MATH 113, FALL 2012
HOMEWORK 1
1. Let : I R3 (where I R is an interval) be a parametrized curve and let v R3 be a xed
vector. Assume that (t) is orthogonal to v for all t I and that (0) is also orthogonal to v .
Prove that (t) is orthogonal to v for all t
MATH 113, FALL 2012
HOMEWORK 2, DUE SEPTEMBER 13
1. Let f : R2 R be a C 1 function (i.e., f has continuous partial derivatives) such that the
gradient f (p) of f is nonzero for every p R2 .
(a) Show that every level curve of f is orthogonal to the gradien
MATH 113, FALL 2012
HOMEWORK 6, DUE NOVEMBER 20
1. Exercise 8.1.9 from Pressley
2. Exercise 8.2.2 from Pressley
3. Show that at the origin of the hyperboloid z = axy we have K = a2 and H = 0.
4. Show that at a point p of a surface S the sum of normal curv
MATH 113, FALL 2012
HOMEWORK 4, DUE OCTOBER 18
1. Exercise 4.1.2 from Pressley.
2. Show that the antipodal map A : S 2 S 2 of the unit sphere S 2 dened by
A(x, y, z ) = (x, y, z )
is a dieomorphism.
3. Show that the paraboloid P : z = x2 + y 2 is dieomorp
MATH 113, FALL 2012
HOMEWORK 5, DUE NOVEMBER 1
1. Exercise 6.1.1 from Pressley
2. Exercise 6.1.3 from Pressley
3. Exercise 6.1.4 from Pressley
4. Let S be a surface of revolution with axis of revolution . Show that rotations about
isometries of S.
are
5.
MATH 113, FALL 2012
HOMEWORK 3, DUE SEPTEMBER 27
1. Let : I R3 be a smooth curve parametrized by arc-length whose curvature is always
positive. Let (s) = (x(s), y(s), z(s). Suppose that 0 I, (0) = (0, 0, 0), t(0) = (1, 0, 0) and
n(0) = (0, 1, 0). Show tha