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| Read 7.1-7.5 (pages 1-5 of Chapter 7); Do Problem 7-1 and make sure you explain why the representation is unique; Suppose x,y,z are vectors in 3-space and z dot (x cross y)=0. Assuming x and y are linearly independent, what can you conclude about z (prove it)? What can you conclude if x and y are linearly dependent? | ||
| no classes - Martin Luther King Day-Optional Thought Problem: If Martin Luther King, Ghandi and Jesus Christ were "alive" today, what would they find horrifying about our society and what would they find encouraging? | ||
| Read pages 6-10 of Chapter 7; Prove that among all parallelopipeds with fixed side lengths a=||x||, b=||y||, c=||u||, the maximum volume is abc and prove that this is achieved only by a rectangular prism. Do Problems 7-7, 7-8, 7-10. For the definition of an equivalence relation see http://en.wikipedia.org/wiki/Equivalence_relation. For example, for SYMMETRY, you must show: If a frame Phi has the same orientation as a frame Psi then Psi has the same orientation as Phi. | ||
| Read pages 10-12 #1. Suppose S is a set of n by n matrices. Suppose we say, for x,y vectors in n-space, that x~y if there is a matrix A in S such that x=Ay (thinking of x,y as a column vectors). What is a set of natural conditions on the set S that will ensure that ~ is an equivalence relation? Prove, under your conditions on S, that ~ is an equivalence relation. #2. Now suppose S=O(n). How many equivalence classes are there under the equivalence relation ~ above ? What are the equivalence classes (the sets that consist entirely of vectors all equivalent to each other)? Refer to Chapter 4 to review information about O(n). | ||
| Read pages 11-16; Do 7-12, 7-13, 7-14 (be careful to define NEW u,v and relate them to the old u,v). Look over but don't do 7-16 (the (w_iw_j) stands for the 3 by 3 matrix whose ij entry is w_i times w_j). Then, using 7-16, compute the matrix for the counter-clockwise rotation of pi/3 about the axis w=(1/(square root of 3),1/(square root of 3), 1/(square root of 3)). | ||
| Read pages 17-20; Do 7-22, 7-24 (think geometrically), 7-25 (use a calculator) | ||
| Read pages 20-22; Do: 7-26 (for parts c,d it might be easiest to use the result of Problem #2b of this problem set-see below), Additional Problems: #1 Suppose J is the n by n matrix with all 1's down the diagonal except for the (n,n) entry which is -1. Show (briefly) that J represents a reflection along w and find the unit vector w. #2 Derive a matrix formula for S= arbitrary reflection along a unit vector w in n-space, analogous to the starred formula at the bottom of page 14 where S takes the place of the rotation matrix R(w, theta). This should be done by mimicking the proof of the starred formula (see page 14) by choosing a nice basis {u_1,u_2,....u_(n-1),w}. Deduce that any reflection matrix is "conjugate" to J, i.e. S=CJ(inverse of C) for some invertible matrix C. #3a. Show that any matrix A has the same characteristic polynomial and hence same eigenvalues as any conjugate of A, i.e., CA(inverse of C). #3b. What are the possible eigenvalues of an n by n reflection matrix S? #4. Extra Credit: Find a specific matrix in O(3) but not in SO(3) that is NOT a reflection matrix and say why it is not. Also find such a matrix for any n at least 3. Turn this problem on a separate paper. | ||
| Read Chapter 8 pages 1-5 and 11-12. Read over but don't DO Problem 8-10. Do: Additional Problem #1: Let x=(1,0) and y=(0,1) points (or vectors) in the plane. Draw a nice picture of the plane (yes graph paper would be best) and draw a picture of the parallelogram determined by x and y. Color it some nice color. Compute its area. Yes this really is just like in 8th grade but there will be a simple lesson here and I don't want it lost in all the theory. Now consider the matrix A whose first column is (4,1) and whose second column is (2,3). Thus A represents a linear transformation of the plane wherein Ax is the point (4,1) and Ay is point (2,3). On another picture of the plane, draw the parallelogram determined by Ax and Ay. Note that it