (7.11.11)

from R^{n−1} into R^{n} defined by

ϕ_{0}(x) = e_{1} + ∑_{j=1}^{n−1} x_{j} (e_{j+1} − e_{1}), (7.11.12)

as in (7.7.12). One can use (7.11.9) and (7.11.12) to verify that (7.11.6) corresponds to the definition of the boundary in (4) on p142 of [184].

If ϕ is an affine mapping from R^{n} into R^{m}, then Φ is a parameterized affine n-simplex in R^{m}. In this case, one can check that (7.11.6) is equivalent to the definition of the boundary of Φ as an affine (n − 1)-chain in R^{m}, as in Section 7.7. This corresponds to a remark on p270 of [155].

7.11.2 Boundaries of C^{r} chains

If Γ = ∑_{l=1}^{k} ϵ_{l} Φ_{l} is a C^{r} or C^{∞} n-chain in V, then the boundary of Γ may be defined as a C^{r} or C^{∞} (n − 1)-chain in V, as appropriate, by

∂Γ = ∑_{l=1}^{k} ϵ_{l} ∂Φ_{l}, (7.11.13)

as in Subsection 7.7.1. This corresponds to (90) on p270 of [155], and to a remark on p142 of [184].

7.12 Compositions and affine simplices

Let m and n be positive integers, and let p_{0}, p_{1}, …, p_{n} be elements of R^{m}. Using this, we get an affine mapping ϕ_{p_{0},p_{1},...,p_{n}} from R^{n} into R^{m} as in Section 7.3, and the corresponding parameterized affine n-simplex Φ(p_{0}, p_{1}, …, p_{n}) in R^{m}. Let V be a nonempty open subset of R^{m}, and suppose that

ϕ_{p_{0},p_{1},...,p_{n}}(Q^{n}) ⊆ V, (7.12.1)

so that Φ(p_{0}, p_{1}, …, p_{n}) may be considered as a parameterized affine n-simplex in V.

Let q be another positive integer, and let η be a mapping from V into R^{q} that is r-times continuously differentiable for some nonnegative integer r, or infinitely differentiable. Thus

ψ = η ∘ ϕ_{p_{0},p_{1},...,p_{n}} (7.12.2)

defines a mapping from ϕ_{p_{0},p_{1},...,p_{n}}^{−1}(V) (7.12.3)

into R^{q} that is r-times continuously differentiable or infinitely differentiable, as appropriate. Let W be a nonempty open subset of R^{q} with η(V) ⊆ W, so that

ψ(Q^{n}) = η(ϕ_{p_{0},p_{1},...,p_{n}}(Q^{n})) ⊆ η(V) ⊆ W. (7.12.4)

Using ψ, we get a parameterized C^{r} or C^{∞} n-simplex

Ψ = η ∘ Φ(p_{0}, p_{1}, …, p_{n}) (7.12.5)

in W, as appropriate, as in Section 7.10.

7.13. SOME REMARKS ABOUT N-CHAINS 165

7.12.1 Defining ≃ for C^{r} simplices

Let τ be a permutation on {0, 1, …, n}, and remember that ϕ_{p_{τ(0)},p_{τ(1)},...,p_{τ(n)}}(Q^{n}) = ϕ_{p_{0},p_{1},...,p_{n}}(Q^{n}), (7.12.6) as in Sections 7.3 and 7.4. In particular, ϕ_{p_{τ(0)},p_{τ(1)},...,p_{τ(n)}}(Q^{n}) ⊆ V, (7.12.7) so that η ∘ Φ(p_{τ(0)}, p_{τ(1)}, …, p_{τ(n)}) (7.12.8) may be considered as a parameterized C^{r} or C^{∞} n-simplex in W, as appropriate, as before. Under these conditions, let us put η ∘ Φ(p_{τ(0)}, p_{τ(1)}, …, p_{τ(n)}) ≃ sgn(τ) (η ∘ Φ(p_{0}, p_{1}, …, p_{n})). (7.12.9) This extends the notation in Subsection 7.5.1 from parameterized affine nsimplices to parameterized C^{r} and C^{∞} n-simplices. If η is an affine mapping from R^{m} into R^{q}, then (7.12.5) and (7.12.8) are parameterized affine n-simplices in R^{q}, and (7.12.9) is equivalent to the analogous notation in Subsection 7.5.1.

7.12.2 Defining ≃ for C^{r} chains

Suppose that

Γ = ∑_{l=1}^{k} ϵ_{l} Φ_{l}, Γ̃ = ∑_{l=1}^{k} ϵ̃_{l} Φ̃_{l} (7.12.10)

are C^{r} or C^{∞} n-chains in W, as in Subsection 7.9.1. If ϵ_{l} Φ_{l} ≃ ϵ̃_{l} Φ̃_{l} (7.12.11) for each l = 1, …, k, then we put Γ ≃ Γ̃, (7.12.12) as in Subsection 7.6.2. In this case, one can check that ∂Γ ≃ ∂Γ̃, (7.12.13) as in Subsection 7.8.1.

7.13 Some remarks about n-chains

Let m and n be positive integers, and let V be a nonempty open subset of R^{m}. If r is a nonnegative integer, then (7.13.1) the space of C^{r} n-chains in V may be defined as the free abelian group generated by the parameterized C^{r} nsimplices in V. This is often used when r = 0 in algebraic topology, as on p192 of [184]. Similarly, the spaces of affine and C^{∞} n-chains in V may be defined as the free abelian groups generated by the parameterized affine and C^{∞} nsimplices in V, respectively. Although the details of this will not be discussed here, we would like to mention some related aspects of this in this section.

One could also consider finite linear combinations of parameterized affine, C^{r}, or C^{∞} n-chains in V with coefficients in the real numbers, as on p142 of [184], and as mentioned in Subsection 7.9.1. The spaces of these finite linear combinations may be defined as the free vector spaces over the real numbers generated by the parameterized affine, C^{r}, or C^{∞} n-simplices, as appropriate. We shall not get into the details of this here either, aside from some related aspects in this section.

7.13.1 Linear functionals on differential forms

Let Φ be an n-surface in V, as in Section 6.15. If α is a differential n-form on V that is continuous on V, then the integral of α over Φ may be defined as a real number as in Subsection 6.15.1. Thus

α ↦ ∫_{Φ} α (7.13.2)

defines a real-valued function on the space

C(V, AM_{n}(R^{m})) (7.13.3)

of all differential n-forms on V that are continuous on V, as mentioned in Section 10.28 on p268 of [155]. More precisely, (7.13.4) defines a linear functional on (7.13.3), which is to say a linear mapping from (7.13.3) into R.

Similarly, if Γ is a C^{1} n-chain in V, then the integral of α over Γ may be defined as a real number as in Subsection 7.9.1. This means that

α ↦ ∫_{Γ} α (7.13.4)

defines a real-valued function on (7.13.3) too, which is a linear functional as well. By construction, this is a linear combination of linear functionals on (7.13.3) associated to finitely many parameterized C^{1} n-simplices in V as in (7.13.2).

7.13.2 Defining ≅ for C^{1} n-chains

If Γ_{1}, Γ_{2} are C^{1} n-chains in V, then put

Γ_{1} ≅ Γ_{2} (7.13.5)

when ∫_{Γ_{1}} α = ∫_{Γ_{2}} α (7.13.6) for every differential n-form α on V that is continuous on V. This is the same as saying that the functions on (7.13.3) associated to Γ_{1} and Γ_{2} as in (7.13.4) are the same.

One could simply look at a C^{1} n-chain in V as a type of linear functional on (7.13.3), as in Section 10.28 on p268 of [155]. If the space of C^{1} n-chains in V is considered as a commutative group with respect to addition, then we get a homomorphism from this group into the space of linear functionals on (7.13.3), as another commutative group with respect to addition. If the space of C^{1} n-chains in V is defined using coefficients in R, then we get a linear map from that space into the space of linear functionals on (7.13.3).

7.13.3 Comparing ≃ and ≅

Let Φ, Φ̃ be parameterized C^{1} n-simplices in V such that

ϵ Φ ≃ ϵ̃Φ̃ (7.13.7)

for some ϵ, ϵ̃ = ±1, as in Subsection 7.12.1. Under these conditions, one can check that ϵ Φ ≅ ϵ̃Φ̃. (7.13.8)

This follows from a remark in Section 7.5 when Φ and Φ̃ are parameterized affine n-simplices in V. One can reduce to that case as in Subsection 6.15.3. If Γ_{1}, Γ_{2} are C^{1} n-chains in V that satisfy (7.12.12), then it follows that (7.13.5) holds as well.

7.14 An associativity property

Let m and n be positive integers again, and let U, V be nonempty open subsets of R^{n}, R^{m}, respectively, with Q^{n} ⊆ U. Also let ϕ be a mapping from U into R^{m} such that ϕ(Q^{n}) ⊆ V and ϕ is r-times continuously differentiable on U for some nonnegative integer r, or infinitely differentiable on U, so that we get a parameterized C^{r} or C^{∞} n-simplex Φ in V, as appropriate, as in Section 7.9. Let q be another positive integer, let η be a mapping from V into R^{q} that is rtimes continuously differentiable or infintely differentiable, as appropriate, and let W be a nonempty open subset of R^{q} with η(V) ⊆ W. Thus ψ = η ∘ϕ defines a mapping from ϕ^{−1}(V) into R^{q} that is r-times continuously differentiable or infinitely differentiable, as appropriate, which leads to a parameterized C^{r} or C^{∞} n-simplex Ψ = η ∘ Φ in W, as in Section 7.10.

Similarly, let q_{1} be a positive integer, let η_{1} be a mapping from W into R^{q_{1}} that is r-times continuously differentiable or infinitely differentiable, as appropriate, and let W_{1} be a nonempty open subset of R^{q_{1}} with

η_{1}(W) ⊆ W_{1}. (7.14.1)

Under these conditions, ψ_{1} = η_{1} ∘ ψ (7.14.2)

168 CHAPTER 7. SIMPLICES AND CHAINS defines a mapping from ϕ^{−1}(V) into R^{q_{1}} that is r-times continuously differentiable or infinitely differentiable, as appropriate, with ψ_{1}(Q^{n}) = η_{1}(ψ(Q^{n})) ⊆ η_{1}(W) ⊆ W_{1}. (7.14.3) As before, we can use ψ_{1} to get a parameterized C^{r} or C^{∞} n-simplex Ψ_{1} in W_{1}, as appropriate, which may be expressed as Ψ_{1} = η_{1} ∘ Ψ. (7.14.4)

Observe that ζ = η_{1} ∘ η (7.14.5) defines a mapping from V into R^{q_{1}} that is r-times continuously differentiable or infinitely differentiable, as appropriate, with ζ(V) ⊆ W_{1}. (7.14.6) This means that ζ ∘ ϕ (7.14.7) defines a mapping from ϕ^{−1}(V) into R^{q_{1}} that is r-times continuously differentiable or infinitely differentiable, as appropriate. This determines a parameterized C^{r} or C^{∞} n-simplex ζ ∘ Φ in W_{1}, as in Section 7.10 again. Of course, ζ ∘ ϕ = η_{1} ∘ η ∘ ϕ = η_{1} ∘ ψ = ψ_{1}, (7.14.8) so that ζ ∘ Φ = Ψ_{1}. (7.14.9)

7.14.1 Associativity and chains

Let Γ be a C^{r} or C^{∞} n-chain in V, so that η ∘ Γ is a C^{r} or C^{∞} n-chain in W, as appropriate, as in Subsection 7.10.1. Similarly η_{1} ∘ (η ∘ Γ) and ζ ∘ Γ are C^{r} or C^{∞} n-chains in W_{1}, as appropriate. It is easy to see that η_{1} ∘ (η ∘ Γ) = (η_{1} ∘ η) ∘ Γ, (7.14.10) as in (7.14.9).

7.15 Compositions and boundaries

Let us return to the notation and hypotheses at the beginning of the previous section. In particular, Ψ = η ∘ Φ is a parameterized C^{r} or C^{∞} n-simplex in W, as appropriate. Also let e_{1}, …, e_{n} be the standard basis in R^{n}, and put e_{0} = 0, as before. Remember that ∂Φ = ϕ ∘ (∂Φ(e_{0}, e_{1}, …, e_{n})), (7.15.1) as in (7.11.6). Similarly,

∂Ψ = ψ ∘ (∂Φ(e_{0}, e_{1}, …, e_{n})). (7.15.2)

This means that

(7.15.3)  ∂(η ∘ Φ) = (η ∘ ϕ) ∘ (∂Φ(e_{0}, e_{1}, …, e_{n}))
= η ∘ (ϕ ∘ (∂Φ(e_{0}, e_{1}, …, e_{n}))) = η ∘ (∂Φ),

where the second step is as in (7.14.10). This is related to some of the remarks on p270 of [155].

If Γ is a C^{r} or C^{∞} n-chain in V, then η ∘ Γ is a C^{r} or C^{∞} n-chain in W, as appropriate, as in Subsection 7.10.1. Similarly, η ∘ (∂Γ) is a C^{r} or C^{∞} (n − 1)-chain in W, as appropriate. Under these conditions, it is easy to see that ∂(η ∘ Γ) = η ∘ (∂Γ), (7.15.4)

using (7.15.3).

7.15.1 Boundaries of boundaries

If p_{0}, p_{1}, …, p_{n} are elements of R^{m}, then one can check that

∂(∂Φ(p_{0}, p_{1}, …, p_{n})) = 0, (7.15.5)

as in Exercise 16 on p291 of [155]. More precisely, this is trivial when n ≤ 1, because the boundary of a 0-simplex or 0-chain is interpreted as being equal to 0, as in Section 7.7. If ϕ and Φ are as before, then we get that

(7.15.6)  ∂(∂Φ) = ∂(ϕ ∘ (∂Φ(e_{0}, e_{1}, …, e_{n})))
= ϕ ∘ (∂(∂Φ(e_{0}, e_{1}, …, e_{n}))) = 0,

using (7.11.6) in the first step, and (7.15.3) in the second step. This implies that ∂(∂Γ) = 0 (7.15.7)

for every C^{r} or C^{∞} n-chain Γ in R^{m}. This corresponds to the second part of Exercise 16 on p291 of [155], and to (7) on p143 of [184].

Stokes’ theorem and other matters

8.1 Stokes’ theorem

Let m and n be positive integers, and let V be a nonempty open set in R^{m}. Also let (8.1.1) ω be a continuously-differentiable differential (n − 1)-form on V, so that the exterior derivative dω of ω is a differential n-form on V that is continuous on V, as in Section 4.6. If (8.1.2) Γ is a C^{2} n-chain in V, then Stokes’ theorem states that ∫_{∂Γ} ω = ∫_{Γ} dω. (8.1.3) This corresponds to Theorem 10.33 on p272 of [155], and Theorem 4.7 on p144 of [184]. More precisely, if n = 1, then it suffices to ask that (8.1.4) Γ be a C^{1} 1-chain in V, instead of (8.1.2).

In particular, if ∂Γ = 0, (8.1.5) then ∫_{Γ} dω = 0, (8.1.6) by (8.1.4). Similarly, let Γ_{1}, Γ_{2} be C^{2} n-chains in V, or C^{1} 1-chains in V. If ∂Γ_{1} = ∂Γ_{2}, (8.1.7) then ∫_{Γ_{1}} dω = ∫_{Γ_{2}} dω. (8.1.8)

This corresponds to Remark (b) on p276 of [155].

Suppose now that ω is a closed (n − 1)-form on V, so that

dω = 0 (8.1.9)

on V. In this case, (8.1.3) reduces to

∫_{∂Γ} ω = 0. (8.1.10)

This corresponds to Remark (c) on p276 of [155].

8.1.1 Stokes’ theorem and ≅

Let Γ_{1}, Γ_{2} be C^{2} n-chains in V again, or C^{1} 1-chains in V. Suppose that Γ_{1} ≅ Γ_{2}, in the sense of Subsection 7.13.2. If ω is as in (8.1.1), then we have that (8.1.8) holds, by hypothesis. Under these conditions, Stokes’ theorem implies that

∫_{∂Γ_{1}} ω = ∫_{∂Γ_{2}} ω. (8.1.11)

One can use this to get that

∂Γ_{1} ≅ ∂Γ_{2}, (8.1.12)

in the sense of Subsection 7.13.2 again. More precisely, (8.1.12) was defined to mean that (8.1.11) holds for all differential (n−1)-forms ω on V that are continuous on V, and not just when ω is continuously differentiable on V. Sometimes this type of property is defined using smooth differential forms on V, for which (8.1.11) could be used directly. One can also approximate continuous differential forms on V, uniformly on compact subsets of V, by differential forms that are continuously differentiable, or even smooth, using partitions of unity, for instance. Using this, one can get (8.1.11) when ω is continuous from the analogous statement when ω is continuously differentiable.

8.2 More on Stokes’ theorem

Let us continue with the same notation and hypotheses as at the beginning of the previous section. Let U be a open subset of R^{n} with Q^{n} ⊆ U, and let ϕ be a mapping from U into R^{m} such that ϕ(Q^{n}) ⊆ V and

(8.2.1) ϕ is twice continuously differentible on U,

so that we get (8.2.2) a parameterized C^{2} n-simplex Φ in V, as in Section 7.9. In this case, Stokes’ theorem says that

∫_{∂Φ} ω = ∫_{Φ} dω, (8.2.3)

as in (92) on p273 of [155], and (2) on p144 of [184]. If n = 1, then it suffices to ask that (8.2.4) ϕ be continuously differentiable on U,

as before. It is easy to see that this version of Stokes’ theorem implies the previous one for n-chains in V, as in [155, 184].

Let e_{1}, …, e_{n} be the standard basis in R^{n}, put e_{0} = 0, and let

Φ_{0} = Φ(e_{0}, e_{1}, …, e_{n}) (8.2.5)

be the standard parameterized affine n-simplex in R^{n}. Thus

Φ = ϕ ∘ Φ_{0} (8.2.6)

and ∂Φ = ϕ ∘ (∂Φ_{0}), (8.2.7)

as in Subsection 7.11.1. Remember that

U_{0} = ϕ^{−1}(V) (8.2.8)

is an open set in R^{n} that contains Q^{n}, as in Section 7.10.

Of course, ϕ^{∗}(ω) is a differential (n − 1)-form on ϕ^{−1}(V), as in Subsection 4.4.1. More precisely,

(8.2.9) ϕ^{∗}(ω) is continuously differentiable on U_{0},

as in Subsection 4.5.2, because ω is continuously differentiable on V, and ϕ is twice continuously differentiable on U, by hypothesis. If n = 1, then it suffices to ask that ϕ be continuously differentiable on U, as before.

Remember that ∫_{Φ} dω = ∫_{Φ_{0}} ϕ^{∗}(dω), (8.2.10)

as in Subsection 6.15.3. We also have that

ϕ^{∗}(dω) = d(ϕ^{∗}(ω)) (8.2.11)

on ϕ^{−1}(V), as in Section 4.9. Thus

∫_{Φ} dω = ∫_{Φ_{0}} d(ϕ^{∗}(ω)). (8.2.12)

Similarly,

∫_{∂Φ} ω = ∫_{∂Φ_{0}} ϕ^{∗}(ω), (8.2.13)

8.3. STOKES’ THEOREM FOR Φ_{0} 173

because of (8.2.7). If n = 1, then this involves integrals over parameterized 0-simplices, as in Subsection 7.5.2.

This means that (8.2.3) is the same as saying that

∫_{∂Φ_{0}} ϕ^{∗}(ω) = ∫_{Φ_{0}} d(ϕ^{∗}(ω)). (8.2.14)

Thus we can reduce to the case where Φ = Φ_{0}. This corresponds to (94) on p273 of [155], and to (3) on p144 of [184]. If n = 1, then this is the same as the fundamental theorem of calculus, as in [155, 184].

8.3 Stokes’ theorem for Φ_{0}

Let n ≥ 2 be an integer, let e_{1}, …, e_{n} be the standard basis for R^{n}, put e_{0} = 0, and let Φ_{0} be the standard parameterized n-simplex in R^{n}, as in (8.2.5). Also let V be an open set in R^{n} with

Q^{n} ⊆ V, (8.3.1)

and let ω be a continuously-differentiable differential (n − 1)-form on V. We would like to show that ∫_{∂Φ_{0}} ω = ∫_{Φ_{0}} dω, (8.3.2)

which corresponds to (8.2.3) with m = n and Φ = Φ_{0}. This will imply Stokes’ theorem, as in the previous section.

Let k be an integer with 1 ≤ k ≤ n, and suppose that ω is of the form

ω = ω_{k}(x) dx_{1} ∧ ⋯ ∧ dx_{k−1} ∧ dx_{k+1} ∧ ⋯ ∧ dx_{n}, (8.3.3)

where

(8.3.4) ω_{k}(x) is a continuously-differentiable real-valued function on V.

Note that every continuously-differentiable differential (n−1)-form on V can be expressed a sum of differential forms of this type. Thus it suffices to show that (8.3.2) holds in this case, because of the linearity of the integrals on both sides of the equation. This corresponds to (96) on p273 of [155], and to a remark shortly after (5) on p144 of [184].

8.3.1 The integral over ∂Φ_{0}

Remember that

∂Φ_{0} = ∑_{l=0}^{n}(−1)^{l} Φ(e_{0}, …, e_{l−1}, e_{l+1}, …, e_{n}), (8.3.5)

as in Section 7.11. The parameterized affine (n − 1)-simplex

Φ(e_{0}, …, e_{l−1}, e_{l+1}, …, e_{n}) (8.3.6)

174 CHAPTER 8. STOKES’ THEOREM AND OTHER MATTERS

in R^{n} corresponds to the affine mapping ϕ_{l} = ϕ_{e_{0},...,e_{l−1},e_{l+1},...,e_{n}} (8.3.7) from R^{n−1} into R^{n} for each l = 0, 1, …, n, as in Section 7.3. We have seen that

ϕ_{0}(y) = e_{1} + ∑_{j=1}^{n−1} y_{j} (e_{j+1} − e_{1}) (8.3.8)

for every y ∈ R^{n−1}, as in Subsection 7.11.1. Similarly, if 1 ≤ l ≤ n, then

ϕ_{l}(y) = ∑_{j=1}^{l−1} y_{j} e_{j} + ∑_{j=l}^{n−1} y_{j} e_{j+1} (8.3.9)

for every y ∈ R^{n−1}, as before.

Observe that

∫_{∂Φ_{0}} ω = ∑_{l=0}^{n}(−1)^{l} ∫_{Φ(e_{0},...,e_{l−1},e_{l+1},...,e_{n})} ω = ∑_{l=0}^{n}(−1)^{l} ∫_{Q^{n−1}} ϕ_{l}^{∗}(ω), (8.3.10)

where the first step is as in Subsection 7.6.1, and the second step is as in Subsection 6.15.1. If l ≥ 1, then it is easy to see that ϕ_{l}^{∗}(dx_{l}) = 0, (8.3.11) because the lth component of ϕ_{l} is equal to 0, as in (8.3.9). This implies that ϕ_{l}^{∗}(ω) = 0 (8.3.12) when l ≠ 0, k and ω is as in (8.3.3). Thus (8.3.10) reduces to ∫_{∂Φ_{0}} ω = ∫_{Q^{n−1}} ϕ_{0}^{∗}(ω) + (−1)^{k} ∫_{Q^{n−1}} ϕ_{k}^{∗}(ω) (8.3.13)

in this case. This basically corresponds to the first line in (102) on p274 of [155], and to part of (9) on p145 of [184].

8.3.2 The integral over Φ_{0}

If ω is as in (8.3.3), then dω = dω_{k} ∧ dx_{1} ∧ ⋯ ∧ dx_{k−1} ∧ dx_{k+1} ∧ ⋯ ∧ dx_{n}, (8.3.14) as in Section 4.6. This implies that dω = (∂_{k}ω_{k}) dx_{k} ∧ dx_{1} ∧ ⋯ ∧ dx_{k−1} ∧ dx_{k+1} ∧ ⋯ ∧ dx_{n}. (8.3.15) It follows that dω = (−1)^{k−1} (∂_{k}ω_{k}) dx_{1} ∧ ⋯ ∧ dx_{n}. (8.3.16)

8.3. STOKES’ THEOREM FOR Φ_{0} 175 This corresponds to remarks on p275 of [155], and on p144 of [184].

Using (8.3.16), we get that ∫_{Φ_{0}} dω = (−1)^{k−1} ∫_{Q^{n}}(∂_{k}ω_{k})(x) dx, (8.3.17) as in Section 6.12. If x ∈ Q^{n}, then (x_{1}, …, x_{k−1}, x_{k+1}, …, x_{n}) (8.3.18) is an element of Q^{n−1}, as in Section 6.2. Let an element of Q^{n−1} be given as in (8.3.18), and consider

∫_{0}^{1−x_{1}−···−x_{k−1}−x_{k+1}−···−x_{n}}(∂_{k}ω_{k})(x_{1}, …, x_{k−1}, x_{k}, x_{k+1}, …, x_{n}) dx_{k}. (8.3.19) This is equal to the sum of ω_{k}(x_{1}, …, x_{k−1}, 1 − x_{1} − ⋯ − x_{k−1} − x_{k+1} − ⋯ − x_{n}, x_{k+1}, …, x_{n}) (8.3.20) and −ω_{k}(x_{1}, …, x_{k−1}, 0, x_{k+1}, …, x_{n}), (8.3.21) by the fundamental theorem of calculus.

Observe that ∫_{Q^{n}}(∂_{k}ω_{k})(x) dx (8.3.22) is equal to the integral of (8.3.19), as a function of (8.3.18), over Q^{n−1}. This is related to some of the remarks in Section 6.2. Thus (8.3.22) is equal to the sum of the integrals of (8.3.20) and (8.3.21) over Q^{n−1}, as in the preceding paragraph.

8.3.3 Matching a pair of terms

If ω is as in (8.3.3), then it is easy to see that ϕ_{k}^{∗}(ω) = (ω_{k} ∘ ϕ_{k}) dy_{1} ∧ ⋯ ∧ dy_{n−1}, (8.3.23) using (8.3.9), with l = k. This means that (−1)^{k} ∫_{Q^{n−1}} ϕ_{k}^{∗}(ω) (8.3.24)

= (−1)^{k} ∫_{Q^{n−1}} ω_{k}(y_{1}, …, y_{k−1}, 0, y_{k}, …, y_{n−1}) dy,

as in Section 6.12. Of course, this is the same as (−1)^{k−1} times the integral of (8.3.21) over Q^{n−1}.

8.3.4 Matching the other two terms

If y ∈ R^{n−1}, then

ϕ_{0}(y) = (1 − ∑_{j=1}^{n−1} y_{j}) e_{1} + ∑_{j=1}^{n−1} y_{j} e_{j+1}, (8.3.25)

as in (8.3.8). This implies that

ϕ_{0}^{∗}(dx_{1}) = − ∑_{j=1}^{n−1} dy_{j} (8.3.26)

and ϕ_{0}^{∗}(dx_{l}) = dy_{l−1} (8.3.27)

for l = 2, …, n. If ω is as in (8.3.3) and k = 1, then we get that

ϕ_{0}^{∗}(ω) = (ω_{1} ∘ ϕ_{0}) dy_{1} ∧ ⋯ ∧ dy_{n−1}. (8.3.28)

If k ≥ 2, then we obtain that

ϕ_{0}^{∗}(ω) =

−(ω_{k} ∘ ϕ_{0}) ( ∑_{j=1}^{n−1} dy_{j}) ∧ dy_{1} ∧ ⋯ ∧ dy_{k−2} ∧ dy_{k} ∧ ⋯ ∧ dy_{n−1}. (8.3.29)

This means that

ϕ_{0}^{∗}(ω) = −(ω_{k} ∘ ϕ_{0}) dy_{k−1} ∧ dy_{1} ∧ ⋯ ∧ dy_{k−2} ∧ dy_{k} ∧ ⋯ ∧ dy_{n−1}, (8.3.30)

because all but one of the terms in the sum on the right side of (8.3.29) leads to a wedge product that is equal to 0. Thus

ϕ_{o}^{∗}(ω) = (−1)^{k−1} (ω_{k} ∘ ϕ_{0}) dy_{1} ∧ ⋯ ∧ dy_{n−1}, (8.3.31)

by rearranging the wedge product on the right side of (8.3.30) appropriately. Note that this is the same as (8.3.28) when k = 1.

It follows that

∫_{Q^{n−1}} ϕ_{0}^{∗}(ω) = (−1)^{k−1} ∫_{Q^{n−1}} ω_{k}(ϕ_{0}(y)) dy. (8.3.32)

If k = 1, then this corresponds exactly to (−1)^{k−1} times the integral of (8.3.20) over Q^{n−1}. If k ≥ 2, then we basically need an additional change of variables, as on p274 of [155], and p145 of [184]. More precisely, one can use a change of variables as in (10) on p145 of [184] to get that (8.3.32) is equal to (−1)^{k−1} times the integral of (8.3.20) over Q^{n−1}. This is related to some of the remarks in Subsection 7.4.1.

