Differential Geometry of Curves and Surfraces
Math 436 Fall 2009


Official Course Number: MATH 436 (Section 0101) (3 credits)
Grade Method: REG/P-F/AUD.
Lectures: Tuesday-Thursday 9:30 - 10:45, B0421 Math Building
Professor: Bill Goldman (wmg@math.umd.edu)
Office: 3106 Math Building
Phone: 301-405-5124
Office hours: Tuesday/Thursday 1:00- 1:45 or by appointment.

Prerequisites: MATH241; and either MATH240 or MATH461; and two 400-level MATH courses (not including MATH400, 461 and 478).

Curves in the plane and Euclidean space, moving frames, surfaces in Euclidean space, orientability of surfaces; Gaussian and mean curvatures; surfaces of revolution, ruled surfaces, minimal surfaces, special curves on surfaces, "Theorema Egregium"; the intrinsic geometry of surfaces.

Text: Elementary Differential Geometry, First Edition, by Andrew Pressley, Springer Undergraduate Mathematics Series, ISBN 1-85233152-6 (2001)

Tentative Syllabus

  1. Curves
    1. Plane curves, signed curvature
    2. Space curves
    3. Curvature and torsion, fundamental theorem
  2. Regular surfaces
    1. Implicit function theorem
    2. Tangent planes, linearization
  3. Extrinsic geometry of surfaces
  4. Intrinsic geometry of surfaces

Tentative Schedule

  • First problem set: 1.14, 2.2, 2.7-2.9 (plane curves), due Thursday 9/10
  • Second problem set: 2.14, 2.15, 1.9
  • Problem Session: Tuesday, September 22.
  • Third problem set: 4.3, 4.8, 4.17, 4.18, 4.19, 4.28
  • The first test was given Tuesday, October 13.
  • Fourth problem set: 5.1, 5.2, 5.3, 5.4, 5.5, 5.8, 5.11, 5.13, 5.17, 5.19
  • Write up a more conceptual proof of Theorem 5.2 (pp. 108-109) along the lines of the lecture on Thursday, November 5.
  • Read Chapter 6. Fifth problem set: 6.7, 6.9, 6.10, Problems 6.11-6.14 (on geodesic torsion) are recommended, but optional.
  • Read Chapter 7.
  • Read Chapter 8, 10, 11.
  • The final text will be given Thursday, December 11.

Makeups:
There will be no makeups for quizzes or midterms. If illness, a death in the family, car trouble or a faulty alarm clock cause you to miss a midterm -- that is the midterm or section quiz you will drop. So do not decide an earlier midterm or quiz is going to be your bad score -- if you miss a later one, then that is going to be your bad score. When you have compelling reasons for missing an exam, share them with me or your TA.
**If you know BEFORE an exam that you have a conflict, contact me in advance. In this case, it is sometimes possible to arrange an early exam.**

Deadlines:
Deadlines. Late homework will not be accepted. Here are two reasons: we want to use resources well (it is much more time consuming to grade homework when it is late), and you will learn better if you do the work on time.

Emergency closures:
In case of an emergency that closes the University for an extended period of time (for example, due to inclement weather), be sure to access your email for instructions from me. Also check the University's home page.

Expectations/philosophy:
You are expected to come to class, do the homework, and most important of all be actively engaged in trying to understand. Two tips for success: Don't fall behind. Make friends. Help each other (especially after trying alone first).

Religious observances:
If your religion dictates that you cannot take an exam or hand in assigned work on a particular date, then contact me at the beginning of the semester to discuss alternatives. You are responsible for making these arrangements at the beginning of the semester.

Disabilities:
If you have some disability related to testing under the usual timed, in-class conditions, you may contact the office of Disabled Students Services (DSS) in Shoemaker. If they assess you as meriting private conditions and/or extra time, then you may arrange to take your tests at DSS, with extra time as they indicate. You must arrange this well in advance of a test (in particular: no retakes). Click to Disability Support Services for further information.

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wmg@math.umd.edu

Last modified: 2 December 2009