is just the image of the first parallelogram under the map A. (Color it some different nice color.) What is its area? Compute the determinant of A. Do the same for a matrix B where Bx=(2,0), By=(0,1/2). Do the same for a matrix C where Cx=(1,0) and Cy=(1,1). What do you conclude "grasshopper" (1 point extra credit if you know to whom I am referring)? Additional Problem #2: Suppose you are are brain surgeon and suppose somebody has provided you with a nice (3-dimensional) representation of a human brain stored on your computer (colored a nice color!) but of course when you view it you can only see one "side" (one point of view - say looking down along the z-axis) on your screen. Suppose you apply a general 3 by 3 matrix A to alter the picture of the brain in an attempt to see a different "side". What undesired distortions might occur? Suppose the determinant of A were 1. What "distortions" might occur then? Suppose A were in O(3) but not in SO(3). What "distortions" might occur then? Suppose A were in S0(3). What would happen to the picture you see on the screen? Feel free to experiment with a cube in 3-space (or 2-space) with labelled vertices as in #1 instead of an actual human brain, but be careful because a cube is so symmetric, it hides certain distortions. If you draw a picture it should have some nice colors!! | ||
| Read Chapter 9 pages 1-4.5 Do: Review Definitions and Theorems from Chapters 7, 8 (only those pages covered in assignments) for midterm that we should have at some point in the future (let's discuss this Wednesday) | ||
| Do the two problems from the class hand-out. For #2, try to understand the proof of the corresponding result for sup in Wikipedia. However beware that I have used different notation so you will not just be able to copy the proof. It might help to prove that the Approximation Property in #1 is equivalent to: If a* is the inf of a set A then for any small epsilon>0, there is some element a in A such that a* is less than or equal to a and a < a*+ epsilon. | ||
| Sorry but I cannot make my office hour next Monday 2pm. Read Chapter 9 pages 6-9; Look over Problems 9-3, 9-4 ; Do: 9-5 second part, 9-6, 9-7 | ||
| Read Chapter 9 pages 9-12; Do: 9-10 (this time you are not required to give a detailed proof that your example is correct-just explain briefly) and do second part of 9-8 (you can use the integrability criterion) | ||
| Read Chapter 9 pages 14-18; Suppose f is a piecewise uniformly continuous function defined on I, i.e. there is SOME partition I_j , such that f is uniformly continuous on the INTERIOR of each I_j (but may be totally bizarre on the boundary of I_j). Prove that f is integrable by trying to follow the book's proof for a uniformly continuous function. Be careful about the definition of the upper and lower integrals. Go back to the definition of a step function and see that it only has to be constant on the INTERIOR of the I_j, a fine point that I have ignored in class. Observe for yourself that this still doesn't help with example of problem 9-6. | ||
| Read Chapter 9 pages 17-20; Do: 9-14 page 21(easy-use Fubini, but explain steps), 9-16, 9-17 (Hint:use Fubini twice to "change order of integration" | ||
| Read Chapter 9, pages 20-25 Do: Additional Problem #1: Compute the volume of the solid in 3-space bounded by the xz-plane, the yz-plane, the xy-plane, the plane x=1, the plane, y=1 and the surface z=x^2 + y^4 by computing the integral of the height function over the base rectangle. Problem #2: Compute the volume of the region in 3-space bounded by the surface z=siny, the planes x=1, x=0, y=0, y=pi/2 and the xy-plane. | ||
| If you would strongly prefer to have the midterm on Monday after next, send me an email. Read Chapter 9 pages 25 and 29-32; Do: Suppose A is the set of irrational numbers in the interval [0,1] in the real numbers. Find the inner and outer volumes of A (1-dimensional volume we normally call the length of A). Hint: consider the definition of volume and use Problem 9-6. So is A a contented set? | ||