On p274 of [155], one considers a permutation of the standard basis vectors so that the argument is like the k = 1 case before. This will be discussed further in the next section.

8.4. A RELATED ARGUMENT 177

8.4 A related argument

Let us continue with the same notation and hypotheses as in the previous section. Remember that ∫_{Q^{n−1}} ϕ_{0}^{∗}(ω) = ∫_{Φ(e_{1},...,e_{n})} ω, (8.4.1) as in Subsection 6.15.1, which corresponds to the l = 0 term in the second step in (8.3.10), and in the right side of (8.3.13). Let τ be the permutation on {1, …, n} defined by

(8.4.2) τ(l) = k when l = 1
= l − 1 when l = 2, …, k
= l  when l = k + 1, …, n.

One can check that sgn(τ) = (−1)^{k−1}. (8.4.3)

As in Subsection 7.5.1, we have that Φ(e_{τ(1)}, …, e_{τ(n)}) ≃ sgn(τ) Φ(e_{1}, …, e_{n}). (8.4.4) More precisely, the n in Subsection 7.5.1 should be taken to be n − 1 here, so that {0, 1, …, n} before should be taken to be {0, 1, …, n − 1} now. However, here we are basically using {1, …, n} instead. Equivalently, we get that Φ(e_{k}, e_{1}, …, e_{k−1}, e_{k+1}, …, e_{n}) ≃ (−1)^{k−1} Φ(e_{1}, …, e_{n}). (8.4.5) This corresponds to a remark on p274 of [155].

We also have that ∫_{Φ(e_{k},e_{1},...,e_{k−1},e_{k+1},...,e_{n})} ω = (−1)^{k−1} ∫_{Φ(e_{1},...,e_{n})} ω, (8.4.6)

as in Section 7.5. Remember that the parameterized affine (n − 1)-simplex Φ(e_{k}, e_{1}, …, e_{k−1}, e_{k+1}, …, e_{n}) (8.4.7) corresponds to the affine mapping ψ_{k} = ϕ_{e_{k},e_{1},...,e_{k−1},e_{k+1},...,e_{n}} (8.4.8) from R^{n−1} into R^{n} as in Section 7.3. In this case, we have that

ψ_{k}(y) = e_{k} + ∑_{j=1}^{k−1} y_{j} (e_{j} − e_{k}) + ∑_{j=k}^{n−1} y_{j} (e_{j+1} − e_{k}) (8.4.9)

for every y ∈ R^{n−1}, as in Section 7.7. This means that

ψ_{k}(y) = (1 − ∑_{j=1}^{n−1} y_{j}) e_{k} + ∑_{j=1}^{k−1} y_{j} e_{j} + ∑_{k=k}^{n−1} y_{j} e_{j+1} (8.4.10)

for every y ∈ R^{n−1}.

As in Subsection 6.15.1,

∫_{Φ(e_{k},e_{1},...,e_{k−1},e_{k+1},...,e_{n})} ω = ∫_{Q^{n−1}} ψ_{k}^{∗}(ω). (8.4.11)

Observe that

(8.4.12)  ψ_{k}^{∗}(dx_{l}) = dy_{l}  when l = 1, …, k − 1
= − ∑_{j=1}^{n−1} dy_{j} when l = k
= dy_{l−1}  when l = k + 1, …, n.

If ω is as in (8.3.3), then we get that

ψ_{k}^{∗}(ω) = (ω_{k} ∘ ψ_{k}) dy_{1} ∧ ⋯ ∧ dy_{n−1}. (8.4.13)

This means that ∫_{Q^{n−1}} ψ_{k}^{∗}(ω) = ∫_{Q^{n−1}} ω_{k}(ψ_{k}(y)) dy, (8.4.14)

as in Section 6.12. This corresponds exactly to the integral of (8.3.20) over Q^{n−1}.

As before, we get that (8.3.32) corresponds to (−1)^{k−1} times the integral of (8.3.20) over Q^{n−1}, because of the factor of (−1)^{k−1} on the right side of (8.4.6). This shows that (8.3.13) is equal to (8.3.17), as desired.

8.5 Stokes’ theorem on cells

Let n be a positive integer, let a_{1}, …, a_{n} and b_{1}, …, b_{n} be real numbers with a_{j} < b_{j} for each j = 1, …, n, and let C = ∏_{j=1}^{n}[a_{j}, b_{j}] be the corresponding cell in R^{n}, as in Section 5.13. Also let U be an open set in R^{n} with

C ⊆ U, (8.5.1)

and let β be a continuously-differentiable differential (n − 1)-form on U. Thus β may be expressed as

β = ∑_{j=1}^{n} β_{j} dx_{1} ∧ ⋯ ∧ dx_{j−1} ∧ dx_{j+1} ∧ ⋯ ∧ dx_{n} (8.5.2)

on U, as in Section 3.14, where β_{j} is a continuously-differentiable real-valued function on U for each j. The exterior derivative of β can be expressed as

dβ = ∑_{j=1}^{n}(−1)^{j−1} (∂_{j}β_{j}) dx_{1} ∧ ⋯ ∧ dx_{n}, (8.5.3)

as in Subsection 6.12.1.

This means that

(8.5.4)  ∫_{C} dβ = ∫_{C} ∑_{j=1}^{n}(−1)^{j−1} (∂_{j}β_{j})(x) dx

= ∑_{j=1}^{n}(−1)^{j−1} ∫_{C}(∂_{j}β_{j})(x) dx.

In the jth term on the right side, we can integrate in the jth variable first, and use the fundamental theorem of calculus. Let us express the result in terms of integrals of β over suitable (n − 1)-surfaces in R^{n}. We suppose from now on in this section that n ≥ 2, since otherwise one can simply use the fundamental theorem of calculus directly.

If 1 ≤ l ≤ n, then put

C_{l} = [a_{1}, b_{1}] × ⋯ × [a_{l−1}, b_{l−1}] × [a_{l+1}, b_{l+1}] × ⋯ × [a_{n}, b_{n}], (8.5.5)

which is a cell in R^{n−1}. Consider the mappings ϕ_{l} and ψ_{l} from R^{n−1} into R^{n} defined by ϕ_{l}(w) = (w_{1}, …, w_{l−1}, a_{l}, w_{l}, …, w_{n−1}) (8.5.6)

and ψ_{l}(w) = (w_{1}, …, w_{l−1}, b_{l}, w_{l}, …, w_{n−1}) (8.5.7)

for each w = (w_{1}, …, w_{n−1}) ∈ R^{n−1}. Using ϕ_{l} and ψ_{l}, we get (n − 1)-surfaces

Φ_{l} = Φ_{l}(C_{l}) and Ψ_{l} = Ψ_{l}(C_{l}) (8.5.8)

in R^{n} with parameter domain C_{l}, as in Section 6.15. More precisely, these are (n − 1)-surfaces in U, because

ϕ_{l}(C_{l}), ψ_{l}(C_{l}) ⊆ C ⊆ U. (8.5.9)

Observe that

∫_{C}(∂_{l}β_{l})(x) dx = ∫_{C_{l}} β_{l}(ψ_{l}(w)) dw − ∫_{C_{l}} β_{l}(ϕ_{l}(w)) dw, (8.5.10)

by the fundamental theorem of calculus. We also have that ϕ_{l}^{−1}(U), ψ_{l}^{−1}(U) are open subsets of R^{n−1}, with

C_{l} ⊆ ϕ_{l}^{−1}(U), ψ_{l}^{−1}(U), (8.5.11) because of (8.5.9).

It is easy to see that

ϕ_{l}^{∗}(β) = (β_{l} ∘ ϕ_{l}) dw_{1} ∧ ⋯ ∧ dw_{n−1} (8.5.12)

on ϕ_{l}^{−1}(U). This uses the fact that ϕ_{l}^{∗}(dx_{l}) = 0, so that only the j = l term in sum on the right side of (8.5.2) can have a nonzero pull-back with respect to ϕ_{l}. Similarly, ψ_{l}^{∗}(β) = (β_{l} ∘ ψ_{l}) dw_{1} ∧ ⋯ ∧ dw_{n−1} (8.5.13)

on ψ_{l}^{−1}(U).

It follows that ∫_{C}(∂β_{l})(x) dx = ∫_{Ψ_{l}} β − ∫_{Φ_{l}} β. (8.5.14)

More precisely, the two integrals on the right side are defined as in Subsection 6.15.1, with n replaced with n − 1.

Combining this with (8.5.4), we get that

∫_{C} dβ = ∑_{l=1}^{n}(−1)^{l−1} ( ∫_{Ψ_{l}} β − ∫_{Φ_{l}} β). (8.5.15)

This is a version of Stokes’ theorem for cells in R^{n}. This version could be obtained from the previous one, by expressing integrals over cells as integrals over affine chains.

8.6 More on vector fields

Let n and r be positive integers, and let U be a nonempty open subset of R^{n}. Also let ξ be an (r−1)-times continuously-differentiable vector field on U, which is to say an (r − 1)-times continuously-differentiable function on U with values in R^{n}, as in Section 4.3. Thus

X_{ξ}(f) = ∑_{j=1}^{n} ξ_{j} \frac{∂f}{∂x_{j}} (8.6.1)

defines a linear mapping from C^{r}(U, R) into C^{r−1}(U, R), as before. If ξ is infinitely differentiable on U, then this defines a linear mapping from C^{∞}(U, R) into itself.

Note that ξ ↦ X_{ξ} (8.6.2)

defines a linear mapping from C^{r−1}(U, R^{n}) into the space

L(C^{r}(U, R), C^{r−1}(U, R)) (8.6.3)

of linear mappings from C^{r}(U, R) into C^{r−1}(U, R). Similarly, (8.6.2) defines a linear mapping from C^{∞}(U, R^{n}) into the space

L(C^{∞}(U, R)) (8.6.4) of linear mappings from C^{∞}(U, R) into itself. If f_{l}(x) = x_{l} on U for l = 1, …, n, then X_{ξ}(f_{l}) = ξ_{l}. (8.6.5)

In particular, this means that (8.6.2) is injective as a mapping from C^{r−1}(U, R^{n}) into (8.6.3), and from C^{∞}(U, R^{n}) into (8.6.4).

8.6.1 Commutators and Lie brackets

Suppose that r ≥ 2, and let η be another (r−1)-times continuously-differentiable vector field on U. Note that X_{ξ} and X_{η} define linear mappings from C^{r}(U, R) into C^{r−1}(U, R), and from C^{r−1}(U, R) into C^{r−2}(U, R). This implies that the compositions X_{ξ} ∘ X_{η} and X_{η} ∘ X_{ξ} (8.6.6)

of X_{ξ} and X_{η}, in either order, define linear maps from C^{r}(U, R) into C^{r−2}(U, R). Thus [X_{ξ}, X_{η}] = X_{ξ} ∘ X_{η} − X_{η} ∘ X_{ξ} (8.6.7)

defines a linear mapping from C^{r}(U, R) into C^{r−2}(U, R) as well. One can check that there is a unique (r − 2)-times continuously-differentiable vector field ζ on U such that [X_{ξ}, X_{η}] = X_{ζ} (8.6.8)

on C^{r}(U, R).

We may use the notation ζ = [ξ, η] (8.6.9)

in this case. This may be called the Lie bracket of ξ and η, as in Definition 1.44 on p36 of [184]. This defines a bilinear mapping from

C^{r−1}(U, R^{n}) × C^{r−1}(U, R^{n}) (8.6.10)

into C^{r−2}(U, R^{n}). It is easy to see that this mapping is antisymmetric, in the sense that [ξ, η] = −[η, ξ]. (8.6.11)

Similarly if ξ and η are infinitely differentiable on U, then X_{ξ} ∘X_{η} and X_{η} ∘X_{ξ} define linear mappings from C^{∞}(U, R) into itself, so that their commutator (8.6.7) defines a linear mapping from C^{∞}(U, R) into itself too. As before, there is a unique smooth vector field ζ on U such that (8.6.8) holds on C^{∞}(U, R). This may also be denoted ζ = [ξ, η]_{C^{∞}(U,R^{n})} (8.6.12)

under these conditions. This defines an antisymmetric bilinear mapping from C^{∞}(U, R^{n}) × C^{∞}(U, R^{n}) into C^{∞}(U, R^{n}).

182 CHAPTER 8. STOKES’ THEOREM AND OTHER MATTERS

8.7 Some more compositions

Let n and r be positive integers again, and let U be a nonempty open subset of R^{n}. Suppose now that r ≥ 3, and put V_{k} = C^{r−k}(U, R^{n}) (8.7.1) for k = 0, 1, 2, 3. Thus V_{3} ⊆ V_{2} ⊆ V_{1} ⊆ V_{0}, (8.7.2) as in Section B.3. If α is an (r − 1)-times continuously-differentiable vector field on U, then X_{α} defines linear mappings from C^{r−k}(U, R) into C^{r−k−1}(U, R) for k = 0, 1, 2. This means that X_{α} ∈ L_{3,2}(V_{1}, V_{0}), (8.7.3) in the notation of Section B.3.

If β is another (r − 1)-times continuously-differentiable vector field on U, then X_{α} ∘ X_{β} ∈ L_{3}(V_{2}, V_{0}), (8.7.4) in the notation of Section B.3. Thus [X_{α}, X_{β}] ∈ L_{3}(V_{2}, V_{0}). (8.7.5)

If γ is an (r −1)-times continuously-differentiable vector field on U too, then X_{α} ∘ X_{β} ∘ X_{γ} ∈ L(V_{3}, V_{0}). (8.7.6) Similarly, ([X_{α}, X_{β}]) ∘ X_{γ}, X_{γ} ∘ ([X_{α}, X_{β}]) ∈ L(V_{3}, V_{0}). (8.7.7) Thus [[X_{α}, X_{β}], X_{γ}] = ([X_{α}, X_{β}]) ∘ X_{γ} − X_{γ} ∘ ([X_{α}, X_{β}]) ∈ L(V_{3}, V_{0}). (8.7.8) In fact, [[X_{α}, X_{β}], X_{γ}] + [[X_{β}, X_{γ}], X_{α}] + [[X_{γ}, X_{α}], X_{β}] = 0 (8.7.9) in L(V_{3}, V_{0}), as in Section B.3. This is a version of the Jacobi identity.

8.7.1 The Jacobi identity for brackets

Note that [α, β], [β, γ], and [γ, α] (8.7.10) may be defined as (r − 2)-times continuously-differentiable vector fields on U as in (8.6.9). Similarly, [[α, β], γ] (8.7.11) and analogous expressions may be defined as vector fields on U that are (r − 3)- times continuously differentiable. One can verify that

[[α, β], γ] + [[β, γ], α] + [[γ, α], β] = 0, (8.7.12)

which is essentially the same as (8.7.9).

If α, β, and γ are infinitely differentiable on U, then X_{α}, X_{β}, and X_{γ} define linear mappings from C^{∞}(U, R) into itself, so that their various compositions define linear mappings from C^{∞}(U, R) into itself too. In this case, the analogue of (8.7.9) on C^{∞}(U, R) is a bit simpler, as mentioned at the beginning of Section B.3. This implies that

(8.7.13) C^{∞}(U, R^{n}) is a Lie algebra

over the real numbers with respect to (8.6.12), as in Section B.1. This corresponds to a remark after Proposition 1.45 on p36 of [184], and to example 3.5 (a) on p84 of [184].

8.8 Lie derivatives of functions and vector fields

Let n and r be positive integers, and let U be a nonempty open subset of R^{n}. If ξ and η are r-times continuously-differentiable vector fields on U, then

L_{ξ}(η) = [ξ, η] (8.8.1)

is an (r −1)-times continuously-differentiable vector field on U, as in Subsection 8.6.1. This may be called the Lie derivative of η with respect to ξ. More precisely, this type of Lie derivative is discussed in Definition 2.24 on p69 of [184] for smooth vector fields. The fact that the Lie derivative may be expressed as in (8.8.1) corresponds to part (b) of Theorem 2.25 on p70 of [184].

Similarly, if ξ is an (r −1)-times continuously-differentiable vector field on U and f is an r-times continuously-differentiable real-valued function on U, then

L_{ξ}(f) = X_{ξ}(f) (8.8.2)

is an (r − 1)-times continuously-differentiable real-valued function on U, where X_{ξ} is as in Section 8.6. This may be called the Lie derivative of f with respect to ξ. This type of Lie derivative may be considered as a particular case of those discussed on the top of p70 of [184] for smooth vector fields. The fact that the Lie derivative may be expressed in this way corresponds to part (a) of Theorem 2.25 of [184].

Let r be a nonnegative integer, let a be an r-times continuously-differentiable real-valued function on U, and let ξ be an r-times continuously-differentiable vector field on U. We can define a ξ as an r-times continuously-differentiable vector field on U, using scalar multiplication on R^{n}, so that

(a ξ)(x) = a(x) ξ(x) (8.8.3) for every x ∈ U. If f is an (r + 1)-times continuously-differentiable real-valued function on U, then X_{a ξ}(f) = a X_{ξ}(f) (8.8.4)

on U.

Suppose that r ≥ 1 again, let a, ξ, and η be as before, and let b be another r-times continuously-differentiable real-valued function on U. One can check that [a ξ, b η] = a b [ξ, η] + a X_{ξ}(b) η − b X_{η}(a) ξ. (8.8.5)

This corresponds to part (b) of Proposition 1.45 on p36 of [184]. Equivalently, this means that

L_{a ξ}(b η) = a b L_{ξ}(η) + a L_{ξ}(b) η − b L_{η}(a) ξ. (8.8.6)

In particular, if a ≡ 1 on U, then

L_{ξ}(b η) = L_{ξ}(b) η + b L_{ξ}(η). (8.8.7)

Suppose now that r ≥ 2, let ξ and η be as before, and let η̃ be another r-times continuously-differentiable vector field on U. Under these conditions, one can check that

L_{ξ}([η, η̃]) = [L_{ξ}(η), η̃] + [η, L_{ξ}(η̃)], (8.8.8)

using the Jacobi identity (8.7.12). If ξ is infinitely differentiable on U, then L_{ξ} is a derivation on the Lie algebra of smooth vector fields on U, as in Section B.4.

8.9 ϕ-Related vector fields

Let n and m be positive integers, let U be a nonempty open subset of R^{n}, and let V be a nonempty open subset of R^{m}. Also let ϕ be a differentiable mapping from U into V, let α be a vector field on U, and let ξ be a vector field on V. This means that α is an R^{n}-valued function on U, and ξ is an R^{m}-valued function on V. We say that (8.9.1) α and ξ are ϕ-related

if dϕ_{x}(α(x)) = ξ(ϕ(x)) (8.9.2)

for every x ∈ U. This corresponds to Definition 1.54 on p41 of [184].

Let f be a differentiable real-valued function on V, so that f ∘ ϕ is differentiable on U, as in Section 3.9. Let us use

X_{α} (8.9.3)

for the first-order differential operator associated to α on U as in Subsection 4.3.1 and Section 8.6, and let us use

Y_{ξ} (8.9.4)

8.9. ϕ-RELATED VECTOR FIELDS 185 for the analogous differential operator associated to ξ on V. One can check that X_{α}(f ∘ ϕ) = (Y_{ξ}(f)) ∘ ϕ (8.9.5) on U, using (8.9.2) and the chain rule. Conversely, one can get (8.9.2) from this property, using f_{l}(y) = y_{l} on V for l = 1, …, m.

8.9.1 Brackets of ϕ-related vector fields

Suppose now that ϕ is a twice continuously-differentiable mapping from U into V, α and β are continuously-differentiable vector fields on U, and ξ and η are continuously-differentiable vector fields on V. Note that [α, β] and [ξ, η] are continuous vector fields on U and V, respectively, as in Subsection 8.6.1. Suppose that (8.9.1) holds, and that (8.9.6) β and η are ϕ-related too. We would like to show that (8.9.7) [α, β] and [ξ, η] are ϕ-related under these conditions. This corresponds to Proposition 1.55 on p41 of [184].

Let f be a twice continuously-differentiable real-valued function on V. This implies that f ∘ϕ is twice continuously differentiable on U, as in Subsection 3.9.1, so that X_{α}(f ∘ ϕ) and X_{β}(f ∘ ϕ) are continuously differentiable on U. Similarly, Y_{ξ}(f) and Y_{η}(f) are continuously differentiable on V, and their compositions with ϕ are continuously differentiable on U. Observe that X_{[α,β]}(f ∘ ϕ) = ([X_{α}, X_{β}])(f ∘ ϕ) = X_{α}(X_{β}(f ∘ ϕ)) − X_{β}(X_{α}(f ∘ ϕ)) (8.9.8) on U, as in Subsection 8.6.1. Using (8.9.1) and (8.9.6), we get that the right side is equal to X_{α}(Y_{η}(f) ∘ ϕ) − X_{β}(Y_{ξ}(f) ∘ ϕ), (8.9.9) as in (8.9.5).

This is equal to (Y_{ξ}(Y_{η}(f))) ∘ ϕ − (Y_{η}(Y_{ξ}(f))) ∘ ϕ, (8.9.10) as in (8.9.5) again. This is the same as (Y_{[ξ,η]}(f)) ∘ ϕ, (8.9.11) as in Subsection 8.6.1. Thus we get that X_{[α,β]}(f ∘ ϕ) = (Y_{[ξ,η]}(f)) ∘ ϕ (8.9.12) on U. This implies (8.9.7), as before.

8.10 Polynomial vector fields

Let (A, b) be an algebra in the strict sense over the real numbers, as in Section A.2. Also let δ be a derivation on A with respect to b, as in Subsection A.2.1. One can check that the kernel of δ is a subalgebra of A. If A has a multiplicative identity element e with respect to b, then one can verify that δ(e) = 0.

Let m and n be positive integers. Consider the space

P(R^{n}, R^{m}) (8.10.1)

of functions on R^{n} with values in R^{m} each of whose components is a polynomial on R^{n} with real coefficients. This is a linear subspace of the space C^{∞}(R^{n}, R^{m}) of infinitely-differentiable functions on R^{n} with values in R^{m}.

Let us now take m = n. One may describe

P(R^{n}, R^{n}) (8.10.2)

as the space of polynomial vector fields on R^{n}. This is a linear subspace of the space C^{∞}(R^{n}, R^{n}) of all smooth vector fields on R^{n}. Let ξ be a polynomial vector field on R^{n}, and let X_{ξ} be the corresponding first-order differential operator on R^{n}, as in Section 8.6. It is easy to see that X_{ξ} defines a linear mapping from the space P(R^{n}) = P(R^{n}, R) (8.10.3)

of polynomials on R^{n} with real coefficients into itself.

As before, ξ ↦ X_{ξ} defines a linear mapping from (8.10.2) into the space

L(P(R^{n})) (8.10.4)

of linear mappings from P(R^{n}) into itself. This mapping is injective, for the same reasons as in Section 8.6. If ξ and η are polynomial vector fields on R^{n}, then it is easy to see that

[ξ, η] ∈ P(R^{n}, R^{n}). (8.10.5)

This means that (8.10.2) is a Lie subalgebra of C^{∞}(R^{n}, R^{n}), with respect to the Lie bracket, as in Section B.2.

Of course, P(R^{n}) is a commutative associative algebra over the real numbers with respect to ordinary multiplication. If ξ is a polynomial vector field on R^{n}, then X_{ξ} is a derivation on P(R^{n}). It is not too difficult to show that every derivation on P(R^{n}) is of this type.

8.11 Cell-nice sets in R^{n}

Let n be a positive integer, and let x, y ∈ R^{n} be given. Consider the cell

C(x, y) = ∏_{j=1}^{n}[min(x_{j}, y_{j}), max(x_{j}, y_{j})] (8.11.1)

8.11. CELL-NICE SETS IN R^{N} 187 in R^{n}, as in Section 5.13. Of course, x, y ∈ C(x, y), (8.11.2) by construction. If C is any cell in R^{n} such that x, y ∈ C, (8.11.3) then it is easy to see that C(x, y) ⊆ C. (8.11.4)

Let us say that a subset E of R^{n} is cell-nice about a point w ∈ E if for every x ∈ E, C(w, x) ⊆ E. (8.11.5) Cells in R^{n} are cell-nice about each of their elements, for instance. Remember that the open ball B(w, r) in R^{n} centered at w ∈ R^{n} with radius r > 0 with respect to the standard Euclidean metric may be defined as in Subsection 3.4.1. One can check that (8.11.6) B(w, r) is cell-nice about w. It is easy to see that unions and intersections of families of subsets of R^{n} that are cell-nice about w are cell-nice about w too.

Let p be a positive integer with p ≤ n, and let π_{p} be the obvious projection from R^{n} onto R^{p}, so that π_{p}(x) = (x_{1}, …, x_{p}) (8.11.7) for each x = (x_{1}, …, x_{n}) ∈ R^{n}. If C is a cell in R^{n}, then (8.11.8) π_{p}(C) is a cell in R^{p}. If E ⊆ R^{n} is cell-nice about w ∈ E, then one can verify that (8.11.9) π_{p}(E) is cell-nice about π_{p}(w), as a subset of R^{p}. If A ⊆ R^{p}, π_{p}(w) ∈ W, and A is cell-nice about π_{p}(w) in R^{p}, then it is easy that π_{p}^{−1}(A) is cell-nice about w in R^{n}.

If w ∈ R^{n} and r > 0, then it is easy to see that π_{p}(B(w, r)) (8.11.10) is the same as the open ball in R^{p} centered at π_{p}(w) with radius r, with respect to the standard Euclidean metric on R^{p}. If U is an open set in R^{n}, then it follows that (8.11.11) π_{p}(U) is an open set in R^{p}. This means that π_{p} is an open mapping from R^{n} onto R^{p}, as in Section 5.10.

8.12 Integrating some functions

Let n and p be positive integers with p ≤ n again, and let W be an open set in R^{n} that is cell-nice about a point w ∈ W. Thus

V = π_{p}(W) (8.12.1)

is an open set in R^{p} that is cell-nice about π_{p}(w), where π_{p} is the obvious projection from R^{n} onto R^{p}, as before. Also let f be a continuously-differentiable real-valued function on W, and suppose that

∂_{j}f(x) = 0 (8.12.2)

for each x ∈ W and j = p + 1, …, n. Of course, this condition holds vacuously when p = n. One can check that this implies that f(x) does not depend on x_{j} for j = p + 1, …, n, because W is cell-nice about w. More precisely, if x ∈ W and p < n, then (x_{1}, …, x_{p}, w_{p+1}, …, w_{n}) ∈ W, (8.12.3)

because W is cell-nice about w. One can verify that

f(x) = f(x_{1}, …, x_{p}, w_{p+1}, …, w_{n}) (8.12.4)

for every x ∈ W under these conditions.

This means that f can be expressed as

f(x) = ϕ(π_{p}(x)), (8.12.5)

where ϕ is a real-valued function on V. In fact, ϕ is continuously differentiable on V, because it is the same as the right side of (8.12.4). If y = (y_{1}, …, y_{p}) ∈ V, then (y_{1}, …, y_{p−1}, t) ∈ V (8.12.6)

when w_{p} ≤ t ≤ y_{p} or y_{p} ≤ t ≤ w_{p}, (8.12.7)

as appropriate, because V is cell-nice about π_{p}(w) in R^{p}. Consider the realvalued function Φ defined on V by

Φ(y_{1}, …, y_{p}) = ∫_{w_{p}}^{y_{p}} ϕ(y_{1}, …, y_{p−1}, t) dt  when y_{p} ≥ w_{p}
(8.12.8)  = − ∫_{y_{p}}^{w_{p}} ϕ(y_{1}, …, y_{p−1}, t) dt when y_{p} ≤ w_{p}.

Note that Φ(y_{1}, …, y_{p}) = 0 when y_{p} = w_{p}. (8.12.9)

One can check that (8.12.10) Φ is continuous on V.

This is straightforward when p = 1, and otherwise one can use the uniform continuity of ϕ on compact subsets of V.

Of course, ∂_{p}Φ = ϕ (8.12.11)

on V, by construction. If p ≥ 2 and j = 1, …, p − 1, then ∂_{j}Φ can be obtained by differentiating under the integral sign. More precisely, the difference quotients of Φ used to define ∂_{j}Φ are equal to analogous integrals of difference quotients of ϕ. In order to differentiate under the integral sign, it is enough to have uniform convergence of the difference quoitents of ϕ along the appropriate closed segment. This can be obtained from the uniform continuity of ∂_{j}ϕ on compact subsets of V, and the mean-value theorem of the fundamental theorem of calculus.

If p ≥ 2, then ∂_{j}Φ is continuous on V for each j = 1, …, p − 1. This follows from the continuity of ∂_{j}ϕ on V, in the same way as in (8.12.10). This means that (8.12.12) Φ is continuous differentiable on V,

because (8.12.11) is continuous on V, by hypothesis.

