Differential geometry is the study of geometric figures using the methods of calculus. It has a long and rich history. See the brief biographies in the links to Some Classical Geometers below. With origins in cartography, it now has many applications in various physical sciences, e.g., solid mechanics, computer tomography, or general relativity. In this course we develop much of the language and many of the basic concepts of differential geometry by consideration of curves and surfaces in ordinary3 dimensional Euclidean space. Some of the most striking results in geometry involve relations between local concepts (e.g. the index of a vectorfield or the curvature of a surface at a point) with a global quantity such as the number of handles of a closed surface. For example the Poincare-Hopf theorem implies that any continuous vectorfield tangent to the sphere must vanish somewhere, or the Gauss-Bonnet theorem implies that the integral of the curvature over any donut-shaped surface, however lumpy, must be zero.
Some of the topics we hope to cover include:
Prerequisites for the course include some familiarity with calculus, linear algebra, and ordinary differential equations
Meets: MWF 3-3:50 in Herman Brown 227
Instructor: Robert Hardt, Herman Brown 430; Office hours: 1-2 MWF (and others by appt.),
Email: hardt@rice.edu, Telephone: ext 3280
Homepage: http://math.rice.edu/~hardt/401F07/
Text: Sebastian Montiel and Antonio Ros, Curves and Surfaces, American Mathematics Society, 2005.
Handouts and other notes:
Handout 1. A Fixed Point Theorem
Handout 2. A Very Short Course in Local O.D.E. Theory
Handout 3. The Implicit and Inverse Function Theorems
Calculations for a graph of a function.
Some Remarks on Degree and Gauss-Bonnet.
Chapter 1 : 1.1, 1.2, 1.3, 1.4, 1.5
Chapter 2 : 2.1, 2.2, 2.3, 2.4, 2.5, 2.6
Chapter 3 : 3.1, 3.2, 3.3, 3.4, 3.6
Chapter 5 : 5.1, 5.2, 5.3, 5.4, 5.6, 5.7
Chapter 6 : 6.1, 6.2, 6.3, 6.4, 6.5
Chapter 7 : 7.1, 7.2, 7.3, 7.5
Chapter 8 : 8.1, 8.2, 8.3, 8.4 8.5
Gnuplot: Though we will not be using graphics packages in this class, I wanted to point out a handy one for which Rolf Ryham has kindly provided some information at http://www.owlnet.rice.edu/~rjr1/gnuplot.html.
Substitute: I will be at a conference in Korea from Thursday Sept.6 through Tuesday Sept.11. Dr. Dan Cole will give the lectures on Sept.7 and Sept. 10. For questions about the class, homework, etc. please contact our TA Peter Horn, phorn@rice.edu, Herman Brown 45, (713)348-2785. I'll also be checking my e-mail.
Midterm Exam: This will be distributed (in closed envelopes) at the end of class on Wednesday, Oct. 17 and is due by the beginning of class Friday, Oct. 26. This take-home exam is to be completed during 2 continuous hours, and the use of either the course textbook, Montiel-Ros, or any notes is permitted. It will cover the first 3 chapters of Montiel-Ros. Before beginning the exam, you may wish to wait until you are sure you understand all the homework problems. We will try to get HW7 graded fairly quickly. Solutions to Midterm Exam.
Final Exam: This will be distributed (in closed envelopes) at the end of class on Friday, December 7 and is due by 5PM Wednesday, Dec.19. This take-home exam is to be completed during 3 1/2 continuous hours, and the use of either the course textbook, Montiel-Ros, or any notes is permitted. It will cover the whole semester. See the "End of semester syllabus" above. Before beginning the exam, you may wish to wait until you are sure you understand all the homework problems. We will try to get HW12 graded fairly quickly. Look outside my door.
M. Do Carmo, Differential Geometry of Curves and Surfaces,Prentice-Hall, 1976.
A. Gray, Modern Differential Geometry of Curves and Surfaceswith Mathematica, CRC, 1997.
H. Guggenheimer, Differential Geometry, Dover, 1997.
D. Henderson, Differential Geometry: A Geometric Introduction, Prentice-Hall,1997.
R. Millman and G. Parker, Elements of Differential Geometry ,Prentice-Hall, 1997.
B. O'Neill, Elementary Differential Geometry, 2nd Ed., AcademicPress, New York, 1997.
J. Oprea, Differential Geometry and its Applications, Prentice-Hall, 1996.
D. Struik, Lectures on Classical Differential Geometry, Dover, 1988.
Homework: There will be a homework assignment approximately every Wednesday which will be due the next Wednesday. Here is the latest homework assignment. You are encouraged to discuss and work together on the homework problems, but each student is responsible for the final preparation of his or her own solutions. At most one late homework assignment will be accepted,as long as it is turned in within seven days of its original due date. Homework will count for 40% of the final grade.
Solution to Exercise (1) on P.305.
Exams: There will be one midterm exams worth 25% of the final grade. The final exam will be take-home, and worth 35% of the final grade.
Introducing Curves by C. T. J. Dodson
Introducing Surfaces by C. T. J. Dodson
Dictionaryof special plane curves
*Thanks to Kevin Scannell for these links. Students are welcome to suggest others.
This page is maintained by Robert Hardt ( email)
Last edited 12/06/07.