| Skim Chapter 9 pages 34-35, Read Chapter 10 Sections C, D. Do: Problems 10-23 and 10-26. Study for Friday's in-class exam covering all we have done in Chapters 7,8 and 9. Refer to past reading assignments to see the precise page numbers of what we covered. Inf/Sup from Wickepedia handout is included. Study definitions, basic properties, basic theorems, and properties stated in the form of Problems. Study class notes (if you have any) for extra emphasis. | ||
| Prepare for in-class midterm on Friday; the take-home exam will be handed out on Friday, but not due until the following Wednesday, so you can wait on studying for it until the weekend if you like. Do: Problem 10-30, Problem 10-24 (in this problem you may assume that, if B is a set in (n-1)-dimensional space and aB is the set of points that are just "a times all the points of B" then (n-1)-dimensional-volume of aB is a^(n-1) times the (n-1-dimensional-volume of B) | ||
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In-class closed book short-answer midterm will be held on Friday. The take-home open-book midterm will also be handed out. No other assignment due Friday or Monday. |
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| No assignment is due today, but there will be an assignment due Wednesday (so maybe it is better to do take-home by Monday). | ||
| Read Chapter 10 pages 15-19 Take home test is due in class Wednesday. Do: Use Fubini\Cavilieri to do Problem 1-32, and #2. find the volume of the region in 3-space bounded the planes y=0, y+z=1 and by the surface z=x^2. You will need to first either draw a picture of the region, or else describe the region analytically using INEQUALITIES (the given equations are just the boundaries of the region). | ||
| Read Chapter 10 pages 20-25 Do: Problem 10-39 (recall the actual definitions of the terms involved), 10-41 (recall def. of alpha_n from a previous section) | ||
| #1. Calculate, using polar coordinates, the integral of the function f(x,y)=x^2+y^2 over the annulus in the plane whose outer boundary is the circle of radius 3 centered the origin and whose inner boundary is the circle of radius 1 centered at the origin. Notice that you will have to use Fubini's theorem (Cavalieri) to evaluate the integral in polar coordinates. #2. Cylindrical coordinates for 3-space is where you use r, theta and z instead of x,y, and z. What is the correct change of variables? Calculate the volume of the ball of radius 1 about the origin using cylindrical coordinates. Again you always have to use Fubini-Cavalieri. Which variable (of r, theta, z) will you slice with? Note that a "slice" with theta or "r" does not look like what you might expect. | ||
| Read Chapter 10 pages 27-32 and 41-42 Do: 10-59, 10-60, The average value of a function f(x) over a region A is defined to be the integral of f over A , divided by the volume of A. For example if f(x)=6 on all of A then the average value is 6vol(A)/vol(A)=6. If f=6 on 1/2 of A and f=0 on 1/2 of A then the average value is (6vol(A)/2+0)/vol(A)=3. Calculate the average radius among all points in the disk of radius 1 in the plane. | ||
| Refer to notes from today's class. Do: 10-61, #1. Find the centroid of the upper half of a unit ball in 3-space, i.e. the set given by the intersection of the unit ball with the half-space where z is non-negative. You can deduce the x and y coordinates by "symmetry". Try using spherical coordinates (or not). | ||
| no classes - Midterm Recess-take it easy (but see little assignment below!) | ||
| Read Chapter 11 pages 1-4. Do: 11-1, Find the length of the piece of parabola y=1-x^2 between x=-1 and x=1. You can use a calculator to evaluate (estimate) the integral. | ||
| Read Chapter 11 pages 5-7 Do: #1. Let M be the set of (x,y) in 2-space satisfying the equation y^2-x^2-x^3=0. Decide if M is a manifold using the Implicit Function Theorem. If it is a manifold except at certain points, identify these points and the dimension of the manifold. Sketch a plot of M by hand or using maple or Mathematica. #2. Do the same for N being the set of (x,y,z) satisfying x^m+y^m+z^m=c for some even positive integer m. For what values of c is N a manifold? What does N look like? #3. Suppose M is a 1-dimensional manifold in R^n given by the parametrization F:[a,b]--M where we think of F as a function of the single variable t. Compute the square root of the Gram determinant in terms of F(t). Does this answer look familiar? | ||
