Geometry of Curves and Surfaces, Prentice-Hall, 1976.
Handouts:Handout 1. A Fixed Point Theorem
Midterm Exam: After class on
Thursday Oct 23, envelopes containing a takehome exam 1 will be
distributed. The completed exams should be returned at the beginning of
class the following Thursday, Oct. 30. (Envelopes containing the exam
are also outside of HB430). You should work on the exam for
(continuous) hours. You
may use books, notes, etc., but, of course, work alone. It will
the class lectures through Oct 16, Do Carmo Page 143, and Handouts 1,
2, 3. In Handout 4, understand the statements (but not
necessarily the proofs) of the Inverse Function Theorem (especially
n=2) and Implicit Function Theorem (especially m=2 or 3, n=1)
Final Exam: After class on Thursday, Dec.4,
envelopes containing the takehome final exam will be distributed. The
completed exams should be returned to my office HB430 or the Math.
office HB220 on or before Wed. Dec.17. You should work on the
exam for at most 4(continuous) hours. You may use books, notes, etc.,
but, of course, work alone. It will cover the class
lectures, the handouts, and the DoCarmo sections which we studied
sometime in the course (see the list below) .
Homework 6, due. Thurs.,Oct.16 -- Sec.2.5: 1(b), 2, 5, 9, 11(a)(b), 14(a)(b) (Hint: Find A and B so that grad f = A x_u + B x_v by using the chain rule which gives the formulas f_u = df_p(x_u) = (grad f)* x_u and f_v = (grad f)* x_v.) Sec.2.6: 7
Homework 7, due. Tues.,Oct.23 -- Sec.2.4: 10, 11, Sec.2.5: 12, Sec.3.2: 4, 8(a)(b)(c)
Homework 8, due. Tues.,Nov 11-- Sec.3.2: 5, 14, 17, Sec.3.3: 2, 4, 21
Homework 9, due. Tues.,Nov 18-- Sec.3.4: 5, 8, 10ab, Sec.3.5: 11(a)(b)(c)
(Hint for (b). Part(a) shows the parallel surface has the same normal. One can use equations p.154(1) for the parallel surface to find the corresponding a11, a21, a12, a22 . A shorter proof involves showing that the corresponding principal curvatures of the parallel surface are k1/(1-ak1), k2/(1-ak2) .)
Homework 10, due. Tues.,Nov 25-- Sec.4.2: 1, 14, 15, Sec.4.3: 1, 4, 9
Homework 11, due. Tues.,Dec 2*-- Sec.4.4: 1(a)(Hint: Consider a(s) wedge N(a(s)) where a(s) is a unit speed parameterization of C and N(p) is the normal to the surface at p .)`, 9, 15(a)(b)
Homework 12, due. Tues.,Dec 4*-- Sec.4.5: 1, 2(See p.157), 3 (Assume S is convex), 6(a)(c)
Late homework is accepted, but try not to get too far behind.