8.12.1 Some properties of these functions

Consider the real-valued function F defined on W by

F = Φ ∘ π_{p}. (8.12.13)

Note that (8.12.14) F is continuously differentiable on W,

because Φ is continuously differentiable on V. It is easy to see that

∂_{p}F = f (8.12.15)

on W, because of (8.12.11). If p < n, then F (x) does not depend on x_{j} for j = p + 1, …, n, so that ∂_{j}F (x) = 0 (8.12.16)

for every x ∈ W.

This basically corresponds to Theorem 10.38 on p278 of [155], which is stated for convex open sets in R^{n}. The argument here is essentially the same as in [155], with some simplifications in this case. This is also related to Exercise 29 on p297 of [155], as mentioned in [155].

Let r be a positive integer, and suppose now that f is r-times continuously differentiable on W, which implies that ϕ is r-times continuously differentiable on V. Under these conditions, one can check that

(8.12.17) Φ is r-times continuously differentiable on V,

using the same type of arguments as before. This implies that

(8.12.18) F is r-times continuously differentiable on W.

If f is infinitely differentiable on W, then it follows that (8.12.19) F is infinitely differentiable on W.

8.13 A version of Poincaré’s lemma

Let n be a positive integer, and let W be an open set in R^{n} that is cell-nice about a point w ∈ W again. Also let (8.13.1) β be an infinitely-differentiable differential k-form on W for some k = 1, …, n, and suppose that (8.13.2) β is a closed form on W, so that dβ = 0 on W. We would like to find an infinitely-differentiable differential (k − 1)-form α on W such that dα = β (8.13.3) on W, so that β is exact on W.

This is a version of the famous Poincaré lemma. This basically corresponds to Theorem 10.39 in [155], which is stated for convex open subsets of R^{n}.

More precisely, suppose that β is r-times continuously differentiable on W with r ≥ n − k + 1, (8.13.4) instead of infinite differentiability. Under these conditions, the proof will show that we can find a differential (k − 1)-form α on W such that (8.13.5) α is (r − n + k)-times continuously differentiable on W, and (8.13.3) holds on W.

8.13.1 Invariance under diffeomorphisms

Let U be a nonempty open subset of R^{n}, and suppose that (8.13.6) every infinitely-differentiable differential k-form on U

that is closed on U can be expressed as the exterior derivative of an infinitely differentiable differential (k − 1)-form on U.

If (8.13.7) ϕ is a C^{∞} diffeomorphism from U onto another open set V ⊆ R^{n}, as in Subsection 5.12.2, then one can check that V has the same property. This corresponds to Theorem 10.40 on p280 of [155].

8.14 Some initial steps

Let us continue with the same notation and hypotheses as at the beginning of the previous section. Let p and r be positive integers, with

k ≤ p ≤ n. (8.14.1)

Consider the space Y (k, p, r) (8.14.2)

of r-times continuously-differentiable differential k-forms γ on W of the form

γ = ∑_{I⊆{1,...,p},#I=k} γ_{I} dx^{I}. (8.14.3)

More precisely, the sum on the right is taken over all subsets I of {1, …, p} with exactly k elements.

Suppose for the moment that

γ ∈ Y (k, k, r). (8.14.4)

This means that γ = f dx_{1} ∧ ⋯ ∧ dx_{k} (8.14.5)

for some r-times continuously-differentiable real-valued function f on W. Observe that dγ = 0 (8.14.6)

on W if and only if ∂_{j}f = 0 (8.14.7)

on W for each j = k + 1, …, n, which is vacuous when k = n.

In this case, there is an r-times continuously-differentiable real-valued function F on W such that ∂_{k}F = f (8.14.8)

on W, and ∂_{j}F = 0 (8.14.9)

on W for each j ≥ k + 1, as in Subsection 8.12.1. Note that F does not depend on r. In particular, if f is infinitely differentiable on W, then F is infinitely differentiable on W too, as before.

Thus (−1)^{k−1} F dx_{1} ∧ ⋯ ∧ dx_{k−1} ∈ Y (k − 1, k − 1, r). (8.14.10)

One can check that

d((−1)^{k−1} F dx_{1} ∧ ⋯ ∧ dx_{k−1}) (8.14.11)

= (∂_{k}F) dx_{1} ∧ ⋯ ∧ dx_{k−1} ∧ dx_{k} = γ

uunder these conditions. Note that this implies the Poincaré lemma when k = n.

8.14.1 The induction hypothesis

Suppose now that k + 1 ≤ p ≤ n, (8.14.12)

so that p ≥ 2 in particular. Our induction hypothesis is that

(8.14.13) if r ≥ p − k and an element of Y (k, p − 1, r) is closed as a differential form on W, then it can be expressed as the exterior derivative of an element of Y (k − 1, p − 2, r − p + k + 1).

More precisely, this element of Y (k − 1, p − 2, r) should not depend on r, so that it is infinitely differentiable when the initial element of Y (k, p − 1, r) is infinitely differentiable. Note that

(8.14.14) (8.14.13) holds when p − 1 = k,

as in (8.14.11).

We would like to use this to show that

(8.14.15) if r ≥ p − k + 1 and an element of Y (k, p, r) is closed as a differential form on W, then it can be expressed as the exterior derivative of an element of Y (k − 1, p − 1, r − p + k).

As before, this element of Y (k − 1, p − 1, r − p + k) should not depend on r, so that it is smooth when the initial element of Y (k, p, r) is smooth.

We shall discuss (8.14.15) further in the next section. Of course, (8.14.13) is the same as (8.14.15) with p replaced with p − 1.

8.15 The induction step

Let us continue with the same notation and hypotheses as in the previous section. Suppose that the induction hypothesis (8.14.13) holds, and let us show that (8.14.15) holds as well, as in [155]. Let γ be an element of Y (k, p, r) as in (8.14.3), with r ≥ p − k + 1, (8.15.1)

and suppose that γ is closed on W.

Observe that

(8.15.2)  dγ = ∑_{I⊆{1,...,p},#I=k} dγ_{I} ∧ dx^{I}

= ∑_{I⊆{1,...,p},#I=k} ∑_{j=1}^{n}(∂_{j}γ_{I}) dx_{j} ∧ dx^{I}

on W. This is equal to 0 on W, by hypothesis. It follows that

∑_{I⊆{1,...,p},#I=k}(∂_{j}γ_{I}) dx_{j} ∧ dx^{I} = 0 (8.15.3)

on W for j = p + 1, …, n. In fact, one can use this to get that

∂_{j}γ_{I} = 0 (8.15.4)

on W for every I ⊆ {1, …, p} with exactly k elements and j = p + 1, …, n.

Note that r ≥ 2, because of (8.14.12) and (8.15.1). If I ⊆ {1, …, p} has exactly k elements, then there is an r-times continuously-differentiable realvalued function Γ_{I} on W such that

∂_{p}Γ_{I} = γ_{I} (8.15.5)

and ∂_{j}Γ_{I} = 0 (8.15.6)

on W for j = p + 1, …, n, as in Subsection 8.12.1. Remember that Γ_{I} does not depend on r, so that it is smooth on W when γ is smooth on W.

8.15.1 Using the induction hypothesis

Put

ξ = ∑_{I⊆{1,...,p−1},#I=k} γ_{I} dx^{I} (8.15.7)

and

η = ∑_{I⊆{1,...,p},p∈I,#I=k} γ_{I} dx^{I}, (8.15.8)

so that ξ ∈ Y_{p−1} and γ = ξ + η. (8.15.9)

If I ⊆ {1, …, p} has exactly k elements and p ∈ I, then put

I(p) = I \ {p}, (8.15.10)

so that I(p) ⊆ {1, …, p − 1} has exactly k − 1 elements.

Put

ζ = ∑_{I⊆{1,...,p},p∈I,#I=k} Γ_{I} dx^{I(p)}, (8.15.11)

which is a differential (k − 1)-form on W. We also have that

dζ = ∑_{I⊆{1,...,p},p∈I,#I=k} dΓ_{I} ∧ dx^{I(p)}

(8.15.12) = ∑_{I⊆{1,...,p},p∈I,#I=k} ∑_{j=1}^{p−1}(∂_{j}Γ_{I}) dx_{j} ∧ dx^{I(p)}

+ ∑_{I⊆{1,...,p},p∈I,#I=k} γ_{I} dx_{p} ∧ dx^{I(p)}

= ∑_{I⊆{1,...,p},p∈I,#I=k} ∑_{j=1}^{p−1}(∂_{j}Γ_{I}) dx_{j} ∧ dx^{I(p)} + (−1)^{k−1} η.

194 CHAPTER 8. STOKES’ THEOREM AND OTHER MATTERS

Thus

γ − (−1)^{k−1} dζ
(8.15.13) = ξ − (−1)^{k−1} ∑_{I⊆{1,...,p},p∈I,#I=k} ∑_{j=1}^{p−1}(∂_{j}Γ_{I}) dx_{j} ∧ dx^{I(p)}.

This is (r − 1)-times continuously differentiable on W, because γ and the Γ_{I}’s are r-times continuously differentiable on W. It is easy to see that γ − (−1)^{k−1} dζ ∈ Y (k, p − 1, r − 1), (8.15.14) by construction. We also have that d(γ − (−1)^{k−1} dζ) = 0 (8.15.15) on W, because ζ is closed on W, by hypothesis.

Of course, r − 1 ≥ p − k, (8.15.16) because of (8.15.1). This permits us to apply our induction hypothesis (8.14.13), with r replaced with r − 1. It follows that (8.15.17) γ − (−1)^{k−1} can be expressed as the exterior derivative

of an element of Y (k − 1, p − 2, r − p + k).

Note that ζ ∈ Y (k − 1, p − 1, r), (8.15.18) by construction. This implies that (8.15.19) γ can be expressed as the exterior derivative

of an element of Y (k − 1, p − 1, r − p + k),

as in (8.14.15).

Lie derivatives and tensor fields

9.1 Tensor fields and multilinearity

Let k and n be positive integers, and let E be a nonempty subset of R^{n}. Also let a be a tensor field of type (0, k) on E, as in Section 4.1. If η_{1}, …, η_{k} are vector fields on E, then a(η_{1}, …, η_{k}) (9.1.1)

defines a real-valued function on E, as in Section 4.3. Thus

(η_{1}, …, η_{k}) ↦ a(η_{1}, …, η_{k}) (9.1.2)

may be considered as a mapping from the space of k-tuples of vector fields on E to the space of real-valued functions on E.

The spaces of real-valued functions on E and vector fields on E are vector spaces over the real numbers with respect to pointwise addition and scalar multiplication, as before. In fact, (9.1.2) may be considered as a multilinear mapping from the space of k-tuples of vector fields on E into the space of real-valued functions on E, as in Section 2.2.

Let f_{l} be a real-valued function on E for some positive integer l ≤ k. If η_{l} is a vector field on E, then f_{l} η_{l} (9.1.3)

defines a vector field on E as well, whose value at x ∈ E is f_{l}(x) η_{l}(x). If η_{1}, …, η_{k} are vector fields on E, then

a(η_{1}, …, η_{l−1}, f_{l} η_{l}, η_{l+1}, …, η_{k}) = f_{l} a(η_{1}, …, η_{k}) (9.1.4)

on E. This may be considered as a stronger version of the multilinearity of (9.1.2).

The space of real-valued functions on E is a commutative associative algebra over the real numbers, as in Section A.2. The space of vector fields on E may be considered as a module over the algebra of real-valued functions with respect to multiplication as in (9.1.3), as in Subsections B.7.1 and B.7.2. One does not need to distinguish between left and right modules here, because the algebra is commutative. Note that the algebra of real-valued functions on E may be considered as a module over itself, as in Subsections B.7.1 and B.7.2.

This means that we may consider (9.1.2) as a mapping from the space of k-tuples of vector fields on E into the space of real-valued functions on E, where the spaces of vector fields on E and real-valued functions on E are both considered as modules over the algebra of real-valued functions on E. We may say that this mapping is multilinear over the algebra of real-valued functions on E, because of (9.1.4). This is related to some remarks in Section 2.18 beginning on p64 of [184].

9.1.1 Multilinearity over C(E, R)

Suppose that a is continuous as a tensor field of type (0, k) on E, as in Subsection 4.1.1. If η_{1}, …, η_{k} are continuous vector fields on E, then (9.1.1) is a continuous real-valued function on E. The space C(E, R) of continuous realvalued functions on E is a subalgebra of the space of all real-valued function on E, and in particular it is a commutative associative algebra over R as well. The space C(E, R^{n}) of continuous vector fields on E may be considered as a module over C(E, R), with multiplication as in (9.1.3). We may consider C(E, R) as a module over itself, as before.

Thus we may consider (9.1.2) as a mapping from the space C(E, R^{n})^{k} of k-tuples of continuous vector fields on E into C(E, R), where C(E, R) and C(E, R^{n}) are both considered as modules over C(E, R). This mpping is multilinear over C(E, R), because of (9.1.4).

9.1.2 Multilinearity over C^{r}(U, R)

Let U be a nonempty open subset of R^{n}, let r be a nonnegative integer, and suppose that a is an r-times continuously differentiable tensor field of type (0, k) on U, as in Section 4.2. If η_{1}, …, η_{k} are r-times continuously-differentiable vector fields on U, then (9.1.1) is r-times continuously differentiable as a realvalued function on U, as in Subsection 4.3.1. The space C^{r}(U, R) of r-times continuously-differentiable real-valued functions on U is a subalgebra of the space C(U, R) of continuous real-valued functions on U, and in particular is a commutative associaitve algebra over R. The space C^{r}(U, R^{n}) of r-times continuously-differentiable vector fields on U may be considered as a module over C^{r}(U, R), with multiplication as in (9.1.3). We may consider C^{r}(U, R) as a module over itself, as usual.

We may consider (9.1.2) as a mapping from C^{r}(U, R^{n})^{k} into C^{r}(U, R), where C^{r}(U, R) and C^{r}(U, R^{n}) are considered as modules over C^{r}(U, R). This mapping is multilinear over C^{r}(U, R), because of (9.1.4).

Similarly, suppose that a is infinitely differentiable as a tensor field of type (0, k) on U, as in Section 4.2. If η_{1}, …, η_{k} are infinitely differentiable vector fields on U, then (9.1.1) is infinitely-differentiable as a real-valued function on U, as in Subsection 4.3.1. The space C^{∞}(U, R) of infinitely-differntiable real-valued functions on U is a subalgebra of C^{r}(U, R) for each r ≥ 0, and in particular C^{∞}(U, R) is a commutative associaitve algebra over R. The space C^{∞}(U, R^{n}) of infinitely-differentiable vector fields on U may be considered as a module over C^{∞}(U, R), with multiplication as in (9.1.3), and we may consider C^{∞}(U, R) as a module over itself. If we consider (9.1.2) as a mapping from C^{∞}(U, R^{n})^{k} into C^{∞}(U, R), then this mapping is multilinear over C^{∞}(U, R), because of (9.1.4).

9.2 Spaces of multilinear mappings

Let k be a positive integer, and let V_{1}, …, V_{k}, Z be vector spaces over the real numbers. Consider the space

L(V_{1}, …, V_{k}; Z) (9.2.1)

of multilinear mappings from ∏_{l=1}^{k} V_{l} into Z, as in Section 2.2. This is a linear subspace of the space of all functions on ∏_{l=1}^{k} V_{k} with values in Z. In particular, this is a vector space over the real numbers with respect to pointwise addition and scalar multiplication of functions.

Let V be a vector space over the real numbers, and suppose now that V_{l} = V for each l = 1, …, k. In this case, (9.2.1) is the same as the space of multilinear mappings from the space V^{k} of k-tuples of elements of V into Z. This space may be denoted M_{k}(V, Z), (9.2.2)

and is a linear subspace of the space of all Z-valued functions on V^{k}, as before. This is the same as M_{k}(V) when Z = R, considered as a one-dimensional vector space over itself.

Let μ be a multilinear mapping from V^{k} into Z, and let σ be a permutation on {1, …, k}. Put

μ^{σ}(v_{1}, …, v_{k}) = μ(v_{σ(1)}, …, v_{σ(k)}) (9.2.3)

for all v_{1}, …, v_{k} ∈ V, as in Section 2.4. This is another multilinear mapping from V^{k} into Z, as before. If τ is another permutation on {1, …, k}, then

μ^{τ∘σ} = (μ^{σ})^{τ} , (9.2.4)

as in Sections 1.9 and 2.4. Note that μ ↦ μ^{σ} is a one-to-one linear mapping from (9.2.2) onto itself, as before.

We say that μ is a symmetric multilinear mapping from V^{k} into Z if

μ^{σ} = μ (9.2.5)

for every σ ∈ Sym(k), as in Section 1.9 and 2.4. Similarly, we say that μ is an alternating if μ^{σ} = sgn(σ) μ (9.2.6) for every σ ∈ Sym(k), as in Sections 1.11 and 2.4. As usual, it suffices to check that these conditions hold when σ is a transposition on {1, …, k}. Let

SM_{k}(V, Z) and AM_{k}(V, Z) (9.2.7)

be the spaces of symmetric and alternating multilinear mappings from V^{k} into Z, respectively. It is easy to see that these are linear subspaces of M_{k}(V, Z).

9.2.1 Multilinear mappings and R^{n}

Let n be a positive integer, and let e_{1}, …, e_{n} be the standard basis for R^{n}, as usual. If μ is a multilinear mapping from (R^{n})^{k} into Z, then μ is uniquely determined by its values μ(e_{m_{1}}, …, e_{m_{k}}) (9.2.8)

on k-tuples of standard basis vectors in R^{n}, as in Subsection 1.6.1. These may be arbitrary elements of Z, for the same reasons as before. It is easy to see that μ is symmetric or alternating on (R^{n}) if and only if its values (9.2.8) on k-tuples of standard basis vectors in R^{n} has the analogous property, as in Sections 1.9 and 1.11.

If μ is a multilinear mapping from V^{k} into R^{n}, then the jth component μ_{j} of μ is a k-linear form on V for each j = 1, …, n. Of course, μ is symmetric or alternating on V^{k} if and only if μ_{j} is symmetric or alternating on V^{k}, as appropriate, for each j = 1, …, n.

If μ is a multilinear mapping from (R^{n})^{k} into R^{n}, then

μ_{j}(e_{m_{1}}, …, e_{m_{k}}) (9.2.9)

is a real number for each j = 1, …, n and 1 ≤ m_{1}, …, m_{k} ≤ n. We also have that μ is uniquely determined by this family of real numbers, and that these real numbers may be arbitrary. In particular, this implies that

dim M_{k}(R^{n}, R^{n}) = n^{k+1}. (9.2.10)

It is sometimes convenient to identify M_{k}(R^{n}, R^{n}) with R^{n^{k+1}}, by listing the elements of {1, …, n}^{k+1} (9.2.11)

in a sequence with n^{k+1} terms. The order in which the elements of (9.2.11) will normally not matter, as before. In particular, this leads to a metric on M_{k}(R^{n}, R^{n}), corresponding to the standard Euclidean metric on R^{n^{k+1}}, that does not depend on the order in which the elements of (9.2.11) are listed.

9.3 Tensor fields of type (1, k)

Let k and n be positive integers, and let E be a nonempty subset of R^{n}. A tensor field of type (1, k) on E is a function on E with values in the space 9.3. TENSOR FIELDS OF TYPE (1, K) 199

M_{k}(R^{n}, R^{n}) of multilinear mappings from (R^{n})^{k} into R^{n}. This corresponds to part of Definition 2.15 on p63 of [184]. The space of tensor fields of type (1, k) on E is a vector space over the real numbers with respect to pointwise addition and scalar multiplication.

Let us say that a tensor field of type (1, k) on E is continuous if it corresponds to a continuous mapping from E into R^{n^{k+1}} as in Subsection 9.2.1. The space

C(E, M_{k}(R^{n}, R^{n})) = C(E, R^{n^{k+1}}) (9.3.1)

of continuous tensor fields of type (1, k) on E is a linear subspace of the space of all tensor fields of type (1, k) on E.

Remember that one can get a multilinear mapping from (R^{n})^{k} into R^{n} by multiplying an element of R^{n} by a k-linear form on R^{n}, as in Subsection 2.2.1. Similarly, if a is a tensor field of type (0, k) on E, as in Section 4.1, and ξ is a vector field on E, as in Section 4.3, then

a ξ (9.3.2)

defines a tensor field of type (1, k) on E. If a and ξ are continuous on E, then it is easy to see that (9.3.2) is continuous on E, as a tensor field of type (1, k).

Let μ be a tensor field of type (1, k) on E, so that μ_{x} is a multilinear mapping from (R^{n})^{k} into R^{n} for each x ∈ E. If η_{1}, …, η_{k} are vector fields on E, then

μ(η_{1}, …, η_{k}) (9.3.3)

defines another vector field on E. The value of this vector field at x ∈ E is

μ_{x}(η_{1}(x), …, η_{k}(x)). (9.3.4)

If μ and η_{1}, …, η_{k} are continuous on E, then (9.3.3) is a continuous vector field on E.

Let U be a nonempty open subset of R^{n}, and let r be a nonnegative integer. A tensor field of type (1, k) on U is said to be r-times continuously differentiable if it corresponds to an r-times continuously-differentiable mapping from U into R^{n^{k+1}} as before. This is interpreted as being the same as continuity when r = 0, as usual. The space

C^{r}(U, M_{k}(R^{n}, R^{n})) = C^{r}(U, R^{n^{k+1}}) (9.3.5)

of all r-times continuously-differentiable tensor fields of type (1, k) on U is a linear subspace of the space of all continuous tensor fields of type (1, k) on U.

A tensor field of type (1, k) on U is said to be infinitely differentiable or smooth if it corresponds to an infinitely-differentiable mapping from U into R^{n^{k+1}} as before. The space

C^{∞}(U, M_{k}(R^{n}, R^{n})) = C^{∞}(U, R^{n^{k+1}}) (9.3.6)

of all infinitely-differentiable tensor fields of type (1, k) on U is a linear subspace of (9.3.5) for each r.

Let μ be a tensor field of type (1, k) on U, and let η_{1}, …, η_{k} be vector fields on U. If μ and η_{1}, …, η_{k} are r-times continuously differentiable on U, then

(9.3.7) μ(η_{1}, …, η_{k}) is r-times continuously differentiable on U,

as a vector field on U. Similarly, if μ and η_{1}, …, η_{k} are infinitely differentiable on U, then (9.3.8) μ(η_{1}, …, η_{k}) is infinitely differentiable on U

as well.

9.4 Some more multilinearity conditions

Let k and n be positive integers, let E be a nonempty subset of R^{n}, and let μ be a tensor field of type (1, k) on E, as in the previous section. If η_{1}, …, η_{k} are vector fields on E, then (9.3.3) is another vector field on E, as before. In fact,

(η_{1}, …, η_{k}) ↦ μ(η_{1}, …, η_{k}) (9.4.1)

defines a multilinear mapping from the space of k-tuples of vector fields on E into the space of vector fields on E. If f_{l} is a real-valued function on E for some positive integer l ≤ k, then

μ(η_{1}, …, η_{l−1}, f_{l} η_{l}, η_{l+1}, …, η_{k}) = f_{l} μ(η_{1}, …, η_{k}) (9.4.2)

on E. This may be considered as a stronger version of multilinearity of (9.4.1), as in Section 9.1.

As before, the spaces of real-valued functions on E and vector fields on E may be considered as modules over the algebra of real-valued functions on E. We may say that (9.4.1) is multilinear over the algebra of real-valued functions on E, because of (9.4.2). This is related to some remarks on p65 of [184].

Suppose that (9.4.3) μ is continuous on E,

and remember that the space C(E, R^{n}) of continuous vector fields on E may be considered as a module over C(E, R), as in Subsection 9.1.1. If η_{1}, …, η_{k} are continuous on E, then (9.3.3) is continuous on E, as in the previous section. In this case, (9.4.1) is multilinear over C(E, R), as a mapping from C(E, R^{n})^{k} into C(E, R^{n}), because of (9.4.2).

Let U be a nonempty open subset of R^{n}, let r be a nonnegative integer, and suppose that μ is an r-times continuously-differentiable tensor field of type (1, k) on U, as in the previous section. The space C^{r}(U, R^{n}) of r-times continuouslydifferentiable vector fields on U may be considered as a module over C^{r}(U, R), as in Subsection 9.1.2. If η_{1}, …, η_{k} are r-times continuously-differentiable vector fields on U, then (9.3.3) is an r-times continuously-differentiable vector field on U as well, as in (9.3.7). Under these conditions, (9.4.1) is multilinear over C^{r}(U, R), as a mapping from C^{r}(U, R^{n})^{k} into C^{r}(U, R^{n}), because of (9.4.2).

Similarly, suppose that μ is infinitely differentiable as a tensor field of type (1, k) on U, as in the previous section. The space C^{∞}(U, R^{n}) of infinitelydifferentiable vector fields on U may be considered as a module over C^{∞}(U, R), as in Subsection 9.1.2. If η_{1}, …, η_{k} are infinitely-differentiable vector fields on U, then (9.3.3) is an infinitely-differentiable vector field on U too, as in (9.3.8). If we consider (9.4.1) as a mapping from C^{∞}(U, R^{n})^{k} into C^{∞}(U, R^{n}), then it is multilinear over C^{∞}(U, R), because of (9.4.2).

9.5 Some more Lie derivatives

Let n be a positive integer, and let U be a nonempty open subset of R^{n}. Also let k be a positive integer, and let a be a tensor field of type (0, k) on U that is r-times continuously differentiable on U for some positive integer r, as in Sections 4.1 and 4.2. If ξ is an r-times continuously-dfferentiable vector field on U, then we would like to define the Lie derivative

L_{ξ}(a) (9.5.1)

of a with respect to ξ as a tensor field of type (0, k) on U that is (r − 1)-times continuously differentiable on U. This type of Lie derivative is mentioned on p70 of [184].

If η_{1}, …, η_{k} are vector fields on U that are r-times continuously differentiable, then a(η_{1}, …, η_{k}) (9.5.2)

is a real-valued function on U that is r-times continuously differentiable as well, as in Section 4.3. We would like to define (9.5.1) so that

L_{ξ}(a(η_{1}, …, η_{k})) = (L_{ξ}(a))(η_{1}, …, η_{k})

+ ∑_{l=1}^{k} a(η_{1}, …, η_{l−1}, L_{ξ}(η_{l}), η_{l+1}, …, η_{k}), (9.5.3)

where the Lie deriviatives of (9.5.2) and η_{1}, …, η_{k} are as in Section 8.8. This means that

X_{ξ}(a(η_{1}, …, η_{k})) = (L_{ξ}(a))(η_{1}, …, η_{k})

+ ∑_{l=1}^{k} a(η_{1}, …, η_{l−1}, [ξ, η_{l}], η_{l+1}, …, η_{k}), (9.5.4)

where X_{ξ} and [ξ, η] are as in Sections 4.3 and 8.6.

Thus we would like to define (9.5.1) so that

(L_{ξ}(a))(η_{1}, …, η_{k}) = X_{ξ}(a(η_{1}, …, η_{k}))

− ∑_{l=1}^{k} a(η_{1}, …, η_{l−1}, [ξ, η_{l}], η_{l+1}, …, η_{k}) (9.5.5)

on U. This can be used to define (9.5.1) as a tensor field of type (0, k) on U, by taking η_{1}, …, η_{k} to be constant vector fields on U equal to any combination of standard basis vectors in R^{n}. More precisely,

(9.5.6) L_{ξ}(a) is (r − 1)-times continuously differentiable on U,

because the right side of (9.5.5) is (r − 1)-times continuously differentiable on U. Similarly, if a and ξ are infinitely differentiable on U, then

(9.5.7) L_{ξ}(a) is infinitely differentiable on U.

If (9.5.1) is defined as in (9.5.5) when η_{1}, …, η_{k} are constant vector fields on U, each of which is a standard basis vector in R^{n}, then it is easy to see that (9.5.5) holds for any constant vector fields η_{1}, …, η_{k} on U, using multilinearity. In fact, (9.5.5) holds when η_{1}, …, η_{k} are any continuously-differentiable vector fields on U. This will be discussed further in the next section.

9.6 Using nonconstant vector fields

Let us continue with the same notation and hypotheses as in the previous section. Let η_{1}, …, η_{k} be continuously-differentiable vector fields on U, and let f_{1}, …, f_{k} be continuously-differentiable real-valued functions on U. Note that

f_{1} η_{1}, …, f_{k} η_{k} (9.6.1)

are continuously-differentiable vector fields on U as well, as in Section 8.8. Of course, a(f_{1} η_{1}, …, f_{k} η_{k}) = (f_{1} ⋯ f_{k}) a(η_{1}, …, η_{k}) (9.6.2)

on U. We also have that

[ξ, f_{l} η_{l}] = X_{ξ}(f_{l}) η_{l} + f_{l} [ξ, η_{l}] (9.6.3)

on U for each l = 1, …, k, as in Section 8.8.

