Math 121A Differential Geometry Winter 2004

Updated 3/19/04

Have a great spring break!

Solution suggestions for the take home final. (No one took the in-class, so I didn't write up solutions for it.)

Some practice problems (mostly computational) and some more practice problems (more proofish).
Correction to #1.b in the second set: your answer will probably also depend on the point value alpha(t_0).
Solution suggestions for the second set of practice problems.

INSTRUCTOR INFORMATION

Instructor: Debra Lewis
Office: 213 Kerr
Phone: 459-2718
E-mail: lewis@math.ucsc.edu
Lecture: TTh 12:00-1:45, Oakes 222
Office hours: Monday 3:30-5:00, Thursday 10:00-11:00, and by appointment

TEACHING ASSISTANT INFORMATION

TA: Kurt Hinderer
E-mail: vladimir@cats.ucsc.edu
Discussion section: T 8:00-9:10 PM, Crown Classroom 202
Office hours: W 6:00-7:30 PM

E-HANDOUTS (optional)

Midterm solution suggestions
Practice problems for midterm

Solution suggestions for practice problems for midterm (page 1, 2, 3, 4, 5)
and for a few homework problems: Sec. 1.5, #12 (page 1, 2), Sec. 2.2, #17b (page 1, 2).
Proofs of the Inverse Function Theorem:
Relatively straightforward proof, referenced in text: Buck.
Very slick, pretty proof using the contraction mapping theorem (a very powerful and versatile result, proved in the handout) and some functional analysis: Arnold.
If you have a copy of Marsden and Tromba's Vector Calculus, there is a simple version of the IFT in Section 3.5.

TEXT

Differential Geometry of Curves and Surfaces, by M. Do Carmo

HOMEWORK POLICIES

There will be weekly homework assignments, given in class. Homework assignments will be available online, but assignments are not `locked in' until they've been given in lecture.

Late homework will be discounted and, at the discretion of the grader and/or the instructor, may not be accepted.

Your homework should be neatly written and well-organized, with the pages securely fastened together and your name on every page. Clearly number the exercises and try to submit them in numerical order; if any problems are out of sequence, indicate that very clearly at the beginning of the assignment. (The grader should not have to hunt through several pages to find a particular problem.)

You may use symbolic computation packages to carry out the routine calculations in the homework. If you do so, please write up the key steps of your solutions by hand, indicating which parts were carried out by computer and attaching a hardcopy of your computer work.
Mathematica and Maple are excellent multipurpose mathematical software packages with powerful symbolic capabilities. Mathematica 3.0 is supposedly available in all instructional computing PC labs. However, the computing lab installations of mathematical software tend to be unstable. Computer difficulties do not justify late or incomplete assignments.

EXAMS

Midterm: February 5.
Final exam: March 16, 4-7 PM.
No make-up exams will be given.

GRADING

Your overall score in the course will be the best of three weighted averages of your homework, midterm, final exam, and (optional) final project scores. Your lowest homework score will be dropped.

TENTATIVE LECTURE SCHEDULE:

Tuesday	Thursday
January 6: 1.2-3 Parametrized curves, arc length	January 8: 1.4-5 The cross product, local theory of curves
January 13: 1.5 The local theory of curves (cont.), solutions of ODEs	January 15: 2.2 Regular surfaces
January 20: 2.2 Level sets and the inverse function theorem	January 22: 2.3 Change of parameters
January 27: 2.4 The tangent plane and linear algebra review	January 29: 2.4 The tangent plane (cont.)
February 3: 2.5 The first fundamental form	February 5: MIDTERM
February 10: 3.2 The Gauss map	February 12: 3.2 The Gauss map (cont.)
February 17: 3.3 The Gauss map in local coordinates	February 19: 3.3 The Gauss map in local coordinates (cont.)
February 24: 4.2 Isometries	February 26: 4.2 Conformal maps
March 2: 4.4 Parallel transport	March 4: 4.4 Geodesics
March 9: 4.4 Geodesics (cont.) and/or final project presentations	March 11: Final project presentations