| Read Chapter 11 pages 7-8 D0: 11-6 (Consider positive octant and multiply by 8; use theta as parameter)11-9, 11-12 (parametrize using x,y as parameters; AFTER finding the integral in terms of the parameters, change to polar), | ||
| Read Chapter 11 page 10 and bottom page 12, rest of chapter optional; Do: 11-13, also compute the VOLUME of the region "below" the paraboloid, above x-y plane and inside the cylinder. This was from previous chapter!! , 11-19 | ||
| Read notes from class, Do: #1. Let F be the vector field in the plane given by (y/(x^2+y^2), -x/(x^2+y^2)). Compute the integral of this vector field around the circle of radius one in the plane with the counter-clockwise orientation. Note that this vector field is not defined at the origin so you CANNOT use Green's theorem. #2. Prove that the integral of the F above around a closed curve gamma in the plane is ZERO for any gamma which encloses a region R that DOES NOT CONTAIN the origin!!! #3. Calculate the integral of the vector field F(x,y)= (x^3(sinx)-5y, 4x+e^(y^2)) about the circle of radius 2 in the plane centered at origin, using Green's theorem. Imagine how hard that would have been without Green's theorem!! | ||
| Read Class notes, Do: #1. Suppose f is a real-valued C^1 function of n variables and F= gradient(f), a vector field on R^n. Prove that the integral of F along ANY C^1 curve gamma in R^n depends only on the endpoints of gamma, and also show it is zero if gamma is a loop. Hint: Recall how the derivative, with respect to t, of f(gamma(t)) is related to the gradient of f. Now compute the integral of F along a general gamma. #2.a. Find an expression for ALL functions f(x,y) whose partial derivative with respect to y is 4xy. b. Find a function f whose gradient is equal to F(x,y)=(2y^2+3x^2, 4xy). | ||
| Read Chapter 12 page 4 and pages 12-16 Do: 12-7 (Hint: express your loop as a "sum" of two loops covered by the previous homework assignment), #2. For the following vector fields F, find a potential function f or show that one does not exist: F=(x,y,z), F=(x+z,-y-z,x-y) F=(3y^4z^2, 4x^3z^2, -3x^2y^2). #3. For the following vector field F= (xr^p,yr^p,zr^p) where r is the square root of x^2+y^2+z^2 and p is ANY real number not -2. Prove that F is conservative in the region R^3-(0,0,0) by finding an explicit potential function f. | ||
| Read Chapter 12 pages 18-23 Section G optional and Chapter 13 pages 1-5 Do: Really read Chapter 13, problems 13-1, 13-2 Note that there will be "piecewise" formulas for the unit normals since the surfaces given are NOT C^1 but merely "piecewise" C^1. | ||
| No Class-Spring recess | ||
| Read Class Notes or Chapter 13 Sections C,E,F; DO:Click here for Problem Set. NOTE: In #2 you must do the line integral NOT the surface integral since I asked you to USE STOKES Theorem!! | ||
| Note my error on previous assignment reading was supposed to be from Chapter 13 not 12. Read:Chapter 13 Sections D,E Do: 13-6 assume m>0 (maybe theta is a good parameter, remember your half-angle and double angle formulas)(to use the Stoke's Theorem we quoted in class, you will need to figure out what is the F=(P,Q,R) that leads to the given line integral. You also get to choose your own surface). 13-8 do only the integral on the lower left. You can use Stokes Theorem if you want to. Extra Credit: Due next Monday 13-4 last integral only. Explain in enough detail that it can be read easily. | ||