It follows from (9.6.2) that

(9.6.4)  X_{ξ}(a(f_{1} η_{1}, …, f_{k}η_{k})) = X_{ξ}(f_{1} ⋯ f_{k}) a(η_{1}, …, η_{k})
+(f_{1} ⋯ f_{k}) X_{ξ}(a(η_{1}, …, η_{k}))

on U. We can use (9.6.3) to get that

a(f_{1} η_{1}, …, f_{l−1} η_{l−1}, [ξ, f_{l} η_{l}], f_{l+1} η_{l+1}, …, f_{k} η_{k}) (9.6.5)

= (f_{1} ⋯ f_{l−1} X_{ξ}(f_{l}) f_{l+1} ⋯ f_{k}) a(η_{1}, …, η_{k})
+(f_{1} ⋯ f_{k}) a(η_{1}, …, η_{l−1}, [ξ, η_{l}], η_{l+1}, …, η_{k})

on U for each l = 1, …, k. Combining (9.6.4) and (9.6.5), we obtain that

X_{ξ}(a(f_{1} η_{1}, …, f_{k} η_{k})) (9.6.6)

− ∑_{l=1}^{k} a(f_{1} η_{1}, …, f_{l−1} η_{l−1}, [ξ, f_{l} η_{l}], f_{l+1} η_{l+1}, …, f_{k} η_{k})

= (f_{1} ⋯ f_{k}) (X_{ξ}(a(η_{1}, …, η_{k})) − ∑_{l=1}^{k} a(η_{1}, …, η_{l−1}, [ξ, η_{l}], η_{l+1}, …, η_{k}))

on U.

Note that

(L_{ξ}(a))(f_{1} η_{1}, …, f_{k} η_{k}) = (f_{1} ⋯ f_{k}) (L_{ξ}(a))(η_{1}, …, η_{k}) (9.6.7)

on U, as in (9.6.2). Suppose that η_{1}, …, η_{k} are constant vector fields on U, so that (9.5.5) holds on U, by construction. This means that the right side of (9.6.7) is equal to the right side of (9.6.6), so that

(L_{ξ}(a))(f_{1} η_{1}, …, f_{k} η_{k}) = X_{ξ}(a(f_{1} η_{1}, …, f_{k} η_{k})) (9.6.8)

− ∑_{l=1}^{k} a(f_{1} η_{1}, …, f_{l−1} η_{l−1}, [ξ, f_{l} η_{l}], f_{l+1} η_{l+1}, …, f_{k} η_{k})

on U. This is the same as (9.5.5), with η_{1}, …, η_{k} replaced with f_{1} η_{1}, …, f_{k} η_{k}. One can use this and multilinearity to get that (9.5.5) holds for all continuouslydifferentiable vector fields η_{1}, …, η_{k} on U.

9.7 Another family of Lie derivatives

Let k and n be positive integers, and let U be a nonempty open subset of R^{n}. Also let μ be a tensor field of type (1, k) on U, as in Section 9.3. Suppose that μ is r-times continuously differentiable on U for some positive integer r, and let ξ be an r-times continuously-differentiable vector field on U. We would like to define the Lie derivative L_{ξ}(μ) (9.7.1) of μ with respect to ξ as an (r − 1)-times continuously-differentiable tensor field of type (1, k) on U. This type of Lie deriviative is mentioned on p70 of [184] as well.

Let η_{1}, …, η_{k} be r-times continuously-differentiable vector fields on U, so that μ(η_{1}, …, η_{k}) (9.7.2) is r-times continuously differentiable as a vector field on U, as before. We would like to define (9.7.1) so that

L_{ξ}(μ(η_{1}, …, η_{k})) = (L_{ξ}(μ))(η_{1}, …, η_{k})

+ ∑_{l=1}^{k} μ(η_{1}, …, η_{l−1}, L_{ξ}(η_{l}), η_{l+1}, …, η_{k}), (9.7.3)

204 CHAPTER 9. LIE DERIVATIVES AND TENSOR FIELDS using Lie derivatives of vector fields as in Section 8.8. This is the same as saying that

(9.7.4) [ξ, μ(η_{1}, …, η_{k})] = (L_{ξ}(μ))(η_{1}, …, η_{k})

+ ∑_{l=1}^{k} μ(η_{1}, …, η_{l−1}, [ξ, η_{l}], η_{l+1}, …, η_{k}),

where the brackets of vector fields are as in Section 8.6.1.

As in Section 9.5, we can define (9.7.1) by taking

(L_{ξ}(μ))(η_{1}, …, η_{k}) = [ξ, μ(η_{1}, …, η_{k})]

− ∑_{l=1}^{k} μ(η_{1}, …, η_{l−1}, [ξ, η_{l}], η_{l+1}, …, η_{k}) (9.7.5)

on U when η_{1}, …, η_{k} are constant vector fields on U equal to any combination of standard basis vectors in R^{n}. Note that (9.7.6) L_{ξ}(μ) is (r − 1)-times continuously differentiable on U, because the right side of (9.7.5) is r-times continuously differentiable on U. Similarly, if μ and ξ are infinitely differentiable on U, then (9.7.7) L_{ξ}(μ) is infinitely differentiable on U. It is easy to see that (9.7.5) also holds when η_{1}, …, η_{k} are any constant vector fields on U, using multilinearity. We would like to check that (9.7.5) when η_{1}, …, η_{k} are any continuously-differentiable vector fields on U too.

9.7.1 Using nonconstant vector fields again

Let η_{1}, …, η_{k} be continuously-differentiable vector fields on U, and let f_{1}, …, f_{k} be continuously-differentiable real-valued functions on U, so that f_{1} η_{1}, …, f_{k} η_{k} (9.7.8) are continuously-differentiable vector fields on U, as before. Note that μ(f_{1} η_{1}, …, f_{k} η_{k}) = (f_{1} ⋯ f_{k}) μ(η_{1}, …, η_{k}), (9.7.9) as vector fields on U. This implies that

(9.7.10) [ξ, μ(f_{1} η_{1}, …, f_{k} η_{k})] = X_{ξ}(f_{1} ⋯ f_{k}) μ(η_{1}, …, η_{k})
+(f_{1} ⋯ f_{k}) [ξ, μ(η_{1}, …, η_{k})]

on U, as in Section 8.8.

We also have that

μ(f_{1} η_{1}, …, f_{l−1} η_{l−1}, [ξ, f_{l} η_{l}], f_{l+1} η_{l+1}, ⋯, f_{k} η_{k}) (9.7.11)

= (f_{1} ⋯ f_{l−1} X_{ξ}(f_{l}) f_{l+1} ⋯ f_{k}) μ(η_{1}, …, η_{k})
+(f_{1} ⋯, f_{k}) μ(η_{1}, …, η_{l−1}, [ξ, η_{l}], η_{l+1}, …, η_{k})

on U for each l = 1, …, k, as in the previous section. Combining this with (9.7.10), we get that

[ξ, μ(f_{1} η_{1}, …, f_{k}η_{k})] (9.7.12)

− ∑_{l=1}^{k} μ(f_{1} η_{1}, …, f_{l−1} η_{l−1}, [ξ, f_{l} η_{l}], f_{l+1} η_{l+1}, …, f_{k} η_{k})

= (f_{1} ⋯ f_{k}) ([ξ, μ(η_{1}, …, η_{k})] − ∑_{l=1}^{n} μ(η_{1}, …, η_{l−1}, [ξ, η_{l}], η_{l+1}, …, η_{k}))

on U.

Of course,

(L_{ξ}(μ))(f_{1} η_{1}, …, f_{k} η_{k}) = (f_{1} ⋯ f_{k}) (L_{ξ}(μ))(η_{1}, …, η_{k}) (9.7.13)

on U, as in (9.7.9). If η_{1}, …, η_{k} are constant vector fields on U, then the right side of (9.7.12) is equal to the right side of (9.7.13), as in (9.7.5). This implies that

(L_{ξ}(μ))(f_{1} η_{1}, …, f_{k} η_{k}) = [ξ, μ(f_{1} η_{1}, …, f_{k} η_{k})] (9.7.14)

− ∑_{l=1}^{k} μ(f_{1} η_{1}, …, f_{l−1}, η_{l−1}, [ξ, f_{l} η_{l}].f_{l+1} η_{l+1}, …, f_{k} η_{k})

on U. This is the same as (9.7.5), with η_{1}, …, η_{k} replaced with f_{1} η_{,} …, f_{k} η_{k}. One can use this to get that (9.7.5) holds for all continuously-differentiable vector fields η_{1}, …, η_{k} on U.

9.8 Lie derivatives as representations

Let n and r be positive integers, and let U be a nonempty open subset of R^{n}. Also let ξ, ζ be r-times continuously-differentiable vector fields on U, so that [ξ, ζ] is an (r − 1)-times continuously-differentiable vector field on U, as in Subsection 8.6.1. If f is an r-times continuously-differentiable real-valued function on U, then L_{[ξ,ζ]}(f) = X_{[ξ,ζ]}(f) (9.8.1)

is an (r − 1)-times continuously=differentiable real-valued function on U, as in Section 8.8. If f is twice continuously differentiable on U, then

L_{[ξ,ζ]}(f) = X_{ξ}(X_{ζ}(f)) − X_{ζ}(X_{ξ}(f)) = L_{ξ}(L_{ζ}(f)) − L_{ζ}(L_{ξ}(f)) (9.8.2) on U, where the first step is as in Subsection 8.6.1.

If ξ is an infinitely-differentiable vector field on U, then X_{ξ} defines a linear mapping from the space C^{∞}(U, R) of infinitely-differentiable real-valued functions on U into itself, as in Subsection 4.3.1. Thus we may consider

ξ ↦ L_{ξ} = X_{ξ} (9.8.3)

as a linear mapping from the space C^{∞}(U, R^{n}) of infinitely-differentiable vector fields on U into the space L(C^{∞}(U, R)) (9.8.4)

of linear mappings from C^{∞}(U, R) into itself. Remember that C^{∞}(U, R^{n}) is a Lie algebra over the real numbers with respect to the Lie bracket, as mentioned in Section 8.7.1. Using (9.8.2), we get that (9.8.3) defines a representation of C^{∞}(U, R^{n}), as a Lie algebra over R, on C^{∞}(U, R), as in Section B.7.

Suppose that r ≥ 2, and let η be another r-times continuously-differentiable vector field on U. Note that

L_{[ξ,ζ]}(η) = [[ξ, ζ], η] (9.8.5)

on U, as in Section 8.8. One can check that

L_{[ξ,ζ]}(η) = [ξ, [ζ, η]] − [ζ, [ξ, η]] = L_{ξ}(L_{ζ}(η)) − L_{ζ}(L_{ξ}(η)) (9.8.6)

on U, using the Jacobi identity in the first step, as in Subsection 8.7.1.

If ξ is an infinitely-differentiable vector field on U, then L_{ξ} defines a linear mapping from C^{∞}(U, R^{n}) into itself, as in Subsection 8.6.1. In fact,

ξ ↦ L_{ξ} = [ξ, ·] (9.8.7)

defines a linear mapping from C^{∞}(U, R^{n}) into the space

L(C^{∞}(U, R^{n})) (9.8.8)

of linear mappings from C^{∞}(U, R^{n}) into itself. This defines a representation of C^{∞}(U, R^{n}), as a Lie algebra over R, on itself, as a vector space over the real numbers. This is the same as the adjoint representation of C^{∞}(U, R^{n}), as in Section B.8.

9.9 Some more representations

Let n, k, and r be positive integers, let U be a nonempty open subset of R^{n}, and let a be a tensor field of type (0, k) on U that is r-times continuously differentiable on U. Suppose that r ≥ 2, and that ξ, ζ are r-times continuouslydifferentiable vector fields on U. Thus [ξ, ζ] is an (r − 1)-times continuouslydifferentiable vector field on U, as in Subsection 8.6.1, and one can check that

L_{[ξ,ζ]}(a) = L_{ξ}(L_{ζ}(a)) − L_{ζ}(L_{ξ}(a)) (9.9.1) on U, where these Lie derivatives of a are as in Section 9.5. This uses the analogous statements for Lie derivatives of functions and vector fields mentioned in the previous section.

Remember that C^{∞}(U, M_{k}(R^{n})) is the space of all infinitely-differentiable tensor fields of type (0, k) on U, as in Section 4.2. If ξ is infinitely differentiable on U, then L_{ξ} defines a linear mapping from C^{∞}(U, M_{k}(R^{n})) into itself, as in Section 9.5. Note that ξ ↦ L_{ξ} (9.9.2)

defines a linear mapping from C^{∞}(U, R^{n}) into the space

L(C^{∞}(U, M_{k}(R^{n}))) (9.9.3)

of linear mappings from C^{∞}(U, M_{k}(R^{n})) into itself. This is a representation of C^{∞}(U, R^{n}), as a Lie algebra over the real numbers with respect to the Lie bracket, on C^{∞}(U, M_{k}(R^{n})), because of (9.9.1).

Similarly, let μ be a tensor field of type (1, k) on U that is r-times continuously differeneitable on U. One can check that

L_{[ξ,ζ]}(μ) = L_{ξ}(L_{ζ}(μ)) − L_{ζ}(L_{ξ}(μ)) (9.9.4)

on U, where these Lie derivatives are as in Section 9.7. This uses the analogous statement for vector fields in the previous section.

The space of all infinitely-differentiable tensor fields of type (1, k) on U is denoted C^{∞}(U, M_{k}(R^{n}, R^{n})), as in Section 9.3. If ξ is infinitely differentiable on this space, then L_{ξ} defines a linear mapping from this space into itself, as in Section 9.7. As before, ξ ↦ L_{ξ} defines a linear mapping from C^{∞}(U, R^{n}) into the space L(C^{∞}(U, M_{k}(R^{n}, R^{n}))) (9.9.5)

of linear mappings from C^{∞}(U, M_{k}(R^{n}, R^{n})) into itself. This is a representation of C^{∞}(U, R^{n}), as a lie algebra over R with respect to the Lie bracket, on C^{∞}(U, M_{k}(R^{n}, R^{n})), by (9.9.4).

9.10 Some properties of Lie derivatives

Let n, k, and r be positive integers, let U be a nonempty open subset of R^{n}, let a be a tensor field of type (0, k) on U that is r-times continuously differentiable on U, and let ξ be an r-times continuously-differentiable vector field on U. If σ is a permutation on {1, …, k}, then let a^{σ} be the tensor field of type (0, k) defined on U by (a^{σ})_{x} = (a_{x})^{σ} (9.10.1)

for every x ∈ U. More precisely, if x ∈ U, then a_{x} is a k-linear form on R^{n}, so that (a_{x})^{σ} may be defined as a k-linear form on R^{n} as in Section 1.9. It is easy to see that a^{σ} is r-times continuously differentiable on U as well, as in Section 4.2.

One can verify that L_{ξ}(a^{σ}) = (L_{ξ}(a))^{σ}, (9.10.2) where these Lie derivatives are as in Section 9.5. If a_{x} is a symmetric k-linear form on R^{n} for every x ∈ U, then it follows that (9.10.3) (L_{ξ}(a))_{x} is a symmetric k-linear form on R^{n} for every x ∈ U. Similarly, if a_{x} is an alternating k-linear form on R^{n} for every x ∈ U, then (9.10.4) (L_{ξ}(a))_{x} is an alternating k-linear form on R^{n} for every x ∈ U.

This means that (9.10.5) if a is a differential k-form on U,

then L_{ξ}(a) is a differential k-form on U too.

Lie derivatives of smooth differential forms are discussed on p70 of [184]. The definition of the Lie derivative of a differential form used here corresponds to part (e) of Proposition 2.25 on p70 of [184].

9.10.1 Lie derivatives of products

Let f be an r-times continuously-differentiable real-valued function on U, and note that f a defines an r-times continuously-differentiable tensor field of type (0, k) on U. If η_{1}, …, η_{k} are vector fields on U, then (f a)(η_{1}, …, η_{k}) = f a(η_{1}, …, η_{k}), (9.10.6) as real-valued functions on U. Remember that L_{ξ}(f) = X_{ξ}(f) on U, as in Section 8.8. One can check that L_{ξ}(f a) = L_{ξ}(f) a + f L_{ξ}(a), (9.10.7) as tensor fields of type (0, k) on U.

Let k′ be another positive integer, and let b be a tensor field of type (0, k′) on U that is r-times continuously differentiable on U. Remember that a⊗b may be defined as a tensor field of type (0, k_{1} + k_{2}) on U that is r-times continuously differentiable on U, as in Sections 4.1 and 4.2. If η_{1}…. , η_{k+k′} are vector fields on U, then (a ⊗ b)(η_{1}, …, η_{k+1}) = a(η_{1}, …, η_{k}) b(η_{k+1}, …, η_{k+k′}), (9.10.8) as real-valued functions on U. One can verify that L_{ξ}(a ⊗ b) = (L_{ξ}(a)) ⊗ b + a ⊗ (L_{ξ}(b)), (9.10.9) as tensor fields of type (0, k + k′) on U.

If a, b are differential k, k′-forms on U, respectively, then L_{ξ}(a ∧ b) = (L_{ξ}(a)) ∧ b + a ∧ (L_{ξ}(b)), (9.10.10) as differential (k + k′)-forms on U. This can be obtained from (9.10.9) using (9.10.2). This corresponds to part of part (c) of Proposition 2.25 on p70 of [184].

9.10.2 Lie derivatives and exterior differentiation

Let f be an (r + 1)-times continuously-differentiable real-valued function on U, so that a = df (9.10.11)

is an r-times continuously-differentiable differential one-form on U, or equivalently a tensor field of type (0, 1) on U. If η is an r-times continuouslydifferentiable vector field on U, then

a(η) = (df)(η) = X_{η}(f) (9.10.12)

is an r-times continuously-differentiable real-valued function on U. Observe that

(L_{ξ}(a))(η) = X_{ξ}(a(η)) − a([ξ, η]) = X_{ξ}(X_{η}(f)) − X_{[ξ,η]}(f)
(9.10.13)  = X_{η}(X_{ξ}(f)) = (d(X_{ξ}(f)))(η),

where the first step is as in Section 9.5, and the second step is as in Subsection 8.6.1. This means that L_{ξ}(df) = d(L_{ξ}(f)), (9.10.14)

as differential one-forms on U.

Of course, L_{ξ}(f) = X_{ξ}(f) is an r-times continuously-differentiabl real-valued function on U. If r ≥ 2, then it follows that

d(L_{ξ}(df)) = d(d(L_{ξ}(f))) = 0, (9.10.15)

as in Section 4.8.

Let α be a differential k-form on U that is twice continuously differentiable on U, so that dα is a differential (k+1)-form on U that is continuously differentiable on U. If ξ is twice continuously differentiable on U, then one can check that

L_{ξ}(dα) = d(L_{ξ}(α)) (9.10.16)

as differential (k + 1)-forms on U. More precisely, if k = 0, then this is the same as (9.10.14), and it is enough to ask that ξ be continuously differentiable. If k ≥ 1, then this can be obtained using the earlier remarks about Lie derivatives of products. This corresponds to another part of part (c) of Proposition 2.25 on p70 of [184].

9.10.3 Tensors of type (1, k)

Let μ be a tensor field of type (1, k) on U that is r-times continuously differentiable on U. If σ ∈ Sym(k), then let μ^{σ} be the tensor field of type (1, k) on U defined by (μ^{σ})_{x} = (μ_{x})^{σ} (9.10.17)

for each x ∈ U, where the right side is as in Section 9.2. Note that μ^{σ} is r-times continuously differentiable on U as well.

210 CHAPTER 9. LIE DERIVATIVES AND TENSOR FIELDS

One can check that L_{ξ}(μ^{σ}) = (L_{ξ}(μ))^{σ}, (9.10.18) where these Lie derivatives are as in Section 9.7. If μ_{x} is symmetric as a multilinear mapping from (R^{n})^{k} into R^{n} for each x ∈ U, then it follows that (9.10.19) (L_{ξ}(μ))_{x} is symmetric as a multilinear mapping

from (R^{n})^{k} into R^{n} for every x ∈ U.

Similarly, if μ_{x} is alternating as a multilinear mapping from (R^{n})^{k} into R^{n} for each x ∈ U, then (9.10.20) (L_{ξ}(μ))_{x} is alternating as a multilinear mapping

from (R^{n})^{k} into R^{n} for every x ∈ U.

Let a be a tensor field of type (0, k) on U that is r-times continuously differentiable on U agagin, and let ζ be a vector field on U that is r-times continuously differentiable on U. Consider μ = a ζ, (9.10.21) as in Section 9.3, which is also r-times continuously differentiable on U. If η_{1}, …, η_{k} are vector fields on U, then μ(η_{1}, …, η_{k}) = a(η_{1}, …, η_{k}) ζ, (9.10.22) as vector fields on U. One can check that L_{ξ}(a ξ) = (L_{ξ}(a)) ζ + a L_{ξ}(ζ), (9.10.23) as tensor fields of type (1, k) on U.

9.11 Some interior products

Let W be a vector space over the real numbers, and let u be an element of W. Also let k and p be positive integers, with p ≤ k. If μ is a k-linear form on W, then let (i_{k,p}(u))(μ) (9.11.1) be the (k − 1)-linear form on W defined by ((i_{k,p}(u))(μ))(w_{1}, …, w_{k−1}) = μ(w_{1}, …, w_{p−1}, u, w_{p}, …, w_{k−1}). (9.11.2) This defines a linear mapping i_{k,p}(u) (9.11.3) from M_{k}(W) into M_{k−1}(W). This linear mapping may be described as the pth interior product by u on M_{k}(W).

If k = 1, then μ is a linear functional on W. In this case, p = 1, and

(i_{1,1}(u))(μ) = μ(u), (9.11.4)

which is a real number. Remember that M_{0}(W) = R, as in Section 2.12. Put

i_{k,tot}(u) = ∑_{p=1}^{k} i_{k,p}(u), (9.11.5)

which is another linear mapping from M_{k}(W) into M_{k−1}(W). This may be described as the total interior product by u on M_{k}(W). It is convenient to take i_{k,tot}(u) to be 0 on M_{0}(W) when k = 0.

9.11.1 Connection with products of forms

Let k_{1} and k_{2} be positive integers, and let μ_{1}, μ_{2} be k_{1}, k_{2}-linear forms on W, respectively. Thus μ_{1} ⊗ μ_{2} may be defined as a (k_{1} + k_{2})-linear form on W, as in Section 2.8. One can check that

(i_{k_{1}+k_{2},p}(u))(μ_{1} ⊗ μ_{2}) (9.11.6)

= ((i_{k_{1},p}(u))(μ_{1})) ⊗ μ_{2}  when 1 ≤ p ≤ k_{1}
= μ_{1} ⊗ ((i_{k_{2},p−k_{1}}(u))(μ_{2})) when k_{1} + 1 ≤ p ≤ k_{1} + k_{2}.

Using this, one can verify that

(i_{k_{1}+k_{2},tot}(u))(μ_{1} ⊗ μ_{2}) (9.11.7)

= ((i_{k_{1},tot}(u))(μ_{1})) ⊗ μ_{2} + μ_{1} ⊗ ((i_{k_{2},tot}(u))(μ_{2})).

This also holds when k_{1} or k_{2} is 0, with suitable interpretations.

The algebra M(W) of multilinear forms on W may be defined as the direct sum of M_{k}(W) over all nonnegative integers k, as in Section A.6. One can define a linear mapping i_{tot}(u) (9.11.8)

from M(W) into itself, using i_{k,tot}(u) on M_{k}(W) for each k ≥ 0. This may be described as the total interior product by u on M(W). It is easy to see that

(9.11.9) i_{tot}(u) is a derivation on M(W),

as in Subsection A.2.1.

9.12 Interior products and symmetrizations

Let W be a vector space over the real numbers, let u be an element of W, and let k be a positive integer. If μ is a symmetric k-linear form on W, then it is easy to see that (i_{k,p}(u))(μ) = (i_{k,q}(u))(μ) (9.12.1)

212 CHAPTER 9. LIE DERIVATIVES AND TENSOR FIELDS

for all positive integers p, q ≤ k. This implies that k (i_{k,p}(u))(μ) = (i_{k,tot}(u))(μ) (9.12.2) for each p ≤ k. Note that (9.12.3) (i_{k,p}(u))(μ) is a symmetric (k − 1)-linear form on W in this case. Thus the restriction of i_{k,p}(u) to SM_{k}(W) defines a linear mapping into SM_{k−1}(W).

Suppose that k ≥ 2, let μ be any k-linear form on W, and let τ be a permutation on {1, …, k − 1}. Observe that

((i_{k,p}(u))(μ))^{τ} (w_{1}, …, w_{k−1}) = ((i_{k,p}(u))(μ))(w_{τ(1)}, …, w_{τ(k−1)})
(9.12.4)  = μ(w_{τ(1)}, …, w_{τ(p−1)}, u, w_{τ(p)}, …, w_{τ(k−1)})

for each p ≤ k and w_{1}, …, w_{k−1} ∈ W, where the left side is as in Section 2.4. We also have that

S_{k−1}((i_{k,tot}(u))(μ)) = \frac{1}{(k − 1)!} ∑_{p=1}^{k} ∑_{τ∈Sym(k−1)}((i_{k,p}(u))(μ))^{τ} , (9.12.5)

where S_{k−1} is as in Subsection 2.4.1. It follows that S_{k−1}((i_{k,tot}(u))(μ))(w_{1}, …, w_{k−1}) (9.12.6)

= \frac{1}{(k − 1)!} ∑_{p=1}^{k} ∑_{τ∈Sym(k−1)}((i_{k,p}(u))(μ))^{τ} (w_{1}, …, w_{k−1})
= \frac{1}{(k − 1)!} ∑_{p=1}^{k} ∑_{τ∈Sym(k−1)} μ(w_{τ(1)}, …, w_{τ(p−1)}, u, w_{τ(p)}, …, w_{τ(k−1)})

for all w_{1}, …, w_{k−1} ∈ W.

One can check that this is equal to k (S_{k}(μ))(u, w_{1}, …, w_{k−1}) (9.12.7) for all w_{1}, …, w_{k−1} ∈ W. One can look at this in terms of shuffles, as in Section 2.15, with r = 2, k_{1} = 1, and k_{2} = k − 1. This means that S_{k−1}((i_{k,tot}(u))(μ)) = k (i_{k,1}(u))(S_{k}(μ)). (9.12.8) Equivalently, S_{k−1}((i_{k,tot}(u))(μ)) = (i_{k,tot}(u))(S_{k}(μ)), (9.12.9) because of (9.12.2). Of course, this is the same as saying that S_{k−1} ∘ i_{k,tot}(u) = i_{k,tot}(u) ∘ S_{k} (9.12.10) on M_{k}(W).

9.12.1 Connection with symmetric products

Let k_{1}, k_{2} be positive integers, and let μ_{1}, μ_{2} be symmetric k_{1}, k_{2}-linear forms on W, respectively. Remember that μ_{1} ⊙ μ_{2} may be defined as a symmetric (k_{1} + k_{2})-linear form on W as in Section 2.9. Observe that

(i_{k_{1}+k_{2},tot}(u))(μ_{1} ⊙ μ_{2}) = (i_{k_{1}+k_{2},tot}(u))(S_{k_{1}+k_{2}}(μ_{1} ⊗ μ_{2}))
(9.12.11)  = S_{k_{1}+k_{2}−1}((i_{k_{1}+k_{2},tot}(u))(μ_{1} ⊗ μ_{2})),

using (9.12.9) in the second step. Combining this with (9.11.7), we get that

(i_{k_{1}+k_{2},tot}(u))(μ_{1} ⊙ μ_{2}) = S_{k_{1}+k_{2}−1}(((i_{k_{1},tot}(u))(μ_{1})) ⊗ μ_{2})

+S_{k_{1}+k_{2}−1,tot}(μ_{1} ⊗ ((i_{k_{2},tot}(u))(μ_{2}))) (9.12.12)

= ((i_{k_{1},tot}(u))(μ_{1})) ⊙ μ_{2} + μ_{1} ⊙ ((i_{k_{2},tot}(u))(μ_{2})).

The algebra SM(W) of symmetric multilinear forms on W may be defined as the direct sum of SM_{k}(W) over all nonnegative integers k, as in Subsection A.6.1. This may be considered as a linear subspace of M(W). Note that the restriction of i_{k,tot}(u) to SM_{k}(W) defines a linear mapping into SM_{k−1}(W) for each k, so that (9.12.13) i_{tot}(u) maps SM(W) into itself.

In fact, (9.12.14) i_{tot}(u) is a derivation on SM(W),

as in Subsection A.2.1, because of (9.12.12).

9.13 Interior products and polynomials

Let k and n be positive integers, and let us now take W = R^{n}. If μ is a k-linear form on R^{n}, then P_{μ}(x) = μ(x, …, x) defines a homogeneous polynomial of degree k on R^{n}, as in Section 1.8. Let u ∈ R^{n} be given, so that (i_{k,q}(u))(μ) is defined as a (k − 1)-linear form on R^{n} as in Section 9.11 for each q = 1, …, k. Thus P_{(i_{k,q}(u))(μ)} (9.13.1)

defines a homogeneous polynomial of degree k − 1 on R^{n} for each q. Similarly, (i_{k,tot}(u))(μ) is a (k − 1)-linear form on R^{n}, which corresponds to the homogeneous polynomial

P_{(i_{k,tot}(u))(μ)} = ∑_{q=1}^{k} P_{(i_{k,q}(u))(μ)} (9.13.2)

of degree k − 1 on R^{n}.

One can check that

P_{(i_{k,tot}(u))(μ)}(x) = (P′_{μ}(x))(u) = d(P_{μ})_{x}(u) (9.13.3) for each x ∈ R^{n}. This is the same as the directional derivative of P_{μ} at x in the direction u, as in Subsection 3.8.1.

Similarly, if μ ∈ M(R^{n}), then P_{μ} may be defined as a polynomial on R^{n} with real coefficients as in Subsection A.6.2. Remember that (i_{tot}(u))(μ) may be defined as an element of M(R^{n}) as in Subsection 9.11.1. As before, we have that P_{(i_{tot}(u))(μ)}(x) = (P′_{μ}(x))(u) = d(P_{μ})_{x}(u) (9.13.4)

for every x ∈ R^{n}.

9.14 Interior multiplication on AM(W)

Let W be a vector space over the real numbers, let u be an element of W, and let k be a positive integer. If μ is an alternating k-linear form on W, then let

(i_{k}(u))(μ) (9.14.1)

be the (k − 1)-linear form defined on W by

((i_{k}(u))(μ))(w_{1}, …, w_{k−1}) = μ(u, w_{1}, …, w_{k−1}). (9.14.2)

It is easy to see that this is an alternating (k−1)-linear form on W. This defines a linear mapping from AM_{k}(W) into AM_{k−1}(W), which is known as interior multiplication by u. This corresponds to part of Section 2.11 on p61 of [184].

Equivalently, i_{k}(u) is the same as the restriction of i_{k,1}(u), as in Section 9.11, to AM_{k}(W). Similarly,

i_{k,p}(u) = (−1)^{p−1} i_{k}(u) on AM_{k}(W) (9.14.3)

for each p ≤ k. It is convenient to take i_{k}(u) to be 0 on AM_{0}(W) = R when k = 0. We can define i(u) as a linear mapping from AM(W) into itself, which corresponds to i_{k}(u) on AM_{k}(W) for each k ≥ 0. This uses the definition of AM(W) in Subsection 2.12.1 when W has finite dimension, or the definition in Section A.7 otherwise.

Let k_{1} and k_{2} be nonnegative integers, and let μ_{1}, μ_{2} be alternating k_{1}, k_{2}-linear forms on W, respectively. We would like to show that

(i_{k_{1}+k_{2}}(u))(μ_{1} ∧ μ_{2}) (9.14.4)

= ((i_{k_{1}}(u))(μ_{1})) ∧ μ_{2} + (−1)^{k_{1}} μ_{1} ∧ ((i_{k_{2}}(u))(μ_{2})).

This means that i(u) is an anti-derivation on AM(W), as in Section 2.11 on p61 of [184]. This corresponds to Proposition 2.12 on p61 of [184]. It is easy to see that (9.14.4) holds when k_{1} or k_{2} is 0, and so we suppose now that k_{1}, k_{2} ≥ 1.

9.14.1 Checking the anti-derivation property

Put E_{1} = {1, …, k_{1}} and E_{2} = {k_{1} + 1, …, k_{1} + k_{2}}, (9.14.5)

9.14. INTERIOR MULTIPLICATION ON AM(W) 215

as in Section 2.13, with r = 2. Remember that τ ∈ Sym(k_{1} + k_{2}) is said to be a shuffle with respect to k_{1}, k_{2} if the restriction of τ to each of E_{1} and E_{2} is increasing, as in Section 2.15. Let Ξ be the set of τ ∈ Sym(k_{1} + k_{2}) that are shuffles with respect to k_{1}, k_{2}, as in Subsection 2.15.1. Thus

μ_{1} ∧ μ_{2} = ∑_{τ∈Ξ} sgn(τ) (μ_{1} ⊗ μ_{2})^{τ} , (9.14.6)

as in Subsection 2.15.2. Of course, (μ_{1} ⊗ μ_{2})^{τ} is defined as in Sections 2.4 and 2.8.

If v_{1}, …, v_{k_{1}+k_{2}} ∈ W, then

(μ_{1} ⊗ μ_{2})^{τ} (v_{1}, …, v_{k_{1}+k_{2}}) = (μ_{1} ⊗ μ_{2})(v_{τ(1)}, …, v_{τ(k_{1}+k_{2})})
(9.14.7)  = μ_{1}(v_{τ(1)}, …, v_{τ(k_{1})}) μ_{2}(v_{τ(k_{1}+1)}, …, v_{τ(k_{1}+k_{2})})

for every τ ∈ Sym(k_{1} + k_{2}). If τ is a shuffle with respect to k_{1}, k_{2}, then it is easy to see that τ^{−1}(1) = 1 or k_{1} + 1. (9.14.8)

This means that

(9.14.9)  μ_{1} ∧ μ_{2} = ∑_{τ∈Ξ,τ^{−1}(1)=1} sgn(τ) (μ_{1} ⊗ μ_{2})^{τ}

+ ∑_{τ∈Ξ,τ^{−1}(1)=k_{1}+1} sgn(τ) (μ_{1} ⊗ μ_{2})^{τ} .

In order to get (9.14.4), it suffices to verify that

∑_{τ∈Ξ,τ^{−1}(1)=1} sgn(τ) (i_{k_{1}+k_{2}}(u))((μ_{1} ⊗ μ_{2})^{τ}) = ((i_{k_{1}}(u))(μ_{1})) ∧ μ_{2} (9.14.10)

and

∑_{τ∈Ξ,τ^{−1}(1)=k_{1}+1} sgn(τ) (i_{k_{1}+k_{2}}(u))((μ_{1} ⊗ μ_{2})^{τ}) (9.14.11)

= (−1)^{k_{1}} μ_{1} ∧ ((i_{k_{2}}(u))(μ_{2})).

This uses (9.14.7) with v_{1} = u in both cases.

Note that the wedge products on the right sides of (9.14.10) and (9.14.11) may be expressed as in Subsection 2.15.2 as well, using shuffles with respect to k_{1} − 1, k_{2} and k_{1}, k_{2} − 1, respectively. More precisely, this works when k_{1} ≥ 2 or k_{2} ≥ 2, as appropriate, and otherwise these wedge products reduce to the product of a real number and an alternating multilinear form. If τ^{−1}(1) = 1 and k_{1} ≥ 2, then τ corresponds to a shuffle with respect to k_{1} − 1, k_{2} in a simple way, which can be used to obtain (9.14.10). This is easier when k_{1} = 1, so that (i_{k_{1}}(u))(μ_{1}) is a real number. Similarly, if τ^{−1}(1) = k_{1} + 1 and k_{2} ≥ 2, then τ corresponds to a shuffle with respect to k_{1}, k_{2} − 1 in a nice way. This can be used to obtain (9.14.11), because the shuffle with respect to k_{1}, k_{2} − 1 corresponding to τ has sign equal to (−1)^{k_{1}} sgn(τ). This is easier when k_{2} = 1, as before.

One may focus on finite-dimensional vector spaces, as on p54 of [184], which is often of primary interest anyway. One can also reduce to that case, by considering finite-dimensional linear subspaces of W.

9.15 More on interior multiplication

Let n, k, and r be positive integers, let U be a nonempty open subset of R^{n}, let α be a differential k-form on U that is r-times continuously differentiable on U, and let ξ be an r-times continuously-differentiable vector field on U. We can define (i_{k}(ξ))(α) (9.15.1)

as a differential (k − 1)-form on U using interior multiplication pointwise on U. This means that for each x ∈ U,

((i_{k}(ξ))(α))_{x} = (i_{k}(ξ(x)))(α_{x}), (9.15.2)

where the right side is defined as an alternating (k − 1)-linear form on R^{n} as in the previous section. One can check that (9.15.1) is r-times continuously differentiable on U as well. It is convenient to interpret (9.15.1) as being equal to 0 when k = 0, as before.

9.15.1 Some related linear mappings

Put L_{k,ξ}(α) = (i_{k+1}(ξ))(dα) + d((i_{k}(ξ))(α)), (9.15.3)

which is a differential k-form on U that is (r−1)-times continuously differentiable on U. If k = 0, then this is interpreted as being

L_{0,ξ}(α) = (i_{k}(ξ))(dα) = (dα)(ξ) = X_{ξ}(α). (9.15.4)

Let k′ be another positive integer, and let β be a differential k′-form on U that is r-times continuously differentiable on U. One can check that

L_{k+k′,ξ}(α ∧ β) = (L_{k,ξ}(α)) ∧ β + α ∧ (L_{k′,ξ}(β)), (9.15.5)

as differential (k + k′)-forms on U. This also works when k or k′ is 0, with the usual interpretations. This uses the anti-derivation properties of interior multiplication and exterior differentiation. This corresponds to a remark on p72 of [184], related to the proof of part (d) of Proposition 2.25 on p70 of [184].

Suppose that α is (r + 1)-times continuously differentiable on U, so that dα is a differential (k + 1)-form on U that is r-times continuously differentiable on U. In this case, L_{k+1,ξ}(dα) = d((i_{k+1}(ξ))(dα)) (9.15.6) on U, which also works when k = 0. If ξ is (r + 1)-times continuously differentiable on U too, then (9.15.1) is (r + 1)-times continuously differentiable on U as well, and d(L_{k,ξ}(α)) = d((i_{k+1}(ξ))(dα)) (9.15.7)

on U. This means that

d(L_{k,ξ}(α)) = L_{k+1,ξ}(dα) (9.15.8)

on U under these conditions. This corresponds to another remark on p72 of [184].

9.15.2 Connections with Lie derivatives

If f is an (r + 1)-times continuously-differentiable real-valued function on U, then df is an r-times continuously-differentiable differential 1-form on U, and

L_{1,ξ}(df) = d((i_{1}(ξ))(df)) = d((df)(ξ)) = d(X_{ξ}(f)) (9.15.9)

on U. One can check that L_{k,ξ}(α) = L_{ξ}(α) (9.15.10)

on U, where the right side is the Lie derivative of α, as in Section 9.5. This is the same as (9.15.4) when k = 0, and it follows from (9.15.9) when k = 1 and α = df, as in Subsection 9.10.2. One can reduce to these cases using (9.15.5) and the analogous statement for Lie derivatives, as in Subsection 9.10.1. This corresponds to part (d) of proposition 2.25 on p70 of [184].

Appendix A

Some linear and abstract algebra

A.1 Some remarks about vector spaces

Let V be a vector space over the real numbers. We shall not get into the abstract definition of a vector space too much here, but it is sometimes convenient to use the terminology. Basically V should be a set with a distinguished element called 0, and with operations of addition and scalar multiplication defined on V that satisfy some standard properties. These include commutativity and associativity of addition, and distributivity of scalar multiplication with respect to addition.

If n is a positive integer, then the space R^{n} of n-tuples of real numbers is a vector space over R with respect to coordinatewise addition and scalar multiplication. Similarly, if X is a nonempty set, then the space of all real-valued functions on X is a vector space over R with respect to pointwise addition and scalar multiplication.

If V is a vector space over the real numbers, then a linear subspace of V is a subset V_{0} of V that satisfies the following three conditions. First,

0 ∈ V_{0}. (A.1.1)

Second, if u, v ∈ V_{0}, then u + v ∈ V_{0}. (A.1.2)

Third, if v ∈ V_{0} and t ∈ R, then

t v ∈ V_{0}. (A.1.3)

Note that (A.1.3) implies (A.1.1) when V_{0} ≠ ∅, by taking t = 0.

Under these conditions, V_{0} is a vector space over the real numbers too, with respect to the restrictions of the operations of addition and scalar multiplication on V to V_{0}. Many of the vector spaces with which we shall be concerned here are linear subspaces of R^{n} for some n, or linear subspaces of the space of all real-valued functions on some nonempty set X.

Let X be a nonempty set again, and let W be a vector space over the real numbers. The space of all W -valued functions on X is a vector space over the real numbers with respect to pointwise addition and scalar multiplication. Thus linear subspaces of this space are vector spaces over R as well.

A.2 Algebras over the real numbers

Let A be a vector space over the real numbers, and let b be a bilinear mapping from A × A into A, as in Section 2.2. Under these conditions, (A, b) is said to be an algebra in the strict sense over the real numbers. This corresponds to the definition of an F -algebra, with F = R, in Section 1.3 on p4 of [86]. This also corresponds to the definition of a k-algebra, with k = R, on p2 of [160]. One may describe b as a bilinear operation on A as well, as in Definition 3.53 on p117 of [184].

If b(x, y) = b(y, x) (A.2.1)

for every x, y ∈ A, then (A, b) is said to be commutative as an algebra in the strict sense over R. If b(b(x, y), z) = b(x, b(y, z)) (A.2.2)

for every x, y, z ∈ A, then (A, b) is said to be an associative algebra over the real numbers.

An element e of A is said to be a multiplicative identity element with respect to b if b(e, x) = b(x, e) = x (A.2.3)

for every x ∈ A. It is easy to see that this is uique when it exists.

Let A_{0} be a linear subspace of A. If

b(x, y) ∈ A_{0} (A.2.4)

for every x, y ∈ A, then A_{0} is called a subalgebra of A with respect to b. This means that A_{0} is an algebra in the strict sense over the real numbers with respect to the restriction of b(x, y) to x, y ∈ A.

A.2.1 Algebra homomorphisms

Let (A_{1}, b_{1}) and (A_{2}, b_{2}) be algebras over the real numbers in the strict sense, and let ϕ be a linear mapping from A_{1} into A_{2}. We say that ϕ is a homomorphism from A_{1} into A_{2}, as algebras in the strict sense with respect to b_{1}, b_{2}, respectively, if ϕ(b_{1}(x, y)) = b_{2}(ϕ(x), ϕ(y)) (A.2.5)

for every x, y ∈ A_{1}.

If ϕ is a homomorphism from A_{1} into A_{2} with respect to b_{1} and b_{2}, respectively, then it is easy to see that

(A.2.6) ϕ(A_{1}) is a subalgebra of A_{2} with respect to b_{2}.

If A_{0} is a subalgebra of A with respect to b, then the obvious inclusion mapping from A_{0} into A is a homomorphism, with respect to b on A and its restriction to A_{0}.

Let (A_{3}, b_{3}) be another algebra in the strict sense over the real numbers. If ϕ is a homomorphism from A_{1} into A_{2} with respect to b_{1} and b_{2}, respectively, and ψ is a homomorphism from A_{2} into A_{3} with respect to b_{2} and b_{3}, respectively, then one can check that

(A.2.7) ψ ∘ ϕ is a homomorphism from A_{1} into A_{3},

with respect to b_{1} and b_{3}, respectively.

Suppose that ϕ is a homomorphism from A_{1} into A_{2} with respect to b_{1} and b_{2}, respectively, again. If ϕ is a one-to-one mapping from A_{1} onto A_{2}, then it is easy to see that the inverse mapping ϕ^{−1} is a homomorphism from A_{2} into A_{1} with respect to b_{2} and b_{1}, respectively. In this case, ϕ is said to be an isomorphism from A_{1} onto A_{2} with respect to b_{1} and b_{2}, respectively.

If ϕ is an isomorphism from A_{1} onto A_{2} with respect to b_{1} and b_{2}, respectively, and ψ is an isomorphism from A_{2} onto A_{3} with respect to b_{2} and b_{3}, respectively, then

(A.2.8) ψ ∘ ϕ is an isomorphism from A_{1} onto A_{3},

with respect to b_{1} and b_{3}, respectively.

An isomorphism from A onto itself with respect to b is said to be an automorphism of A with respect to b. Note that the identity mapping on A is automatically an automorphism of A with respect to b. The set of automorphisms of A with respect to b is a subgroup of the group of all one-to-one mappings from A onto itself, as in Section 2.6.

A linear mapping δ from A into itself is said to be a derivation with respect to b if δ(b(x, y)) = b(δ(x), y) + b(x, δ(y)) (A.2.9)

for every x, y ∈ A.

A.3 More on vector spaces

Let V and W be vector spaces over the real numbers, and let ϕ be a one-to-one linear mapping from V onto W. It is well known and easy to see that ϕ^{−1} is a linear mapping from W onto V in this case. One may say that ϕ is an isomorphism from V onto W as vector spaces over R under these conditions.

The space of one-to-one linear mappings from V onto itself is denoted GL(V). This is a subgroup of the group of all one-to-one mappings from V onto itself, as in Section 2.6. In particular, GL(V) is a group with respect to composition of mappings on V, which is known as the general linear group of V.

Let V^{∗} be the space of all linear functionals on V, which is to say the space of all linear mappings from V into R, as a vector space over itself. This is known as the dual of V, which is sometimes denoted V′. This is a linear subspace of the space of all real-valued functions on V, as a vector space over the real numbers with respect to pointwise addition and scalar multiplication. In particular, V^{∗} is a vector space over the real numbers.

If V has finite dimension, then it is well known that

dim V^{∗} = dim V. (A.3.1)

If n is a positive integer, then the n standard coordinate functions on R^{n} form a basis for the dual of R^{n}.

A.3.1 Dual linear mappings

Let T be a linear mapping from V into W. If λ is a linear functional on W, then T^{∗}(λ) = λ ∘ T (A.3.2)

is a linear functional on V. This defines a linear mapping T^{∗} from W^{∗} into V^{∗}, which is the dual linear mapping corresponding to T. This corresponds to the k = 1 case of an analogous definition in Section 2.3. If the dual spaces are denoted V′, W′, then one may use the notation T′ for the dual linear mapping associated to T.

Let Z be another vector space over the real numbers, and let R be a linear mapping from W into Z. One can check that

(R ∘ T)^{∗} = T^{∗} ∘ R^{∗}, (A.3.3)

as linear mappings from Z^{∗} into V^{∗}, as in Section 2.3. If T is a one-to-one linear mapping from V onto W, then T^{∗} is a one-to-one linear mapping from W^{∗} onto V^{∗}, with (T^{∗})^{−1} = (T^{−1})^{∗}, (A.3.4)

as before.

A.3.2 Direct sums and dual spaces

Let N be a positive integer, and let V_{1}, …, V_{N} be N vector spaces over the real numbers. The direct sum of the V_{j}’s may be defined as a vector space over R as in Section 2.1. Similarly, the direct sum

⨁_{j=1}^{n} V_{j}^{∗} (A.3.5)

of the corresponding dual spaces is defined as a vector space over R. The dual

( ⨁_{j=1}^{N} V_{j})^{∗} (A.3.6)

of ⨁_{j=1}^{N} V_{j} is defined as a vector space over R as well. There is a natural isomorphism between (A.3.5) and (A.3.6), as vector spaces over the real numbers, as follows.

If λ_{j} is a linear functional on V_{j} for each j = 1, …, N, then

v_{1} ⊕ ⋯ ⊕ v_{V} ↦ ∑_{j=1}^{N} λ_{j}(v_{j}) (A.3.7)

defines a linear functional on ⨁_{j=1}^{N} V_{j}. This defines a linear mapping from (A.3.5) into (A.3.6), which sends λ_{1} ⊕ ⋯ ⊕ λ_{N} to the linear functional (A.3.7). One can check that this mapping is a bijection. More precisely, let λ be a linear functional on ⨁_{j=1}^{N} V_{j}, and let ι_{l} be the natural mapping from V_{l} into ⨁_{j=1}^{N} V_{j} for each l = 1, …, N. Thus λ ∘ ι_{l} (A.3.8)

is a linear functional on V_{l} for each l = 1, …, N. This can be used to define a linear mapping from (A.3.6) into (A.3.5). It is easy to see that this is the inverse of the previous mapping.

A.4 Some direct sums and products

Let V_{j} be a vector space over the real numbers for each nonnegative integer j. Of course, one could restrict one’s attention to positive integers here, or use any nonempty set of indices. Consider the Cartesian product

∏_{j=0}^{∞} V_{j}, (A.4.1)

which is the set of infintie sequences {v_{j}}_{j=0}^{∞} with v_{j} ∈ V_{j} for each j ≥ 0. This is a vector space over the real numbers, with respect to coordinatewise addition and scalar multiplication. This is known as the direct product of the V_{j}’s, j ≥ 0.

The direct sum of the V_{j}’s, j ≥ 0, is defined by

(A.4.2) ⨁_{j=0}^{∞} V_{j} = {{v_{j}}_{j=0}^{∞} ∈ ∏_{j=0}^{∞} V_{j} : v_{j} = 0 for all but finitely many j}.

This is a linear subspace of the direct product. The direct sum of finitely many vector spaces over the real numbers, as in Section 2.1, may also be considered as a direct product.

If l is a nonnegative integer, then there is an obvious mapping ι_{l} from V_{l} into the direct sum (A.4.2). This mapping sends w_{l} ∈ V_{l} to the sequence whose lth term is equal to w_{l}, and whose jth term in V_{j} is equal to 0 when j ≠ l. This is a one-to-one linear mapping from V_{l} into (A.4.2) for each l ≥ 0, and one may wish to identify V_{l} with its image in the direct sum for each l. The direct sum is the same as the linear subspace of the direct product generated by the images of the V_{l}’s.

A.4.1 Sums, products, and dual spaces

Consider the direct product

∏_{j=0}^{∞} V_{j}^{∗} (A.4.3)

of the duals of the V_{j}’s. Let us also consider the dual

( ⨁_{j=0}^{∞} V_{j})^{∗} (A.4.4)

of the V_{j}’s. There is a natural isomorphism between (A.4.3) and (A.4.4), as vector spaces over the real numbers, which is analogous to the corresponding statement for finite direct sums in Subsection A.3.2. If λ_{j} is a linear functional on V_{j} for each j, v_{j} ∈ V_{j} for each j, and v_{j} = 0 for all but finitely many j, then

(A.4.5) λ_{j}(v_{j}) = 0 for all but finitely many j.

This means that

∑_{j=0}^{∞} λ_{j}(v_{j}) (A.4.6)

may be defined as a real number, which is the same as

∑_{j=0}^{n} λ_{j}(v_{j}) (A.4.7)

when n is sufficiently large. This defines a linear functional on ⨁_{j=0}^{∞} V_{j}. Using this, we get a linear mapping from (A.4.3) into (A.4.4).

One can check that this mapping is a bijection. More precisely, if λ is a linear functional on ⨁_{j=0}^{∞} V_{j}, then

λ ∘ ι_{l} (A.4.8)

is a linear functional on V_{l} for each l ≥ 0. This leads to a linear mapping from (A.4.4) into (A.4.3). This is the inverse of the mapping described in the preceding paragraph, as before.

A.5 Some remarks about linear mappings

If V and W are vector spaces over the real numbers, then let

L(V, W) (A.5.1)

be the space of all linear mappings from V into W. This is a linear subspace of the space of all W -valued functions on V, with respect to pointwise addition and scalar multiplication of functions. If V and W have finite dimension, then it is well known that the dimension of L(V, W) is given by

(A.5.2) dim L(V, W) = (dim V) (dim W).

Indeed, one can use bases for V and W to describe linear mappings from V into W in terms of matrices to get (A.5.2).

Suppose for the moment that W = R, considered as a one-dimensional vector space over itself. In this case, L(V, W) = L(V, R) is the same as the dual space V^{∗} of linear functionals on V, as in Section A.3.

A.5.1 Linear mappings and direct sums

Let V_{1}, …, V_{N} be finitely many vector spaces over the real numbers. If T_{j} is a linear mapping from V_{j} into W for each j = 1, …, N, then one can get a linear mapping T from ⨁_{j=1}^{N} V_{j} into W as in Subsection 2.1.1. Conversely, every linear mapping from ⨁_{j=1}^{N} V_{j} into W corresponds to unique linear mappings from V_{j} into W, 1 ≤ j ≤ N, in this way, as before. This leads to an isomorphism from

⨁_{j=1}^{N} L(V_{j}, W) (A.5.3)

onto

L( ⨁_{j=1}^{N} V_{j}, W ), (A.5.4)

as vector spaces over the real numbers.

Similarly, let W_{1}, …, W_{M} be finitely many vector spaces over the real numbers. If R_{l} is a linear mapping from V into W_{l} for each l = 1, …, M, then

R(v) = R_{1}(v) ⊕ ⋯ ⊕ R_{M}(v) (A.5.5)

defines a linear mapping from V into ⨁_{l=1}^{M} W_{l}. Conversely, it is easy to see that every linear mapping R from V into ⨁_{l=1}^{M} W_{l} corresponds to unique linear mappings from V into W_{l}, 1 ≤ l ≤ M, in this way. This leads to an isomorphism from

⨁_{l=1}^{M} L(V, W_{l}) (A.5.6)

onto

L(V, ⨁_{l=1}^{M} W_{l}), (A.5.7)

as vector spaces over the real numbers.

A.5.2 Linear mappings and direct products

Now let V_{j} be a vector space over the real numbers for each nonnegative integer j. If T_{j} is a linear mapping from V_{j} into W for each j ≥ 0, then one can get a linear mapping T from ⨁_{j=0}^{∞} V_{j} into W in basically the same way as before. This defines a linear mapping from

∏_{j=0}^{∞} L(V_{j}, W) (A.5.8)

into

L( ⨁_{j=0}^{∞} V_{j}, W ). (A.5.9)

Conversely, if T is any linear mapping from ⨁_{j=0}^{∞} V_{j} into W, then one can get a linear mapping T_{l} from V_{l} into W for each l ≥ 0, by composing the standard embedding of V_{l} into ⨁_{j=0}^{∞} V_{j} with T. This defines a linear mapping from (A.5.9) into (A.5.8), which is the inverse of the previous mapping.

Similarly, let W_{l} be a vector space over the real numbers for every nonnegative integer l. If R_{l} is a linear mapping from V into W_{l} for each l ≥ 0, then we get a linear mapping R from V into ∏_{l=0}^{∞} W_{l}, where the lth coordinate of R(v) is equal to R_{l}(v) for every v ∈ V and l ≥ 0. This defines a linear mapping from

∏_{l=0}^{∞} L(V, W_{l}) (A.5.10)

into

L(V, ∏_{l=0}^{∞} W_{l}). (A.5.11)

Conversely, if R is any linear mapping from V into ∏_{l=0}^{∞} W_{l}, then one can get a linear mapping R_{k} from V into W_{k} for each k ≥ 0, by composing R with the kth standard coordinate projection from ∏_{l=0}^{∞} W_{l} onto W_{k}. This defines a linear mapping from (A.5.11) into (A.5.10), which is the inverse of the previous mapping.

A.5.3 Some more direct sums

Of course,

⨁_{l=0}^{∞} L(V, W_{l}) (A.5.12)

is a linear subspace of (A.5.10), as in the previous section. We also have that

L(V, ⨁_{l=0}^{∞} W_{l}) (A.5.13)

is a linear subspace of (A.5.11), because ⨁_{l=0}^{∞} W_{l} is a linear subspace of ∏_{l=0}^{∞} W_{l}. The restriction of the linear mapping from (A.5.10) into (A.5.11) defined in the preceding paragraph is a one-to-one linear mapping from (A.5.12) into (A.5.13).

Let R be a linear mapping from V into ⨁_{l=0}^{∞} W_{l}, and let R_{k} be the composition of R with the kth standard coordinate projection from ⨁_{l=0}^{∞} W_{l} onto W_{k} for each k ≥ 0. If v ∈ V, then

(A.5.14) R_{k}(v) = 0 for all but finitely many k ≥ 0,

by the definition of ⨁_{l=0}^{∞} W_{l}. If V has finite dimension, then one can check that

(A.5.15) R_{k} = 0 for all but finitely many k ≥ 0.

This means that the linear mapping from (A.5.12) into (A.5.13) defined in the previous paragraph is surjective when V has finite dimension.

A.6 The algebra of multilinear forms

Let W be a vector space over the real numbers. Let us define M(W) initially as a vector space over the real numbers by

M(W) = ⨁_{k=0}^{∞} M_{k}(W), (A.6.1)

where the right side is as in Section A.4. We shall normally identify elements of M_{l}(W) with their images in M(W) under the embeddings defined before, so that every element of M(W) corresponds to a finite sum of elements of M_{l}(W), l ≥ 0.