| Read Chapter 13 Read pages 14-27 Note this is a lot!!! Do: 13-16, #2 In example at the bottom of page 25 called delta theta, draw the force vectors at the points (1,0),(0,1),(-1,0),(0,-1). Observe how it looks like F is "circulating" around the origin. But we know that curlF=0 so it is not circulating with respect to any axis at any point. How can this be? The answer is that curlF dot normal at a point p gives a measure of the INFINITESSIMAL circulation at p about N. (consider N is the unit vector in +z or -z direction). Consider the circulation near the point (1,0) by drawing a few force vectors at points on the x-axis NEAR (1,0). Note they all have the same length. So in a little neighborhood about (1,0) one DOESN'T see circulation!! The reason that we seem to see a "global" circulation, yet there is no "local circulation is due to the singularity at the origin (because if F WERE defined inside the region enclosed by a big loop and its curl were zero everywhere (i.e. locally) then the line integral would have to be zero). That is normally Stokes theorem is saying that the sum of the local circulations over the region adds up to give the "global" circulation around the loop. In this case this is foiled due to the singularity. Now run the same exercise as above with the field F=(-y,x,0) at bottom of page 25. These fields agree at the first four points I mentioned. But now you should "see" the circulation at the point (1,0) this time. #3. (extra credit) Where in the proof of Stokes theorem did we really need that the surface was orientable? #4. Find the line integral of F=(5y^2,0,0) around the (edges of) the square of side length 2 in the plane, centered at the origin, with sides perpendicular/parallel to the axes. Orient it CLOCKWISE. | /tr>||
| Read Chapter 14 pages 1-4; Here is the assignment but this will be due Wednesday with the next assignment: 14-1, 14-2 | ||
| Read Chapter 14 pages 5-9 Do: #1. Calculate the flux of F=(x^3,y^3,z^3) across the unit sphere in 3-space using the outward normal. #2. Use the divergence theorem (in an indirect way) to evaluate the integral of 2x+2y+z^2 over the unit sphere in 3-space. This is a surface integral of a scalar function. | ||
| Do: #1. As regards the hand-out from class today: For each planar vector field A,B,...,E in R^3and each point p and p' state whether the divergence is positive, zero or negative and state the same for the curl and state the general direction of the curl.#2. If F is a vector field in R^3, a VECTOR POTENTIAL for F is a vector field such that F=curl(G). Use a previous HW problem in Chapter 14 to deduce a NECESSARY condition for such a G (C^2) to exist. Now suppose F=(y,z,x). Find a vector potential of the form G=(f,g,0) by solving the differential equations. Note that you get to make some arbitrary choices. #3. Suppose G is a vector potential for F. Show that G+gradf is also a vector potential for F for any C^2 scalar function f. (Thus a vector potential, when it exists, is highly non-unique). | ||
| Chapter 14 Read pages 9-10; Optional: Read rest of proof of Gauss' theorem on pages 11-12 Do: 14-22 (The gravitational field F (inverse square radial field-see #14-21), as we have seen, is undefined at the origin, but has zero divergence elsewhere. Yet it has no vector potential G defined on R^3-origin, as this problem shows. Thus the NECESSARY condition that divF=0 is not sufficient on a general region). #2. Let F=(x^2,y^2,z^2). Let D be the unit CUBE in the positive octant where x,y, and z each vary between 0 and 1. Compute the flux of F "out" through the boundary of D in two different ways (one using Gauss, theorem). Draw a picture for yourself. | ||
| To study for final exam: Review book and class notes and make a written outline for yourself. Don't forget equivalence relations and sup/inf. The test is heavily weighted to Chapters 10-14 (material after the midterm), Give preference to your class notes on these later chapters. Especially emphasize change of variables (10 page 28), integration on manifolds including line integrals, lengths of curves, and surface integrals, Green's, Stoke,s, Gauss' Theorems, conservative vector fields (11 page 11-22) and potentials. Know the answers to the questions on the in-class midterm. Study old problem sets, especially the problems I assigned that were not in the book. Here are some especially good problems: 9-17, 11-7, 11-12, the problems on the special assignment sheet from 4/10 (link above). Here are some other random problems on material from the later chapters. Here are some solutions/td> | ||
| Finals Exam Period | ||