If μ, ν ∈ M(W), then we can define

μ ⊗ ν (A.6.2)

as an element of M(W) in an obvious way. More precisely, if μ ∈ M_{l_{1}}(W) and ν ∈ M_{l_{2}}(W) for some positive integers l_{1}, l_{2}, then (A.6.2) may be defined as an element of M_{l_{1}+l_{2}}(W) as in Section 2.8. The cases where l_{1} or l_{2} is equal to 0 may be included as in Section 2.12. Otherwise, μ and ν correspond to finite sums of multilinear l_{1}, l_{2}-linear forms on W, l_{1}, l_{2} ≥ 0, and (A.6.2) may be defined as the corresponding sum of finitely many (l_{1} + l_{2})-linear forms on W.

This defines a bilinear mapping from M(W) × M(W) into M(W). This makes M(W) into an associative algebra over the real numbers, because of the associativity property of products of multilinear forms on W mentioned in Subsection 2.8.1. Remember that we take M_{0}(W) = R, as in Section 2.12, which corresponds to a subalgebra of M(W). In fact, the real number 1 corresponds to the multiplicative identity element of M(W) in this way.

A.6.1 The definition of SM(W)

Similarly, we can define SM(W) as a vector space over the real numbers by

SM(W) = ⨁_{k=0}^{∞} SM_{k}(W), (A.6.3)

which is a linear subspace of M(W). Using the symmetrization mapping S_{k} from M_{k}(W) onto SM_{k}(W) defined for each k ≥ 1 in Subsection 2.4.1, we get a symmetrization mapping S from M(W) onto SM(W). More precisely, we can take S_{k} to be the identity mapping on M_{0}(W) = SM_{0}(W) = R when k = 0, as in Section 2.12.

If μ, ν ∈ SM(W), then we can define

μ ⊙ ν (A.6.4)

in an obvious way, to get a bilinear mapping from SM(W) × SM(W) into SM(W). This uses the bilinear mapping from SM_{l_{1}}(W) × SM_{l_{2}}(W) into SM_{l_{1}+l_{2}}(W) defined in Section 2.9. This makes SM(W) into a commutative associative algebra over the real numbers, because of the analogous properties discussed in Section 2.9. Remember that SM_{0}(R) = R, as in Section 2.12, which corresponds to a subalgebra of SM(W). The real number 1 corresponds to the multiplicative identity element of SM(W), as before.

If μ, ν ∈ M(W), then

S(μ ⊗ ν) = S(S(μ) ⊗ S(ν)) = S(μ) ⊙ S(ν). (A.6.5)

The first step corresponds to a property of multilinear forms mentioned in Subsection 2.9.1, and the second step follows from the definition of the right side. This shows that S is an algebra homomorphism from M(W) onto SM(W).

A.6.2 Polynomials and M(R^{n})

If W = R^{n} for some positive integer n and μ ∈ M(R^{n}), then we can define

P_{μ} (A.6.6)

as a polynomial on R^{n} with real coefficients, using the earlier definition for multilinear forms on R^{n}. If ν ∈ M(R^{n}) too, then

P_{μ⊗ν} = P_{μ} P_{ν}, (A.6.7)

because of the analogous property of multilinear forms on R^{n} mentioned in Subsection 2.8.1. Thus μ ↦ P_{μ} defines an algebra homomorphism from M(R^{n}) onto the space P(R^{n}) of all polynomials on R^{n} with real coefficients, with respect to pointwise multiplication of polynomials on R^{n}.

Similarly, the restriction of μ → P_{μ} to SM(R^{n}) defines an algebra isomorphism onto P(R^{n}), with respect to ⊙ on SM(R^{n}). This uses some of the remarks in Section 1.10.

A.7 More on AM(W)

Let W be a vector space over the real numbers again. Let us define AM(W) initially as a vector space over the real numbers by

AM(W) = ⨁_{k=0}^{∞} AM_{k}(W), (A.7.1)

where the right side is as in Section A.4. If W has dimension n for some positive integer n, then AM_{k}(W) = {0} when k > n, and this is essentially the same as the definition of AM(W) in Subsection 2.12.1. We can use the alternatization mapping A_{k} from M_{k}(W) onto AM_{k}(W) defined for k ≥ 1 in Subsection 2.4.1 to get an alternatization mapping A from M(W) onto AM(W). As in Section 2.12, we take A_{0} to be the the identity mapping on M_{0}(W) = AM_{0}(W) = R.

If μ, ν ∈ AM(W), then we can define

μ ∧_{0} ν (A.7.2)

as an element of AM(W) in an obvious way, using the analogous definition for alternating multilinear forms on W in Subsection 2.10.1. This defines a bilinear mapping from AM(W) × AM(W) into AM(W). This makes AM(W) into an associative algebra over the real numbers, because of the associativity property of ∧_{0} mentioned in Subsection 2.10.1. As before, AM_{0}(W) = R corresponds to a subalgebra of AM(W) with respect to ∧_{0}, and the real number 1 corresponds to the multiplicative identity element in AM(W) with respect to ∧_{0}.

If μ, ν ∈ M(W), then

A(μ ⊗ ν) = A(A(μ) ⊗ A(ν)) = A(μ) ∧_{0} A(ν), (A.7.3)

where the first step corresponds to a property of multilinear forms mentioned in Section 2.10, and the second step follows from the definition of the right side. This means that A is an algebra homomorphism from M(W) onto AM(W), with respect to ∧_{0} on AM(W).

If μ, ν ∈ AM(W), then we can define

μ ∧ ν (A.7.4)

as an element of AM(W) in an obvious way, using the analogous definition for alternating multilinear forms on Section 2.11. This defines a bilinear mapping from AM(W) × AM(W) into AM(W), which makes AM(W) into an associative algebra over the real numbers, because of the associativity property mentioned in Section 2.11. Note that AM_{0}(W) = R also corresponds to a subalgebra of AM(W) with respect to ∧, and that the real number 1 corresponds to the multiplicative identity element in AM(W) with respect to ∧.

If μ ∈ M_{k}(W) for some nonnegative integer k, then put

Ã_{k}(μ) = k! A_{k}(μ). (A.7.5)

Using these mappings, we get a linear mapping Ã from M(W) onto AM(W).

If μ, ν are k_{1}, k_{2}-linear forms on W for some nonnegative integers k_{1}, k_{2}, then μ ⊗ ν is a (k_{1} + k_{2})-linear form on W, and we have that

Ã_{k_{1}+k_{2}}(μ ⊗ ν) = (k_{1} + k_{2})! A_{k_{1}+k_{2}}(μ ⊗ ν) = (k_{1} + k_{2})! A_{k_{1}}(μ) ∧_{0} A_{k_{2}}(ν)
(A.7.6)  = \frac{(k_{1} + k_{2})!}{k_{1}! k_{2}!} Ã_{k_{1}}(μ) ∧_{o} Ã_{k_{2}}(ν) = Ã_{k_{1}}(μ) ∧ Ã_{k_{2}}(ν),

using (A.7.3) in the second step, and the definition of ∧ in the fourth step. This implies that Ã(μ ⊗ ν) = Ã(μ) ∧ Ã(ν) (A.7.7)

for all μ, ν ∈ M(W). Thus Ã is an algebra homomorphism from M(W) onto AM(W), with respect to ∧ on AM(W).

A.8 More on inner products

Let W be a vector space over the real numbers with an inner product ⟨·, ·⟩_{W}, and let ∥ · ∥_{W} be the corresponding norm on W, as in Section 3.3. We say that u, w ∈ W are orthogonal with respect to ⟨·, ·⟩_{W} if

⟨u, w⟩_{W} = 0. (A.8.1)

It is easy to see that this happens if and only if

∥u + w∥_{W}^{2} = ∥u∥_{W}^{2} + ∥w∥_{W}^{2} . (A.8.2)

Let n be a positive integer, and let w_{1}, …, w_{n} be n pairwise-orthogonal vectors in W. We say that w_{1}, …, w_{n} ∈ W are orthonormal with respect to ⟨·, ·⟩_{W} if ∥w_{j}∥_{W} = 1 (A.8.3)

for each j = 1, …, n. It is well known and not difficult to check that this implies that w_{1}, …, w_{n} are linearly independent in W. Note that the standard basis vectors e_{1}, …, e_{n} in R^{n} are orthonormal with respect to the standard inner product ⟨·, ·⟩_{R^{n}} on R^{n}.

If W has finite dimension, then it is well known that W has an orthonormal basis. In fact, an orthonormal basis for W can be obtained from any basis for W using the Gram–Schmidt process.

Let V be a vector space over the real numbers with an inner product ⟨·, ·⟩_{V} and associated norm ∥ · ∥_{V}. Also let T be a linear mapping from V into W, and remember that T is an isometry with respect to these norms if and only if T preserves the inner products, as in Section 3.15. Suppose that V has dimension n, and let v_{1}, …, v_{n} be an orthonormal basis for V. One can check that T is an isometry if and only if

(A.8.4) T (v_{1}), …, T (v_{n}) are orthonormal in W with respect to ⟨·, ·⟩_{W} .

Any sequence of n vectors in W may be realized as

L(e_{1}), …, L(e_{n}), (A.8.5)

where L is a linear mapping from R^{n} into W. These vectors are linearly independent in W exactly when L is one-to-one, and these vectors span W exactly when L maps R^{n} onto W. If W has dimension n, then we can use an orthonormal basis for W to get an isometric linear mapping from R^{n} onto W, with respect to the standard inner product and norm on R^{n}.

A.9 Inner products and adjoints

Let V be a vector space over the real numbers with an inner product ⟨·, ·⟩_{V}. If v ∈ V, then λ_{v}(u) = ⟨u, v⟩_{V} (A.9.1)

defines a linear functional on V. It is easy to see that

v ↦ λ_{v} (A.9.2)

defines a linear mapping from V into the dual space of linear functionals on V. Note that λ_{v}(v) = ⟨v, v⟩_{V} (A.9.3)

is equal to 0 if and only if v = 0. This implies that (A.9.2) is one-to-one.

If V has finite dimension, then it is well known that every linear functional on V may be expressed as in (A.9.1) for a unique v ∈ V. Equivalently, this means that (A.9.2) is a one-to-one linear mapping from V onto the dual space of V. This can be verified directly when V = R^{n} for some positive integer n, with the standard inner product. Otherwise, one can use an orthonormal basis for V to reduce to that case, or to essentially the same argument. Alternatively, one can use the fact that the dimension of the dual space of V is the same as the dimension of V when the dimension of V is finite, as in Section A.3.

Let W be another vector space over the real numbers with an inner product ⟨·, ·⟩_{W}, and let T be a linear mapping from V into W. If w ∈ W, then

μ_{w}(u) = ⟨T (u), w⟩_{W} (A.9.4)

defines a linear functional on V. Suppose that V has finite dimension, so that there is a unique element T^{∗}(w) of V such that

⟨T (u), w⟩_{W} = ⟨u, T^{∗}(w)⟩_{V} (A.9.5)

for every u ∈ V, as in the preceding paragraph. One can check that T^{∗} is a linear mapping from W into V. This is called the adjoint of T with respect to the inner products on V and W.

This is not quite the same as the dual linear mapping associated to T as in Subsection A.3.1, although we are using the same notation here. Sometimes one may use other notation for dual linear spaces and the corresponding dual linear mappings, such as V′, W′, and T′, to distinguish the dual linear mapping from the adjoint. If W also has finite dimension, then one can characterize linear functionals on W in terms of the inner product in the same way, so that the dual linear mapping corresponds more closely to the adjoint.

If W has finite dimension too, then one can use orthonormal bases for V and W to characterize T in terms of a matrix. In this case, T^{∗} corresponds to the transpose of this matrix, with respect to the same orthonormal bases for V and W.

A.9.1 Some properties of the adjoint

Let Z be a third vector space over the real numbers with an inner product ⟨·, ·⟩_{Z}, and let R be a linear mapping from W into Z. If V and W have finite dimension, then the adjoints R^{∗}, (R ∘ T)^{∗} of R, R ∘ T may be defined as linear mappings from Z into W, V, respectively, as before. In particular,

⟨R(w), z⟩_{Z} = ⟨w, R^{∗}(z)⟩_{W} (A.9.6)

for every w ∈ W and z ∈ Z. If u ∈ V and z ∈ Z, then

⟨R(T (u)), z⟩_{Z} = ⟨T (u), R^{∗}(z)⟩_{W} = ⟨u, T^{∗}(R^{∗}(z))⟩_{V} . (A.9.7)

This implies that (R ∘ T)^{∗} = T^{∗} ∘ R^{∗}, (A.9.8)

as linear mappings from Z into V.

Remember that T is an isometric linear mapping from V into W, with respect to the norms associated to the inner products, if and only if

⟨T (u), T (v)⟩_{W} = ⟨u, v⟩_{V} (A.9.9)

for every u, v ∈ V, as in Section 3.15. If V has finite dimension, then this is the same as saying that ⟨u, T^{∗}(T (v))⟩_{V} = ⟨u, v⟩_{V} (A.9.10)

for every u, v ∈ V. One can check that this holds if and only if

T^{∗} ∘ T = I_{V} , (A.9.11)

where I_{V} is the identity mapping on V.

If V has finite dimension and T is a one-to-one linear mapping from V onto itself, then T^{∗} ∘ (T^{−1})^{∗} = (T^{−1} ∘ T)^{∗} = I_{V} ∗ = I_{V} , (A.9.12)

using (A.9.8) in the first step. Similarly,

(T^{−1})^{∗} ∘ T^{∗} = (T ∘ T^{−1})^{∗} = I_{W}^{∗} = I_{W} , (A.9.13) where I_{W} is the identity mapping on W. This implies that T^{∗} is a one-to-one linear mapping from W onto V, with

(T^{∗})^{−1} = (T^{−1})^{∗}, (A.9.14)

as linear mappings from V into W, using (A.9.8). Note that T is an isometric linear mapping from V onto W if and only if

T^{−1} = T^{∗}, (A.9.15)

because of (A.9.11). This is related to a remark in Section 4.12.

A.10 Nonnegativity and self-adjointness

Let n be a positive integer, and let V be an n-dimensional vector space over the real numbers with an inner product ⟨·, ·⟩_{V} and its associated norm ∥ · ∥_{V}. Of course, this includes R^{n} with its standard inner product and norm, and one can often reduce to this case using an isometric linear mapping from R^{n} onto V. This basically corresponds to choosing an orthonormal basis for V, as before.

A linear mapping B from V into itself is said to be self-adjoint if

B^{∗} = B. (A.10.1)

This means that the matrix associated to B with respect to an orthonormal basis of V is symmetric, in the sense that it is equal to its transpose.

Under these conditions, it is well known that

(A.10.2) there is an orthonormal basis for V consisting of eigenvectors for B.

This is often stated for real symmetric matrices, or self-adjoint linear mappings from R^{n} into itself, and one can reduce to that case.

A self-adjoint linear mapping B from V into itself is said to be nonnegative if ⟨B(v), v⟩_{V} ≥ 0 (A.10.3)

for every v ∈ V. One can check that this implies that the eigenvalues of B are nonnegative real numbers. Conversely, if the eigenvalues of B are nonnegative real numbers, then one can use diagonalization of B mentioned in the preceding paragraph to get that B is nonnegative in this sense. We also get that

det B ≥ 0 (A.10.4)

when B is nonnegative, because the determinant of B is equal to the product of its eigenvalues.

A.10.1 A family of examples

Let W be another vector space over the real numbers with an inner product ⟨·, ·⟩_{W} and associated norm ∥ · ∥_{W}, and let T be a linear mapping from V into W. Let us check that (T^{∗})^{∗} = T, (A.10.5)

as linear mappings from V into W. If u ∈ V and w ∈ W, then

⟨T (u), w⟩_{W} = ⟨u, T^{∗}(w)⟩_{V} = ⟨(T^{∗})^{∗}(u), w⟩_{W} , (A.10.6)

using the definition of T^{∗} in the first step, and the definition of (T^{∗})^{∗} in the second step. This implies (A.10.5).

Note that T^{∗} ∘ T is a linear mapping from V into itself. This mapping is self-adjoint, because

(T^{∗} ∘ T)^{∗} = T^{∗} ∘ (T^{∗})^{∗} = T^{∗} ∘ T. (A.10.7)

If v ∈ V, then

⟨(T^{∗} ∘ T)(v), v⟩_{V} = ⟨T^{∗}(T (v)), v⟩_{V} = ⟨T (v), T (v)⟩_{W} = ∥T (v)∥_{W}^{2} . (A.10.8)

Thus T^{∗} ∘ T is nonnegative as a self-adjoint linear mapping from V into itself. This implies that det(T^{∗} ∘ T) ≥ 0, (A.10.9)

as before.

Of course, if V = W, then T and T^{∗} are linear mappings from V into itself, and (A.10.10) det(T^{∗} ∘ T) = (det T^{∗}) (det T).

If ⟨·, ·⟩_{V} = ⟨·, ·⟩_{W} too, then det T^{∗} = det T. (A.10.11)

This is because the matrix associated to T^{∗} with respect to an orthonormal basis for V is the same as the transpose of the matrix associated to T with respect to that basis. This implies that

det(T^{∗} ∘ T) = (det T)^{2} (A.10.12)

in this case. This is another way to get (A.10.9) under these conditions.

A.11 Adjoints and volumes

Let n be a positive integer, and let W be an n-dimensional vector space over the real numbers with an inner product ⟨·, ·⟩_{W} and associated norm ∥ · ∥_{W}. We can use an orthonormal basis for W to get an isometric linear mapping L from R^{n} onto W, as in Section A.8, with respect to the standard inner product ⟨·, ·⟩_{R^{n}} and norm ∥ · ∥_{R^{n}} on R^{n}. If E is a reasonably nice subset of W, then put

Λ_{W} (E) = Vol_{n}(L^{−1}(E)), (A.11.1)

234 APPENDIX A. SOME LINEAR AND ABSTRACT ALGEBRA as in Section 4.15. Remember that this depends only on the inner product on W, and not the particular choice of L.

Let T be a linear mapping from R^{n} into W. If E_{0} is a reasonably nice subset of R^{n}, then (A.11.2) Λ_{W} (T (E_{0})) = Vol_{n}(L^{−1}(T (E_{0}))) = | det(L^{−1} ∘ T)| Vol_{n}(E_{0}), where the second step is as in Section 4.12. In particular, if e_{1}, …, e_{n} is the standard basis in R^{n}, then P_{W} (T (e_{1}), …, T (e_{n})) = T (P_{R^{n}}(e_{1}, …, e_{n})) = T ([0, 1]^{n}), (A.11.3) as in Section 4.14. If we take E_{0} = [0, 1]^{n} in (A.11.2), then we get that Λ_{W} (P_{W} (T (e_{1}), …, T (e_{n}))) = | det(L^{−1} ∘ T)|. (A.11.4)

Observe that (L^{−1} ∘ T)^{∗} = T^{∗} ∘ (L^{−1})^{∗} = T^{∗} ∘ L, (A.11.5) where the second step uses the fact that L is an isometry from R^{n} onto W. This means that (L^{−1} ∘ T)^{∗} ∘ (L^{−1} ∘ T) = T^{∗} ∘ L ∘ L^{−1} ∘ T = T^{∗} ∘ T. (A.11.6) It follows that (A.11.7) det(T^{∗} ∘ T) = det((L^{−1} ∘ T)^{∗} ∘ (L^{−1} ∘ T)) = (det(L^{−1} ∘ T))^{2}, where the second step is as in (A.10.12). This shows that | det(L^{−1} ∘ T)| = (det(T^{∗} ∘ T))^{1/2}. (A.11.8)

If E_{0} is a reasonably nice subset of R^{n}, then we get that (A.11.9) Λ_{W} (T (E_{0})) = (det(T^{∗} ∘ T))^{1/2} Vol_{n}(E_{0}), by (A.11.2). In particular, Λ_{W} (P_{W} (T (e_{1}), …, T (e_{n}))) = (det(T^{∗} ∘ T))^{1/2}, (A.11.10) by (A.11.4). This is very close to Theorem 7 on p328 of [20], and we shall return to that in a moment.

A.11.1 Another inner product space

Let Z be another vector space over the real numbers with an inner product ⟨·, ·⟩_{Z}. Suppose that W is a linear subspace of Z, and that ⟨·, ·⟩_{W} is the same as the restriction of ⟨·, ·⟩_{Z} to W. Let T_{1} be the same as T, considered as a linear mapping from R^{n} into Z. The adjoint T_{1}^{∗} of T_{1} may be defined as before, as a linear mapping from Z into R^{n}. Let us check that

T_{1}^{∗} ∘ T_{1} = T^{∗} ∘ T, (A.11.11)

as linear mappings from R^{n} into itself.

If u, v ∈ R^{n}, then

⟨(T^{∗} ∘ T)(u), v⟩_{R^{n}} = ⟨T^{∗}(T (u)), v⟩_{R^{n}} = ⟨T (u), T (v)⟩_{W}
(A.11.12)  = ⟨T_{1}(u), T_{1}(v)⟩_{Z} = ⟨T_{1}^{∗}(T_{1}(u)), v⟩_{R^{n}}
= ⟨(T_{1}^{∗} ∘ T_{1})(u), v⟩_{R^{n}}.

This implies (A.11.11). Of course, it follows that

det(T_{1}^{∗} ∘ T_{1}) = det(T^{∗} ∘ T). (A.11.13)

In Theorem 7 on p328 of [20], one basically takes Z to be another Euclidean space, with the standard inner product. It is also stated in terms of the matrix associated to T_{1}.

More precisely, the volume of a parallelepiped in a Euclidean space is defined on p328 of [20], and Theorem 7 there expresses the square of the volume as a determinant. This is used to determine the effect of a linear mapping from R^{n} into itself on the n-dimensional volumes of parallelepipeds or other subsets of R^{n}.

If W is an n-dimensional linear subspace of a Euclidean space, then the restriction of the standard inner product on the Euclidean space to W defines an inner product on W. Remember that an admissible volume on W is determined by its value at a parallelepiped corresponding to a basis for W, as in Subsection 4.14.1. If the n-dimensional volume of a parallelepiped in a Euclidean space has already been defined, then one can use that to determine Λ_{W}.

A.12 Traces and transposes of matrices

Let m and n be positive integers, and let a = (a_{j,l})_{j,l=1}^{m,n} be an m×n matrix of real numbers. The transpose of a is defined to be the n × m matrix a^{t} = (a_{l,j}^{t})_{l,j=1}^{n,m} of real numbers with a_{l,j}^{t} = a_{j,l} (A.12.1)

for each j = 1, …, m and l = 1, …, n. If m = n, then the trace of a is defined as usual by

tr a = ∑_{j=1}^{n} a_{j,j}. (A.12.2)

This defines a linear functional on the space M_{n,n}(R) of n × n matrices with entries in R. Clearly tr a^{t} = tr a (A.12.3) in this case.

Let r be another positive integer, and let b be an n×r matrix of real numbers. If a is an m × n matrix of real numbers again, then a b is defined as an m × r matrix of real numbers, by matrix multiplication. It is well known and not difficult to check that (a b)^{t} = b^{t} a^{t}, (A.12.4)

as r × m matrices of real numbers.

Suppose that m = r, so that a b is an m × m matrix of real numbers, and b a is an n × n matrix of real numbers. Under these conditions, it is well known and not difficult to verify that

tr(a b) = tr(b a) = ∑_{j=1}^{m} ∑_{l=1}^{n} a_{j,l} b_{l,j}. (A.12.5)

Let I_{n} be the identity matrix in M_{n,n}(R), whose diagonal entries are equal to 1, and whose other entries are equal to 0. If a ∈ M_{n,n}(R) and t ∈ R, then put p_{a}(t) = det(I_{n} + t a), (A.12.6)

which is a polynomial in t with real coefficients. It is well known and not too difficult to check that p′_{a}(0) = tr a. (A.12.7)

A.12.1 Traces of linear mappings

Let V be a vector space over the real numbers of dimension n, and let v_{1}, …, v_{n} be a basis for V. If T is a linear mapping from V into itself, then T corresponds to an n × n matrix c(T) of real numbers with respect to this basis for V in a standard way. The trace of T may be defined as a real number by

tr T = tr c(T). (A.12.8)

It is well known that this does not depend on the basis for V, and we shall return to this in a moment.

Let W be a vector space over the real numbers of dimension m, and let w_{1}, …, w_{m} be a basis for W. One can use this basis to define the trace of a linear mapping from W into itself, as in the preceding paragraph. Let A be a linear mapping from V into W, and let B be a linear mapping from W into V. Using the bases for V and W, we can associate m × n and n × m matrices a and b of real numbers to A and B, respectively.

Of course, A ∘ B is a linear mapping from W into itself, and B ∘ A is a linear mapping from V into itself. These linear mappings correspond to the product matrices a b and b a, respectively, with respect to the given bases on V and W. It follows that tr(A ∘ B) = tr(B ∘ A), (A.12.9)

because of (A.12.5).

In particular, this holds with V = W, and v_{j} = w_{j}. If R is a one-to-one linear mapping from V onto itself, then it follows that

tr(R ∘ T ∘ R^{−1}) = tr T. (A.12.10)

One can use this to get that (A.12.8) does not depend on the particular basis for V, by standard arguments.

Appendix B

Lie algebras

B.1 The definition and some examples

Let A be a vector space over the real numbers, and let [·, ·] = [·, ·]_{A} be a bilinear mapping from A×A into A. Thus, if x, y ∈ A, then [x, y] = [x, y]_{A} is an element of A too, and this is linear in each of x and y. We say that A is a Lie algebra over the real numbers with respect to [·, ·] if the following two additional conditions hold, as in Definition 3.4 on p84 of [184]. The first condition is that

[x, y] = −[y, x] (B.1.1)

for every x, y ∈ A. The second condition is that

[[x, y], z] + [[y, z], x] + [[z, x], y] = 0 (B.1.2)

for every x, y, z ∈ A, which is known as the Jacobi identity.

Sometimes the Jacobi identity is formulated as saying that

[x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0 (B.1.3)

for every x, y, z ∈ A. It is easy to see that this is equivalent to (B.1.2) when the first condition (B.1.1) holds.

Of course, (B.1.1) implies that

[x, x] = 0 (B.1.4)

for every x ∈ A. Conversely, if (B.1.4) holds for every x ∈ A, then one can check that (B.1.1) holds for every x, y ∈ A, as in Section 1.7. The definition of a Lie algebra is often formulated using (B.1.4) instead of (B.1.1), as on p1 of [86], and Definition 1 on p2 of [160]. These formulations are used more broadly, where (B.1.4) implies (B.1.1), but the converse may not hold.

If A is a Lie algebra with respect to [·, ·], then [·, ·] may be called the Lie bracket on A. Of course, a Lie algebra over R is a particular type of algebra over R in the strict sense, as in Section A.2. In particular, the notions of homomorphisms and isomorphisms between Lie algebras may be defined in the same way as before.

B.1.1 Some examples of Lie algebras

Suppose for the moment that A is an associative algebra over the real numbers, where multiplication of x, y ∈ A is expressed as x y. In this case, one can verify that A is a Lie algebra with respect to

[x, y] = x y − y x. (B.1.5)

This corresponds to Example (iii) on p2 of [160].

Let V be a vector space over the real numbers, and let

L(V) = L(V, V) (B.1.6)

be the space of linear mappings from V into itself. This is a vector space over the real numbers, as in Section A.5. In fact, L(V) is an associative algebra over R, with composition of linear mappings as multiplication.

If R, T ∈ L(V), then put

[R, T] = [R, T]_{L(V)} = R ∘ T − T ∘ R, (B.1.7)

which is another element of L(V). As before, L(V) is a Lie algebra over R with respect to (B.1.7). This is known as the general linear algebra associated to V, and denoted gl(V), (B.1.8)

as on p2 of [86]. Of course, this is the same as L(V) as a vector space over R.

Let n be a positive integer, and let M_{n,n}(R) be the space of n × n matrices with entries in R, as in Sections 5.1 and 5.2. This is a vector space over the real numbers with respect to entrywise addition and scalar multiplication, and an associative algebra over R with respect to matrix multiplication, as before. The general linear algebra gl(n, R) (B.1.9)

of n×n matrices with entries in R is the same as M_{n,n}(R) as a vector space over the real numbers, considered as a Lie algebra with respect to the corresponding commutator bracket (B.1.5). This is example 3.5 (c) on p84 of [184].

There is a standard one-to-one correspondence between linear mappings from R^{n} into itself, and n × n matrices of real numbers, using the standard basis for R^{n}. This defines an isomorphism between L(R^{n}) and M_{n,n}(R), as vector spaces over R, and associative algebras over R. This also defines an isomorphism between gl(R^{n}) and gl(n, R), as Lie algebras over R.

If A is any vector space over the real numbers, then A is a Lie algebra with respect to the Lie bracket defined by putting

[x, y] = 0 (B.1.10)

for every x, y ∈ A. In this case, we may say that A is abelian or commutative as a Lie algebra, as in Section 1.4 on p4 of [86], Example (ii) on p2 of [160], and example 3.5 (b) on p84 of [184].

If (A, [·, ·]) is any Lie algebra over R, then A is commutative as an algebra in the strict sense if and only if

[x, y] = [y, x] (B.1.11)

for every x, y ∈ A, as in Section A.2. Of course, (B.1.1) and (B.1.11) imply (B.1.10). However, this does not always work when considering vector spaces over other fields, or modules over other commutative rings. In those cases, a Lie algebra is considered to be abelian or commutative if the corresponding Lie bracket is identically 0, and not merely symmetric. Thus a Lie algebra over other fields or commutative rings could be commutative as an algebra in the strict sense, and not commutative as a Lie algebra.

B.2 Subalgebras of Lie algebras

Let (A, [·, ·]) be a Lie algebra over the real numbers. If A_{0} is a linear subspace of A such that [x, y] ∈ A_{0} (B.2.1)

for every x, y ∈ A_{0}, then A_{0} is said to be a Lie subalgebra of A. This is the same as saying that A_{0} is a subalgebra of A as an algebra in the strict sense over R, as in Section A.2. It is easy to see that A_{0} is also a Lie algebra with respect to the restriction of [x, y] to x, y ∈ A_{0} in this case.

Suppose for the moment that A is an associative algebra over R, with multiplication of x, y ∈ A expressed as x y. If A_{0} is a subalgebra of A, then A_{0} is also a subalgebra of A as a Lie algebra with respect to the corresponding commutator bracket (B.1.5).

If n is a positive integer, then the special linear algebra

sl(n, R) (B.2.2)

consists of the n × n matrices a with entries in R such that

tr a = 0. (B.2.3)

It is easy to see that this is a Lie subalgebra of gl(n, R), using (A.12.5). More precisely, if a, b are any n × n matrices of real numbers, then

a b − b a ∈ sl(n, R). (B.2.4)

Similarly, if V is a finite-dimensional vector space over the real numbers, then the special linear algebra sl(V) (B.2.5)

associated to V consists of the linear mappings T from V into itself such that

tr T = 0. (B.2.6)

If R and T are any linear mappings from V into itself, then

R ∘ T − T ∘ R ∈ sl(V), (B.2.7)

because of (A.12.9). In particular, sl(V) is a Lie subalgebra of gl(V), as on p2 of [86]. The isomorphism between gl(R^{n}) and gl(n, R), as Lie algebras over the real numbers, mentioned in Subsection B.1.1 also maps sl(R^{n}) onto sl(n, R).

The orthogonal algebra o(n, R) (B.2.8)

consists of the n × n matrices a of real numbers that are anti-symmetric, in the sense that a^{t} = −a. (B.2.9)

One can check that this is a Lie subalgebra of gl(n, R), using (A.12.4). More precisely, this is a Lie subalgebra of sl(n, R).

Let V be a finite-dimensional vector space over the real numbers with an inner product ⟨·, ·⟩_{V}. If T is a linear mapping from V into itself, then the adjoint T^{∗} of T with respect to ⟨·, ·⟩_{V} may be defined as a linear mapping from V into itself as in Section A.9. Under these conditions, we say that T is anti-self-adjoint if T^{∗} = −T. (B.2.10)

The orthogonal algebra o(V) (B.2.11)

associated to ⟨·, ·⟩_{V} on V consists of the anti-self-adjoint linear mappings from V into itself. One can check that this is a Lie subalgebra of gl(V), using (A.9.8).

If V = R^{n} with the standard inner product, then o(R^{n}) corresponds to o(n, R) under the Lie algebra isomorphism between gl(R^{n}) and gl(n, R) mentioned in Subsection B.1.1. This follows from the fact that the adjoint of a linear mapping from R^{n} into itself with respect to the standard inner product corresponds to the transpose of the associated n × n matrix, as in Section A.9.

Let V be a vector space over the real numbers, and let b be a bilinear form on V. Let us say that a linear mapping T from V into itself is anti-symmetric with respect to b if b(T (v), w) = −b(v, T (w)) (B.2.12)

for every v, w ∈ V. One can check that the collection of these linear mappings is a Lie subalgebra of gl(V). This corresponds to part of Theorem 3.56 on p119 of [184].

B.3 Some linear mappings

Let V_{0} be a vector space over the real numbers. If R and T are linear mappings from V_{0} into itself, then let [R, T] put

[R, T] = R ∘ T − T ∘ R, (B.3.1)

242 APPENDIX B. LIE ALGEBRAS as in (B.1.7). If U is another linear mapping from V_{0} into itself, then [[R, T], U] + [[T, U], R] + [[U, R], T] = 0, (B.3.2) by the Jacobi identity, as in Section B.1.

Let V_{1}, V_{2}, V_{3} be linear subspaces of V, with V_{3} ⊆ V_{2} ⊆ V_{1}. (B.3.3) Consider the space L_{3,2}(V_{1}, V_{0}) (B.3.4) of linear mappings T from V_{1} into V_{0} such that T (V_{3}) ⊆ V_{2}, T (V_{2}) ⊆ V_{1}. (B.3.5) This is a linear subspace of the space L(V_{1}, V_{0}) of all linear mappings from V_{1} into V_{0}.

Similarly, let L_{3}(V_{2}, V_{0}) (B.3.6) be the space of linear mappings B from V_{2} into V_{0} such that B(V_{3}) ⊆ V_{1}. (B.3.7) This is a linear subspace of L(V_{2}, V_{0}). If R, T ∈ L_{3,2}(V_{1}, 0), then R ∘ T ∈ L_{3}(V_{2}, V_{0}). (B.3.8) In this case, the commutator (B.3.1) of R and T is defined as a linear mapping from V_{2} into V_{0}, and in fact [R, T] ∈ L_{3}(V_{2}, V_{0}). (B.3.9)

If R, T, U ∈ L_{3,2}(V_{1}, V_{0}), then R ∘ T ∘ U ∈ L(V_{3}, V_{0}). (B.3.10) Similarly, ([R, T]) ∘ U, U ∘ ([R, T]) ∈ L(V_{3}, V_{0}), (B.3.11) so that [[R, T], U] = ([R, T]) ∘ U − U ∘ ([R, T]) ∈ L(V_{3}, V_{0}). (B.3.12) One can check that (B.3.2) holds in L(V_{3}, V_{0}) under these conditions.

B.4 Some more Lie subalgebras

Let (A, b) be an algebra in the strict sense over the real numbers, as in Section A.2. Thus A is a vector space over the real numbers in particular, so that the space L(A) of all linear mappings from A into itself is a vector space over R, and an associative algebra over R with respect to composition of linear mappings. Similarly gl(A) is a Lie algebra over R with respect to the usual commutator of linear mappings, as in Section B.2.

Let Der(A) = Der(A, b) (B.4.1)

be the space of derivations on A with respect to b, as in Subsection A.2.1. It is easy to see that this is a linear subspace of L(A). In fact, one can check that

(B.4.2) Der(A) is a subalgebra of gl(A),

as a Lie algebra over R. This corresponds to a remark on p4 of [86], to Example (iv) on p2 of [160], and to part of Theorem 3.54 on p117 of [184].

Suppose for the moment that A is an associative algebra, where multiplication of x, y in A is expressed as x y. If a ∈ A, then put

δ_{a}(x) = [a, x] = a x − x a (B.4.3)

for each x ∈ A. One can check that

δ_{a} ∈ Der(A). (B.4.4)

More precisely, δ_{a} is called an inner derivation of A. Note that

a ↦ δ_{a} (B.4.5)

defines a linear mapping from A into Der(A).

If b ∈ A too, then one can verify that

δ_{a} ∘ δ_{b} − δ_{b} ∘ δ_{a} = δ_{[a,b]}. (B.4.6)

This implies that {δ_{a} : a ∈ A} (B.4.7)

is a Lie subalgebra of Der(A).

Suppose now that (A, [·, ·]_{A}) is a Lie algebra over the real numbers. In particular, A is an algebra over R in the strict sense, so that the previous notions and remarks about algebras in the strict sense may be used in this case. Thus a linear mapping δ from A into itself is a derivation of A as a Lie algebra if δ([x, y]_{A}) = [δ(x), y]_{A} + [x, δ(y)]_{A} (B.4.8)

for every x, y ∈ A, as in Subsection A.2.1.

If x ∈ A, then let ad x = ad_{x} be the linear mapping from A into itself defined by ad_{x}(y) = [x, y]_{A} (B.4.9) for every y ∈ A. One can check that

ad_{x} ∈ Der(A), (B.4.10)

using the Jacobi identity. This corresponds to a remark on p4 of [86], and the first part of Theorem 3 on p3 of [160]. In fact, ad_{x} is called an inner derivation of A, as on p4 of [86]. It is sometimes convenient to use the notation ad_{A} x = ad_{A,x} for ad_{x}, to indicate the role of A.

Clearly x ↦ ad_{x} (B.4.11)

is a linear mapping from A into Der(A). One can check that

(B.4.12) ad_{x} ∘ ad_{y} − ad_{y} ∘ ad_{x} = ad_{[x,y]_{A}}

for every x, y ∈ A, using the Jacobi identity. This corresponds to a remark on p8 of [86], and to the second part of Theorem 3 on p3 of [160]. It follows that

{ad_{x} : x ∈ A} (B.4.13)

is a Lie subalgebra of Der(A).

B.5 Some multiplication operators

Let (A, b) be an algebra over the real numbers in the strict sense, as in Section A.2. If a ∈ A, then put L_{a}(x) = b(a, x) (B.5.1)

and R_{a}(x) = b(x, a) (B.5.2)

for every x ∈ A. These define linear mappings from A into itself, which are the left and right multiplication operators on A associated to a, respectively. Note that a ↦ L_{a} (B.5.3)

and a ↦ R_{a} (B.5.4)

define linear mappings from A into the space L(A) of all linear mappings from A into itself.

Suppose for the moment that A has a multipicative identity element e with respect to b. In this case, L_{a}(e) = R_{a}(e) = a (B.5.5)

for every a ∈ A. In particular, this means that (B.5.3) and (B.5.4) are one-to-one mappings from A into L(A).

Let δ be a derivation on A with respect to b. It is easy to see that

δ ∘ L_{a} − L_{a} ∘ δ = L_{δ(a)} (B.5.6)

B.5. SOME MULTIPLICATION OPERATORS 245

for every a ∈ A. Similarly, δ ∘ R_{a} − R_{a} ∘ δ = R_{δ(a)} (B.5.7) for every a ∈ A.

Of course, if A is commutative with respect to b, then L_{a} and R_{a} are the same. If (A, [·, ·]_{A}) is a Lie algebra over R, then L_{a} = ad_{a} (B.5.8) and R_{a} = −ad_{a} (B.5.9) for every a ∈ A.

B.5.1 Multiplication operators on associative algebras

Suppose now that A is an associative algebra over the real numbers, where multiplication of x, y ∈ A is expressed as x y. Thus L_{a}(x) = a x (B.5.10) and R_{a}(x) = a x (B.5.11) for all a, x ∈ A. If a_{1}, a_{2} ∈ A, then L_{a_{1} a_{2}}(x) = (a_{1} a_{2}) x = a_{1} (a_{2} x) = L_{a_{1}}(L_{a_{2}}(x)) (B.5.12) for every x ∈ A. This means that L_{a_{1} a_{2}} = L_{a_{1}} ∘ L_{a_{2}} (B.5.13) on A.

Similarly, R_{a_{1} a_{2}}(x) = x (a_{1} a_{2}) = (x a_{1}) a_{2} = R_{a_{2}}(R_{a_{1}}(x)) (B.5.14) for every x ∈ A. This is the same as saying that R_{a_{1} a_{2}} = R_{a_{2}} ∘ R_{a_{1}} (B.5.15) on A. Observe that L_{a_{1}}(R_{a_{2}}(x)) = a_{1} (x a_{2}) = (a_{1} x) a_{2} = R_{a_{2}}(L_{a_{1}}(x)) (B.5.16) for every x ∈ A. This shows that L_{a_{1}} ∘ R_{a_{2}} = R_{a_{2}} L_{a_{1}} (B.5.17) on A.

If a ∈ A, then put δ_{a} = L_{a} − R_{a}, (B.5.18)

as in (B.4.3). If δ is a derivation on A, then

δ ∘ δ_{a} − δ_{a} ∘ δ = δ_{δ(a)}, (B.5.19)

because of (B.5.6) and (B.5.7).

Suppose that A is commutative as well. If δ is a derivation on A, then it is easy to see that L_{a} ∘ δ ∈ Der(A) (B.5.20)

for every a ∈ A.

B.6 Ideals in algebras over R

Let (A, b) be an algebra over the real numbers in the strict sense again, as in Section A.2. A linear subspace A_{0} of A is said to be a left ideal in A with respect to b if b(x, y) ∈ A_{0} (B.6.1)

for every x ∈ A and y ∈ A_{0}. Similarly, A_{0} is said to be a right ideal in A if (B.6.1) holds for every x ∈ A_{0} and y ∈ A. If A_{0} is both a left and right ideal in A_{0}, then A_{0} is said to be a two-sided ideal in A. Of course, if A is commutative with respect to b, then left and right ideals in A are the same.

If A_{0} is a left or right ideal in A, then A_{0} is a subalgebra of A. It is easy to see that the kernel of a homomorphism from A into another algebra in the strict sense over R is a two-sided ideal in A.

Let A_{0} be a linear subspace of A. One can define the quotient space A/A_{0} as a vector space over the real numbers in a standard way. This includes a quotient mapping q, which is a linear mapping from A onto A/A_{0} with

ker q = A_{0}. (B.6.2)

If A_{0} is a two-sided ideal in A, then it is not too difficult to show that there is a unique bilinear mapping b_{0} from (A/A_{0}) × (A/A_{0}) into A/A_{0} such that

q(b(x, y)) = b_{0}(q(x), q(y)) (B.6.3)

for every x, y ∈ A. This means that A/A_{0} is an algebra in the strict sense over the real numbers with respect to b_{0}, and that q is an algebra homomorphism from A onto A/A_{0}. This corresponds to a remark on p2 of [160].

If A is commutative or associative, then it is easy to see that A/A_{0} has the same property. If A has a multiplicative identity element e with respect to b, then q(e) is the multiplicative identity element in A/A_{0} with respect to b_{0}.

B.6.1 Ideals in Lie algebras

Let (A, [·, ·]_{A}) be a Lie algebra over the real numbers. A linear subspace A_{0} of A is said to be an ideal in A if

[x, y]_{A} ∈ A_{0} (B.6.4)

for every x ∈ A and y ∈ A_{0}, as on p6 of [86]. This is the same as being a left, right, or two-sided ideal in A, because of the antisymmetry of the Lie bracket. It is easy to see that A/A_{0} is also a Lie algebra in this case, as on p7 of [86].

If n is a positive integer, then sl(n, R) is an ideal in gl(n, R). Similarly, if V is a finite-dimensional vector space over R, then sl(V) is an ideal in gl(V).

If B is an associative algebra over R, then the set Der(B) of derivations on B is a Lie subalgebra of gl(B), as in Section B.4. We have also seen that the set of derivations on B defined by commutators with elements of B is a Lie subalgebra of Der(B). In fact, this is an ideal in Der(B), as a Lie algebra over R, as in (B.5.19).

Similarly, the set Der(A) of derivations on A as a Lie algebra is a Lie subalgebra of gl(A), and the set (B.4.13) of derivations on A of the form ad_{x}, x ∈ A, is a Lie subalgebra of Der(A). More precisely, (B.4.13) is an ideal in Der(A), because of (B.5.6). This corresponds to Exercise 1 on p9 of [86].

The center of an associative algebra B is defined to be the set of elements of B that commute witl all other elements of B. It is easy to see that this is a subalgebra of B. The center of A as a Lie algebra is defined by

Z(A) = {x ∈ A : [x, y]_{A} = 0 for all y ∈ A}. (B.6.5)

This is an ideal in A, as mentioned on p6 of [86].

B.7 Representations

Let (A, [·, ·]_{A}) be a Lie algebra over the real numbers, and let V be a vector space over R. A representation of A on V is defined to be a homomorphism from A into the Lie algebra gl(V) of linear mappings from V into itself. If ρ is such a representation, then we may use ρ_{x} for the linear mapping on V corresponding to x ∈ A. In order for ρ to be a representation of A as a Lie algebra on V, we should have that x ↦ ρ_{x} (B.7.1)

is linear as a mapping from A into the space of linear mappings on V, and that

ρ_{[x,y]_{A}} = ρ_{x} ∘ ρ_{y} − ρ_{y} ∘ ρ_{x}, (B.7.2)

as linear mappings from V into itself, for every x, y ∈ A.

Under these conditions, we may say that V is a module over A, as a Lie algebra over R. We may also put

x · v = ρ_{x}(v) (B.7.3) for each x ∈ A and v ∈ V, which is another element of V. Using this notation, the condition that (B.7.1) be a linear mapping from A into the space of linear mappings on V is equivalent to asking that

(x, v) ↦ x · v (B.7.4)

be a bilinear mapping from A × V into V. Similarly, (B.7.2) is the same as saying that ([x, y]_{A}) · v = x · (y · v) − y · (x · v) (B.7.5)

for every x, y ∈ A and v ∈ V.

This corresponds to some remarks on p8, 25 of [86], p31 of [160], and p90 of [184].

B.7.1 Representations of associative algebras

Suppose now that A is an associative algebra over the real numbers, where multiplication of x, y ∈ A is expressed as x y. A representation of A on a vector space V over the real numbers is defined to be a homomorphism from A into the algebra L(V) of linear mappings from V into itself. If ρ is such a representation, then we may use ρ_{x} for the linear mapping on V corresponding to x ∈ A, as before. In order for ρ to be a representation of A as an associative algebra on V, we should have that x ↦ ρ_{x} is linear as a mapping from A into L(V), and that ρ_{x y} = ρ_{x} ∘ ρ_{y}, (B.7.6)

as linear mappings from V into itself, for every x, y ∈ A.

Remember that A may also be considered as a Lie algebra over R with respect to the corresponding commutator bracket, as in Subsection B.1.1. If ρ is a representation of A as an associative algebra on V, then ρ may also be considered as a representation of A as a Lie algebra on V.

If ρ is a representation of A, as an associative algebra over R, on V, then we may say that V is a left module over A. We may use the notation x · v for ρ_{x}(v) when x ∈ A and v ∈ V, as before. With this notation, the linearity of x ↦ ρ_{x} as a mapping from A into L(V) is equivalent to the bilinearity of x·v as a mapping from A×V into V, as before. We also have that (B.7.6) is equivalent to the condition that (x y) · v = x · (y · v) (B.7.7)

for every x, y ∈ A and v ∈ V. If A has a multiplicative identity element e, then one may ask that e · v = v (B.7.8)

for every v ∈ V too, so that ρ_{e} is the identity mapping on V.

We may consider A as a left module over itself, using multiplication in the algebra. This corresponds to the mapping that sends a ∈ A to the left multiplication operator L_{a} on A, as in Subsection B.5.1.

B.7.2 Right modules over associative algebras

Let V be a vector space over the real numbers again, and suppose that v · x is defined as an element of V for every x ∈ A and v ∈ V. More precisely, we ask that (v, x) ↦ v · x (B.7.9)

be a bilinear mapping from V × A into V. If

(v · x) · y = v · (x y) (B.7.10)

for all x, y ∈ A and v ∈ V, then we say that V is a right module over A. If A has a multiplicative identity element e, then one may ask that

v · e = v (B.7.11)

for every v ∈ V as well.

It is easy to see that A may be considered as a right module over itself, using multiplication in the algebra.

Suppose that V is a right module over A, and put

x · v = −v · x (B.7.12)

for every x ∈ X and v ∈ V. One can check that V is a module over A, as a Lie algebra with respect to the corresponding commutator bracket, in this way.

B.8 More on representations

Let (A, [·, ·]_{A}) be a Lie algebra over the real numbers, and let V be a vector space over the real numbers. The trivial representation sends every element of A to the linear mapping equal to 0 on V. In particular, R may be considered as a module over A with respect to the trivial representation. We shall normally only be concerned with trivial representations of Lie algebras on R, unless otherwise indicated, as on p31 of [160].

As in Section B.4, we put ad_{x}(y) = [x, y]_{A} for every x, y ∈ A. This defines a representation of A on itself, as before. This is called the adjoint representation of A. Note that the kernel of x ↦ ad_{x} (B.8.1)

is the center Z(A) of A as a Lie algebra, defined in Subsection B.6.1, as mentioned on p8 of [86].

B.8.1 Subrepresentations

Let ρ be any representation of A on V, and let W be a linear subspace of V. If

ρ_{x}(W) ⊆ W (B.8.2) for every x ∈ A, then the restriction of ρ_{x} to W defines a representation of A on W. This is called a subrepresentation of ρ. Equivalently, W is said to be a submodule of V, as a module over A. An ideal in A is the same as a submodule of A as a module over itself with respect to the adjoint representation.

Now let A be an associative algebra over R, where multiplication of x, y ∈ A is expressed as x y, and let ρ be a representation of A on V, as in Subsection B.7.1. If W is a linear subspace of V that satisfies (B.8.2) for every x ∈ A, then the restriction of ρ_{x} to W defines a representation of A on W, called a subrepresentation of ρ again. We may also call W a submodule of V, as a left module over A. A left ideal in A is the same as a submodule of A as a left module over itself, using multiplication on A, as in Subsection B.7.1.

Suppose that V is a right module over A, as in Subsection B.7.2. If

w · x ∈ W (B.8.3)

for every w ∈ W and x ∈ A, then W is said to be a submodule of V, as a right module over A. A right ideal in A is the same as a submodule of A as a right module over itself, using multiplication on A, as in Subsection B.7.2.

B.8.2 Direct sums of representations

Let V_{1}, …, V_{N} be finitely many vector spaces over the real numbers, so that their direct sum ⨁_{j=1}^{N} V_{j} may be defined as a vector space over the real numbers as in Section 2.1. If (A, [·, ·]_{A}) is a Lie algebra over R and V_{j} is a module over A for each j, then ⨁_{j=1}^{N} V_{j} may be considered as a module over A in a natural way too, so that

(B.8.4) V_{l} corresponds to a submodule of ⨁_{j=1}^{N} V_{j}

for each l. Similarly, if A is an associative algebra over R and V_{j} is a left module over A for each j, then ⨁_{j=1}^{N} V_{j} is a left module over A in a natural way. If V_{j} is a right module over A for each j, then ⨁_{j=1}^{N} V_{j} is a right module over A in a natural way.

If V_{j} is a vector space over R for each nonnegative integer j, then the direct sum ⨁_{j=0}^{∞} V_{j} and direct product ∏_{j=0}^{∞} V_{j} of the V_{j}’s may be defined as vector space over R as in Section A.4. If A is a Lie algebra again and A_{j} is a module over A for each j, then it is easy to see that ∏_{j=0}^{∞} V_{j} is a module over A too, where the action of x ∈ A on an element of ∏_{j=0}^{∞} V_{j} is defined coordinatewise. In this case,

(B.8.5)  ⨁_{j=0}^{∞} V_{j} is a submodule of ∏_{j=0}^{∞} V_{j},

as a module over A. There are analogous statements for left and right modules over associative algebras, as before.

B.8.3 Representations and dual spaces

Let V be a vector space over the real numbers, and let V^{∗} be the dual space of linear functionals on V, as in Section A.3. If T is a linear mapping from V into itself, then T^{∗}(λ) = λ ∘ T defines a linear mapping from V^{∗} into itself, as in Subsection A.3.1. If R is another linear mapping from V into itself, then we have seen that (R ∘ T)^{∗} = T^{∗} ∘ R^{∗}. (B.8.6)

This implies that

(R ∘ T − T ∘ R)^{∗} = −(R^{∗} ∘ T^{∗} − T^{∗} ∘ R^{∗}). (B.8.7)

Note that T ↦ T^{∗} defines a linear mapping from L(V) into itself.

Let A be an associative algebra over R again, and suppose for the moment that V is a left module over A. If x ∈ A and λ ∈ V^{∗}, then put

(λ · x)(v) = λ(x · v) (B.8.8)

for every v ∈ V. This defines λ·x as a linear functional on V, and one can check that V^{∗} is a right module over A in this way, using the remarks in the preceding paragraph. Equivalently, if T_{x} is the linear mapping from V into itself defined by T_{x}(v) = x · v, (B.8.9)

then λ · x = T_{x}^{∗}(λ) (B.8.10)

for every x ∈ A and λ ∈ V^{∗}.

Similarly, suppose that V is a right module over A. If x ∈ A and v ∈ V, then put (x · λ)(v) = λ(v · x) (B.8.11)

for every v ∈ V, which defines a linear functional on V. One can check that V^{∗} is a left module over A in this way.

Let (A, [·, ·]_{A}) be a Lie algebra over R, and let ρ = ρ^{V} be a representation of A on V. If x ∈ A, then ρ_{x} is a linear mapping from V into itself, so that the dual linear mapping (ρ_{x})^{∗} is defined on V^{∗} as before. One can check that

ρ_{x}^{V^{∗}} = −(ρ_{x})^{∗} (B.8.12)

defines a representation of A on V. Equivalently, if x ∈ A acts on V as in (B.7.3), and λ ∈ V^{∗}, then put

(x · λ)(v) = −λ(x · v) (B.8.13)

for each v ∈ V. One can verify that V^{∗} is a module over A in this way, as before. This is the dual representation or module of A associated to V, as on p26 of [86]. This is related to some remarks on p31 of [160], which will be discussed further in the next section.

252 APPENDIX B. LIE ALGEBRAS

B.8.4 Commuting representations of Lie algebras

Let (A, [·, ·]_{A}) be a Lie algebra over R again, and let ρ^{1}, …, ρ^{n} be finitely many representations of A on V. Suppose that these representations commute with each other, in the sense that ρ_{x}^{j} ∘ ρ_{y}^{l} = ρ_{y}^{l} ∘ ρ_{x}^{j} (B.8.14) on V for all x, y ∈ A and j, l = 1, …, n. Under these conditions, one can check that

ρ_{x} = ∑_{j=1}^{n} ρ_{x}^{j} (B.8.15)

defines a representation of A on V.

B.9 Representations and multilinear mappings

Let (A, [·, ·]_{A}) be a Lie algebra over the real numbers, let k be a positive integer, and let V_{1}, …, V_{k} and Z be vector spaces over R. Suppose that ρ^{V_{l}} (B.9.1) is a representation of A on V_{l} for each l = 1, …, k, and that ρ^{Z} (B.9.2) is a representation of A on Z. It will be convenient to express the actions of x ∈ A on V_{1}, …, V_{k} and Z as in (B.7.3).

Remember that L(V_{1}, …, V_{k}; Z) (B.9.3) denotes the space of multilinear mappings from ∏_{l=1}^{k} V_{l} into Z, as in Section 9.2. We would like to define representations ρ^{1}, …, ρ^{k} and ρ^{0} (B.9.4) on (B.9.3) corresponding to ρ^{V_{1}}, …, ρ^{V_{k}} and ρ^{Z}, respectively, as follows. If μ is a multilinear mapping from ∏_{l=1}^{k} V_{l} into Z and x ∈ A, then let ρ_{x}^{0}(μ) be the multilinear mapping from ∏_{l=1}^{k} V_{l} into Z defined by (ρ_{x}^{0}(μ))(v_{1}, …, v_{k}) = x · μ(v_{1}, …, v_{k}) (B.9.5) for each v_{1}, …, v_{k} in V_{1}, …, V_{k}, respectively. Equivalently, ρ_{x}^{0}(μ) = ρ_{x}^{Z} ∘ μ (B.9.6) on ∏_{l=1}^{k} V_{l}. It is easy to see that this defines a representation ρ^{0} of A on (B.9.3).

B.9. REPRESENTATIONS AND MULTILINEAR MAPPINGS 253

Similarly, for each l = 1, …, k, let ρ_{x}^{l}(μ) be the multilinear mapping from ∏_{l=1}^{k} V_{l} into Z defined by

(B.9.7) (ρ_{x}^{l}(μ))(v_{1}, …, v_{k}) = −μ(v_{1}, …, v_{l−1}, x · v_{l}, v_{l+1}, …, v_{k})
= −μ(v_{1}, …, v_{l−1}, ρ_{x}^{V_{l}}(v_{l}), v_{l+1}, …, v_{k})

for every v_{1}, …, v_{k} in V_{1}, …, V_{k}, respectively. One can check that this defines a representation ρ^{l} of A on (B.9.3) as well. This is related to the remarks in Subsection B.8.3.

It is easy to see that these representations on (B.9.3) commute with each other, as in (B.8.14). This implies that

ρ = ∑_{l=0}^{k} ρ^{l} (B.9.8)

defines a representation of A on (B.9.3) too, as in (B.8.15). This corresponds to some remarks on p27 of [86], and on p31 of [160].

This is a bit simpler when k = 1, so that (B.9.3) is the same as the space L(V_{1}, Z) (B.9.9) of linear mappings from V_{1} into Z. In this case, ρ_{x}^{1}(μ) = −μ ∘ ρ_{x}^{V_{1}} (B.9.10) for every x ∈ A and linear mapping μ from V_{1} into Z.

B.9.1 Representations on M_{k}(V, Z)

Let V be a vector space over the real numbers, and suppose that V_{l} = V (B.9.11) for each l = 1, …, k. This means that (B.9.3) is the same as the space M_{k}(V, Z) (B.9.12) of multilinear mappings from V^{k} into Z, as in Section 9.2. Also let ρ^{V} be a representation of A on V, and suppose that ρ^{V_{l}} = ρ^{V} (B.9.13) for each l = 1, …, k.

Let μ be a multilinear mapping from V^{k} into Z, and let σ be a permutation on {1, …, k}. Remember that μ^{σ} may be defined as another element of M_{k}(V, Z) as in Section 9.2. It is easy to see that ρ_{x}^{0}(μ^{σ}) = (ρ_{x}^{0}(μ))^{σ} (B.9.14) for every x ∈ A. If x ∈ A and 1 ≤ l ≤ k, then

(ρ_{x}^{l}(μ^{σ}))(v_{1}, …, v_{k})
(B.9.15) = −μ^{σ}(v_{1}, …, v_{l−1}, x · v_{l}, v_{l+1}, …, v_{k})
= −μ(v_{σ(1)}, …, v_{σ(σ^{−1}(l)−1)}, x · v_{l}, v_{σ(σ^{−1}(l)+1)}, …, v_{σ(k)})
= (ρ_{x}^{σ^{−1}(l)}(μ))(v_{σ(1)}, …, v_{σ(k)}) = (ρ_{x}^{σ^{−1}(l)}(μ))^{σ}(v_{1}, …, v_{k})

for every v_{1}, …, v_{k} ∈ V. Thus

ρ_{x}^{l}(μ^{σ}) = (ρ_{x}^{σ^{−1}(l)}(μ))^{σ}. (B.9.16)

One can use (B.9.14), (B.9.16), and the definition (B.9.8) of ρ to get that

ρ_{x}(μ^{σ}) = (ρ_{x}(μ))^{σ}. (B.9.17)

If μ is symmetric or antisymmetric on V^{k}, as in Section 9.2, then it follows that ρ_{x}(μ) has the same property. This means that

ρ_{x}(SM_{k}(V, Z)) ⊆ SM_{k}(V, Z) (B.9.18)

and ρ_{x}(AM_{k}(V, Z)) ⊆ SM_{k}(V, Z), (B.9.19)

where AM_{k}(V, Z) and SM_{k}(V, Z) are as in Section 9.2. Thus SM_{k}(V, Z) and AM_{k}(V, Z) are submodules of M_{k}(V, Z), as a module over A with respect to ρ, as in Subsection B.8.1.

B.10 Homomorphisms between representations

Let (A, [·, ·]_{A}) be a Lie algebra over the real numbers, let V, W be vector spaces over the real numbers, and let ρ^{V}, ρ^{W} be representations of A on V, W, respectively. A linear mapping ϕ from V into W is said to define a homomorphism between these representations if

ϕ ∘ ρ_{x}^{V} = ρ_{x}^{W} ∘ ϕ (B.10.1)

for every x ∈ A. Equivalently, this means that

ϕ(x · v) = x · ϕ(v) (B.10.2)

for every x ∈ A and v ∈ V, where the actions of x on v and ϕ(v) are defined using ρ_{x}^{V} and ρ_{x}^{W}, as in Section B.7. We may also say that ϕ is a homomorphism from V into W as modules over A under these conditions, as on p25 of [86], and on p31 of [160].

Let W_{0} be a linear subspace of W that is a submodule of W, as a module over A, as in Subsection B.8.1. In this case, the obvious inclusion mapping from W_{0} into W is a module homomorphism.

If ϕ is a homomorphism from V into W, as modules over A, then it is easy to see that (B.10.3) ker ϕ is a submodule of V,

as a module over A. Similarly,

(B.10.4) ϕ(V) is a submodule of W,

as a module over A.

Let Z be another module over A, and let ψ be a homomorphism from W into Z, as modules over A. One can check that

(B.10.5) ψ ∘ ϕ is a homomorphism from V into Z,

as modules over A.

If ϕ is a homomorphism from V into W, as modules over A, and ϕ is a one-to-one mapping from V onto W, then

(B.10.6) ϕ^{−1} is a homomorphism from W onto V,

as modules over A. Under these conditions, we say that ϕ is an isomorphism from V onto W, as modules over A, or equivalently that ϕ is an isomorphism between these representations of A. Of course, this means that ϕ^{−1} is an isomorphism from W onto V, as modules over A. If ψ is also an isomorphism from W onto Z, as modules over A, then

(B.10.7) ψ ∘ ϕ is an isomorphism from V onto Z,

as modules over A.

Let V_{0} be a linear subspave of V that is a submodule of V, as a module over A. The quotient space V/V_{0} can be defined as a vector space over the real numbers in the usual way, with the corresponding quotient mapping q from V onto V/V_{0} with kernel V_{0}. If x ∈ A and v ∈ V, then we would like to put

x · q(v) = q(x · v). (B.10.8)

It is easy to see that the right side depends only on x and q(v), because V_{0} is a submodule of V.

One can check that V/V_{0} is a module over A with respect to (B.10.8). This may be described as a quotient module or quotient representation. Of course, q is a homomorphism from V onto V/V_{0}, as modules over A, by construction.

B.10.1 Modules over associative algebras

Now let A be an associative algebra over the real numbers, where multiplication of x, y ∈ A is expressed as x y, let V, W be vector spaces over the real numbers, and let ϕ be a linear mapping from V into W. If V and W are left nodules over A, and ϕ intertwines the corresponding representations of A, then we say that ϕ is a homomorphism between these representations, or equivalent a homomorphism from V into W, as left modules over A. Similarly, if V and W are right modules over A, and if ϕ(v · x) = ϕ(v) · x (B.10.9)

for every x ∈ A and v ∈ V, then we say that ϕ is a homomorphism from V into W, as right modules over A.

B.11 Invariant elements

Let A be an associative algebra over the real numbers, where multiplication of x, y ∈ A is expressed as x y, and let V be a vector space over R. If v ∈ V and V is a left module over A, then it is easy to see that

{x ∈ A : x · v = 0} (B.11.1)

is a left ideal in A. Similarly, if V is a right module over A, then

{x ∈ A : v · x = 0} (B.11.2)

is a right ideal in A.

Suppose now that (A, [·, ·]_{A}) is a Lie algebra over R, and that V is a module over A. If v ∈ V, then one can check that (B.11.1) is a subalgebra of A.

An element v of V is said to be invariant under the representation of A on V if x · v = 0 (B.11.3)

for every x ∈ A, as on p31 of [160]. The collection of invariant elements of V is a linear subspace of V, and a submodule of V, as a module over A.

B.11.1 Invariant linear mappings

Let V and W be vector spaces over the real numbers that are also modules over A. The space L(V, W) of linear mappings from V into W may be considered as a module over A as well, as in Section B.9. More precisely, if T ∈ L(V, W) and x ∈ A, then x · T ∈ L(V, W) (B.11.4)

is defined by putting

(x · T)(v) = x · (T (v)) − T (x · v) (B.11.5)

for every v ∈ V, as before. It is easy to see that T is an invariant element of L(V, W) with respect to this representation of A if and only if T is a homomorphism from V into W, as modules over A, as in Example 1 on p31 of

B.11.2 Invariant bilinear forms

Let V and W be modules over A again, and consider the space

L(V, W; R) (B.11.6)

of all bilinear mappings from V × W into R. This may be considered as a module over A too, as in Section B.9, using the trivial representation of A on R. If β ∈ L(V, W; R) and x ∈ A, then

x · β ∈ L(V, W; R) (B.11.7)

is defined by (x · β)(v, w) = −β(x · v, w) − β(v, x · w) (B.11.8)

for every v ∈ V and w ∈ W. It follows that β is invariant with respect to this representation of A if and only if

β(x · v, w) = −β(v, x · w) (B.11.9)

for every x ∈ A, v ∈ V, and w ∈ W, as in Example 2 on p31 of [160]. If V = W = A, considered as a module over itself with respect to the adjoint representation, then this corresponds to the condition that a bilinear form on A be associative, as on p21 of [86].

Appendix C

Complex numbers and complex analysis

C.1 Complex numbers

A complex number z can be expressed in a unique way as z = x + y i, (C.1.1) with x, y ∈ R, and where i^{2} = −1. We may call x and y the real and imaginary of Z, respectively, and these may be denoted Re z and Im z, respectively. The set of complex numbers is denoted C, and addition and multiplication on R can be extended to C in a standard way. Note that addition and multiplication on C satisfy the usual commutativity, associativity, and distributivity properties.

If z ∈ C is as in (C.1.1), then the complex conjugate of z is defined by \overline{z} = x − y i. (C.1.2) If w is another complex number, then one can check that \overline{z + w} = \overline{z} + \overline{w} (C.1.3) and \overline{z w} = \overline{z} \overline{w}. (C.1.4) The absolute value or modulus is defined by |z| = (x^{2} + y^{2})^{1/2}, (C.1.5) using the nonnegative square root on the right side. If z ∈ R, then this is the same as the absolute value of z as a real number.

Observe that z \overline{z} = |z|^{2} (C.1.6) for every z ∈ C. One can use this and (C.1.4) to get that

|z w| = |z| |w| (C.1.7)

for every z, w ∈ C. If z ≠ 0, then |z| > 0, one can check that

\frac{1}{z} = \frac{\overline{z}}{|z|^{2}} (C.1.8)

is the multiplicative inverse of z in C. This means that C is a field, although we shall not discuss this in detail here.

If z ∈ C is as in (C.1.1), then we may identify z with (x, y) ∈ R^{2}. Note that addition of complex numbers corresponds to addition on R^{2}, and that (C.1.5) is the same as the standard Euclidean norm of (x, y) in R^{2}. Similarly, we may identify the complex plane C with R^{2} as a metric space, with respect to the standard Euclidean metric.

C.1.1 Complex vector spaces

One can define the notion of a vector space over the complex numbers in essentially the same way as for vector spaces over the real numbers, using complex numbers as the scalars instead of real numbers, although we shall not discuss this in detail here. If n is a positive integer, then the space C^{n} of n-tuples of complex numbers is a vector space over C, with respect to coordinatewise addition and scalar multiplication. Similarly, if X is a nonempty set, then the space of all complex-valued functions on X is a vector space over C, with respect to pointwise addition and scalar multiplication.

Note that any vector space over the complex numbers may be considered as a vector space over the real numbers, by simply not using multiplication by i. In particular, C may be considered as a two-dimensional vector space over R, which may be identified with R^{2}, as before.

If V and W are vector spaces over the complex numbers, then the notion of a linear mapping from V into W, as vector spaces over the complex numbers, may be defined in the usual way. We may also consider linear mappings from V into W as vector spaces over R. In order to be precise, we may refer to a mapping T from V into W as being real-linear if T is linear as a mapping from V into W when they are considered as vector spaces over the real numbers, and we may say that T is complex-linear if T is linear as a mapping from V into W when they are considered as vector spaces over the complex numbers. Thus T is complex-linear if and only if T is real-linear and

T (i v) = i T (v) (C.1.9)

for every v ∈ V. A complex-linear mapping from C into itself corresponds to multiplication by a complex number, for instance, and a real-linear mapping from C into itself is really the same as a linear mapping from R^{2} into itself.

260 APPENDIX C. COMPLEX NUMBERS AND COMPLEX ANALYSIS

C.2 Complex derivatives

Let U be a nonempty open subset of the complex plane, and let f be a complexvalued function on U. The complex derivative of f at a point z ∈ U is defined by

f′(z) = lim_{w→z} \frac{f(w) − f(z)}{w − z}, (C.2.1)

when the limit on the right exists. If the complex derivative of f exists at every z ∈ U, then f is said to be complex analytic or holomorphic on U.

A complex-valued function on U corresponds to a mapping from U into R^{2}, using the usual identification between C and R^{2}. If the complex derivative of f at z ∈ U exists, then one can check that f is differentiable at z as a mapping from U into R^{2}, as in Section 3.8. In this case, the differential of f at z is the linear mapping from R^{2} into itself that corresponds to

(C.2.2) multiplication by f′(z) on C,

which is a complex-linear mapping from C into itself.

Conversely, suppose that f is differentiable at z as a mapping from U into R^{2}, as in Section 3.8, and that

(C.2.3) the differential of f at z is complex-linear,

when considered as a mapping from C into itself. This means that

(C.2.4) the differential of f at z corresponds to multiplication by a complex number on C,

and one can verify that the complex derivative of f at z exists and is equal to that complex number.

The condition (C.2.3) is characterized by a first-order linear system of partial differential equations in the real and imaginary parts of f, which are known as the Cauchy–Rieamann equations.

C.2.1 More on holomorphic functions

If f is complex analytic on U, then it is well known that

(C.2.5) the complex derivatives of f of all orders exist on U.

In particular, this implies that f is infinitely differentiable on U, as a mapping from U into R^{2}.

Let g be another complex-valued function on U. If the complex derivatives of f and g exist at z ∈ U, then one can show that the complex derivatives of f + g and f g exist at z, with

(f + g)′(z) = f′(z) + g′(z) (C.2.6) and (f g)′(z) = f′(z) g(z) + f(z) g′(z). (C.2.7)

If f and g are both holomorphic on U, then it follows that

(C.2.8) f + g and f g are holomorphic on U

as well.

Of course, constant functions are holomorphic on C, with complex derivative equal to 0. It is easy to see that the identity mapping on C is holomorphic, with complex derivative equal to 1. Using this and (C.2.8), we get that

(C.2.9) all polynomials in z with complex coefficients are holomorphic on C.

Suppose that f is not equal to 0 at any point in U, so that 1/f defines a complex-valued function on U too. If the complex derivative of f exists at z ∈ U, then one can show that the complex derivative of 1/f exists at z, with

(1/f)′(z) = −\frac{f′(z)}{f(z)^{2}} . (C.2.10)

This implies that

(C.2.11) 1/f is holomorphic on U when f is holomorphic on U.

One can use this to get that rational functions of z are holomorphic on the set where the denominator is not zero.

C.2.2 Some related differential operators

Let f be a complex-valued function on U again. If U is considered as an open set in R^{2}, then the partial derivatives

∂f/∂x and ∂f/∂y (C.2.12)

may be defined as complex numbers, when they exist, in the same way as for real-valued functions on U. This is the same as taking the partial derivatives of f as a function on U with values in R^{2}, and interpreting the partial derivatives, when they exist, as complex numbers.

Put

\frac{∂f}{∂z} = \frac{1}{2} (\frac{∂f}{∂x} − i \frac{∂f}{∂y} ) (C.2.13)

and

\frac{∂f}{∂\overline{z}} = \frac{1}{2} (\frac{∂f}{∂x} + i \frac{∂f}{∂y} ), (C.2.14)

when the partial derivatives on the right sides of these equations exist. We may be particularly concerned with these expresseions at points where f is differentiable as a mapping from U into R^{2}. One can check that

\frac{∂f}{∂\overline{z}} = 0 (C.2.15)

is equivalent to the system of Cauchy–Riemann equations for the real and imaginary parts of f. If the complex derivative of f at a point z ∈ U exists, then one can verify that

f′(z) = \frac{∂f}{∂z} (z). (C.2.16)

C.3 Complex differential forms

Let k be a positive integer, and let V be a vector space over the real numbers. Consider the space M_{k}(V, C) (C.3.1)

of multilinear mappings from the space V^{k} of k-tuples of eleents of V into C, as a vector space over the real numbers. This is the same as in Section 9.2, with Z = C, as a vector space over the real numbers.

The elements of (C.3.1) may be described as complex-valued k-linear forms on V. Note that a complex-valued function on V^{k} is an element of (C.3.1) if and only if its real and imaginary parts are real-valued k-linear forms on V.

It is easy to see that (C.3.1) is a linear subspace of the space of all complexvalued functions on V^{k}, as a vector space over the complex numbers with respect to pointwise addition and scalar multiplication of functions. This means that (C.3.1) may be considered as a vector space over the complex numbers, with respect to pointwise addition and scalar multiplication on V^{k}.

Basically everything that we did before for ordinary differential forms can be extended to differential forms with complex-valued coefficients.

C.3.1 Complex differential forms on C

In particular, dz = dx + i dy and d\overline{z} = dx − i dy (C.3.2)

may be considered as differential 1-forms on the the complex plane with complex coefficients. Observe that

dz ∧ dz = d\overline{z} ∧ d\overline{z} = 0 (C.3.3)

and dz ∧ d\overline{z} = −d\overline{z} ∧ dz = −2 i dx ∧ dy. (C.3.4)

Of course, dx = (1/2) (dz + d\overline{z}) and dy = (−i/2) (dz − d\overline{z}). (C.3.5)

Thus any differential 1-form on a subset E of C with complex-valued coefficients may be expressed in a unique way as

a dz + b d\overline{z}, (C.3.6)

where a and b are complex-valued functions on E.

Let U be a nonempty open set in C, and let f be a complex-valued function on U. Suppose that f is continuously differentiable as an R^{2}-valued function on U. One can check that

df = \frac{∂f}{∂z} dz + \frac{∂f}{∂\overline{z}} d\overline{z} (C.3.7)

on U, as differential 1-forms on U with complex coefficients.

C.3.2 Some closed forms on C

Similarly,

d(f dz) = df ∧ dz = \frac{∂f}{∂\overline{z}} d\overline{z} ∧ dz = −\frac{∂f}{∂\overline{z}} dz ∧ d\overline{z} (C.3.8)

on U, using (C.3.3) in the second step. This implies that f dz is closed as a differential 1-form on U with complex coefficients if and only if f is holomorphic on U.

Let F be another complex-valued function on U, and suppose that F is continuously-differentiable as an R^{2}-valued function on U. It is easy to see that

dF = f dz (C.3.9)

on U if and only if F is holomorphic on U, with

F′ = f (C.3.10)

on U.

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Index

absolute value of a complex number, 258 of a real number, 46

adjoints of linear mappings, 230 admissible volumes, 89

nondegenerate, 89 affine chains, 153 affine mappings, 145 affine simplices, 149, 153 algebras in the strict sense, 219

automorphisms, 220 commutative, 219 homomorphisms, 219 isomorphisms, 220 quotients, 246 subalgebras, 219

AM(W), 40, 228 AM_{0}(W), 39 AM_{k}(R^{n}), 15 AM_{k}(V, Z), 198 AM_{k}(W), 26 anti-derivations, 79, 214 anti-self-adjoint linear mappings, 241 anti-symmetric linear mappings, 241 anti-symmetric matrices, 241 associative algebras, 219

left modules, 248 direct products, 250 direct sums, 250 homomorphisms, 255 submodules, 250

representations, 248 direct products, 250 direct sums, 250 homomorphisms, 255 subrepresentations, 250

right modules, 249 direct products, 250 direct sums, 250 homomorphisms, 256 submodules, 250

associative bilinear forms, 257

bilinear forms, 6, 24 antisymmetric, 9 symmetric, 9

bilinear functions, 23 bilinear mappings, 23 bilinear operations, 219 binomial coefficients, 1 binomial theorem, 1 boundaries

of affine n-chains, 156 of affine n-simplices, 155 of C^{r} n-chains, 164 of C^{r} n-simplices, 163 of subsets of R^{n}, 103

bounded sets, 103

C, 258 C(E), 50 C(E, AM(R^{n})), 65 C(E, AM_{k}(R^{n})), 66 C(E, L(R^{m_{2}}, R^{m_{1}})), 96 C(E, M_{k}(R^{n})), 72 C(E, M_{k}(R^{n}, R^{n})), 199 C(E, M_{m_{1},m_{2}}(R)), 93 C(E, R^{m}), 50 C^{1}(U, R^{m}), 52 C^{∞}(U, AM(R^{n})), 67 C^{∞}(U, AM_{k}(R^{n})), 67 C^{∞}(U, L(R^{m_{2}}, R^{m_{1}})), 97 C^{∞}(U, M_{k}(R^{n}, R^{n})), 199 C^{∞}(U, M_{k}(R^{n})), 73 C^{∞}(U, M_{m_{1},m_{2}}(R)), 94 C^{∞}(U, R^{m}), 55 C^{r}(U, AM(R^{n})), 66 C^{r}(U, AM_{k}(R^{n})), 67 C^{r}(U, L(R^{m_{2}}, R^{m_{1}})), 97 C^{r}(U, M_{k}(R^{n})), 73 C^{r}(U, M_{k}(R^{n}, R^{n})), 199 C^{r}(U, M_{m_{1},m_{2}}(R)), 94 C^{r}(U, R^{m}), 54 Cauchy–Riemann equations, 260 Cauchy–Schwarz inequality, 49 cell-nice sets, 187 cells, 112 center

of a Lie algebra, 247

of an associative algebra, 247 chain rule, 58 chains

affine, 153 C^{∞}, 160 C^{r}, 160

changing variables, 122, 124 closed differential forms, 82 closed sets, 102 closures of sets, 102 compact sets, 103 complements of sets, 102 complex conjugation, 258 complex derivative, 260 complex numbers, 258 complex-analytic functions, 260 connected sets, 84, 85 continuous differentiability, 52

r-times, 53 two times, 52

continuous functions, 49, 50 L(R^{m_{2}}, R^{m_{1}})-valued, 96 matrix-valued, 93

convex sets, 85

Der(A), 243 derivations, 220, 243 determinants, 20, 27, 28 diffeomorphisms

C^{1}, 109

C^{∞}, 112

C^{r}, 111

differentiable mappings, 56, 68, 75 differential forms, 64

C^{∞} (smooth), 67

C^{r}, 66

continuous, 65 differential k-forms, 64 differential of a mapping, 57 direct products of groups, 42 direct products of vector spaces, 222 direct sums of vector spaces, 22, 222 directional derivatives, 57 discrete metric, 46 dual spaces, 221 dual linear mappings, 221

embeddings, 130

in the strong sense, 130 exact differential forms, 82 exterior derivative, 78 extreme value theorem, 104

general linear algebras, 239

general linear groups, 95, 98, 221 GL(m, R), 95 gl(n, R), 239 GL(R^{m}), 98 GL(V), 220 gl(V), 239 group homomorphisms, 31, 41, 42 group isomorphisms, 42 groups, 29, 95, 98, 221

holomorphic functions, 260 homeomorphisms, 105 homogeneous functions, 2 homogeneous polynomials, 2

ideals left, 246 right, 246 two-sided, 246 identity matrix, 95, 236 imaginary part, 258 imbeddings, 130 immersions, 130 implicit function theorem, 134 infinite differentiability, 55 inner derivations, 243, 244 inner product spaces, 49 inner products, 49 integrals of differential forms, 136, 137

over affine n-chains, 154 over n-surfaces, 142 over oriented 0-simplices, 153

interior multiplication, 214 interior products, 210, 211 invariant elements, 256 inverse function theorem, 100 invertible matrices, 95 isometric linear mappings, 69 isometries, 69

Jacobi identity, 182, 238 Jacobian determinants, 122

k-linear forms, 6, 24 alternating, 14, 26, 29, 39 alternatization, 18, 27, 30, 39 complex valued, 262 k = 0, 39 products, 32, 34, 36, 37, 39 symmetric, 11, 26, 29, 39 symmetrization, 12, 27, 30, 39

k-linear functions, 6, 23 k-linear mappings, 23

L(R^{m}), 98 L(R^{m_{2}}, R^{m_{1}}), 96 L(R^{m_{2}}, R^{m_{1}})-valued functions, 96

C^{∞} (smooth), 97

C^{r}, 97

continuous, 96 L(V), 239 L(V, W), 224 L(V_{1}, …, V_{k}; Z), 197 Lie algebras, 238

abelian, 239

commutative, 239 homomorphisms, 238 ideals, 247 isomorphisms, 238 modules, 247

direct products, 250 direct sums, 250 dual, 251 homomorphisms, 254 isomorphisms, 255 quotients, 255 submodules, 250

quotients, 247

representations, 247 adjoint representation, 249 direct products, 250 direct sums, 250 dual, 251 homomorphisms, 254 isomorphisms, 255 quotients, 255 subrepresentations, 250 trivial, 249 subalgebras, 240 Lie brackets, 181, 238 Lie derivatives

of differential forms, 208 of functions, 183 of tensor fields

of type (0, k), 201 of type (1, k), 203 of vector fields, 183 limit point property, 103 limit points, 101 linear functionals, 3, 24, 221 linear subspaces, 218 locally constant functions, 85

M(W), 226 M_{0}(W), 39 M_{k}(R^{n}), 7 M_{k}(V), 24 M_{k}(V, Z), 197 M_{m,m}(R), 95 M_{m_{1},m_{2}}(R), 93 manifolds, 130 matrices, 59 matrix multiplication, 59 matrix-valued functions, 93 C^{∞} (smooth), 94 C^{r}, 94 continuous, 93

metric spaces, 46 metrics, 46 modulus of a complex number, 258 monomials, 2 multi-indices, 2 multilinear forms, 6, 24, 39

alternating, 14, 26, 29, 39 alternatization, 18, 27, 30, 39 products, 32, 34, 36, 37, 39 symmetric, 11, 26, 29, 39 symmetrization, 12, 27, 30, 39

multilinear functions, 6, 23 multilinear mappings, 23, 195, 196, 200

alternating, 197 over C(E, R), 196, 200 over C^{∞}(U, R), 197, 201 over C^{r}(U, R), 196, 200 symmetric, 197

multinomial coefficients, 18 multinomial theorem, 18 multiplication operators, 244 multiplicative identity elements, 219

n-surfaces, 142 C^{∞}, 142 C^{r}, 142

negative orientations, 152 nonnegativity, 232 norms, 47

associated to inner products, 49

o(n, R), 241 o(V), 241 open balls, 51 open mappings, 106, 107 open sets, 51 orientation-preserving, 140 orientation-reversing, 140 orientations

negative, 152 opposite, 152 positive, 152 same, 152

oriented affine simplices, 149, 153 oriented simplices

0-simplices, 153, 160 affine, 149, 153 C^{∞}, 159 C^{r}, 159

orthogonal algebras, 241 orthogonal transformations, 88 orthogonal vectors, 229 orthonormal vectors, 229

P(R^{n}), 3 P(R^{n}, R^{m}), 186 P_{d}(R^{n}), 4 parallelepipeds, 90

nondegenerate, 90 parameter domains, 142 parameterized affine simplices, 149 parameterized simplices

0-simplices, 153, 160 affine, 149 C^{∞}, 159 C^{r}, 159

partial derivatives, 51, 261 partitions of unity, 119 path-connected components, 87 path-connected sets, 86 permutations, 4

even, 5 odd, 5

Poincaré lemma, 190 polynomials, 2, 3 positive orientations, 152 pull-backs, 75, 138

R, 1 R^{n}, 1 (R^{n})^{k}, 6 real part, 258 relatively closed sets, 102 relatively open sets, 102

self-adjointness, 232 sequential compactness, 104 shuffles, 43 signum of a permutation, 4 sl(n, R), 240 sl(V), 240 SM(W), 227 SM_{0}(W), 39 SM_{k}(R^{n}), 14 SM_{k}(V, Z), 198 SM_{k}(W), 26 smooth functions, 55 special linear algebras, 240 standard basis, 7 standard Euclidean metrics, 46, 48 standard Euclidean norms, 47 standard inner products, 49 standard simplex

0-simplex, 153 in R^{n}, 119 parameterized affine, 162 positively oriented bounary, 162

Stokes’ theorem, 170, 180 submanifolds, 130 supports of functions, 114

of differential forms, 136 Sym(k), 11 Sym(X), 11

subgroups, 29 symmetric matrices, 232 symmetric group, 11

tangent spaces, 130 tensor fields

of type (0, k), 71 C^{∞} (smooth), 73 C^{r}, 72 continuous, 72 of type (1, 0), 74 of type (1, k), 198 C^{∞} (smooth), 199 C^{r}, 199 continuous, 199 of type (r, s), 71

trace of a linear mapping, 236 of a matrix, 235

transpose of a matrix, 235 transpositions, 5 triangle inequality, 46, 47 uniform continuity, 104 unit cibe in R^{n}, 120

V′, 221 V^{∗}, 221 V^{k}, 24 vector fields, 74

ϕ-related, 184 polynomial, 186

vector space isomorphisms, 220 vector spaces

complex, 259 complex-linear mappings, 259 linear mappings, 259 real-linear mappings, 259

real, 218 vertices of affine simplices, 149

wedge product, 37, 39