Math 4061 Homework 6
Due Thursday, 4/8/10
1. Consider the paraboloid S, dened by the equation z = x2 + y 2.
(a) Compute the principal curvatures at a point (x, y, z) of S.
(b) What are the umbilic points, where 1 = 2 ?
(c) Let (t) = (cos t, sin t, 1) be a
Math 4061 Homework 1
Due Thursday, 1/28/10
1. For t R, let (t) be a parametrized curve which does not pass through the origin. Let t0
be a time at which comes closest to the origin. Prove that (t0 ) (t0 ) = 0.
2. Let be a parametrized curve such that (t)
Math 4061 Homework 8
Due Thursday, 4/29/10
1. Let S be a compact surface in R3 , and let p S be the point of S that is furthest from the
origin. Prove that the Gaussian curvature at p is positive.
2. Let S be a compact surface in R3 , and suppose that its
Math 4061 Homework 3
Due Thursday, 2/18/10
1. For each of the following subsets of R3 , decide (i) whether that set is a surface, and (ii)
whether it is a smooth surface. Justify your answer (i.e., if the set is a surface, write down an
atlas of coordinat
Math 4061 Homework 5
Due Tuesday, 3/30/10
1. Let S be the unit sphere in R3 , and let : R2 S be the stereographic projection
(x, y) =
2y
1 + x2 + y 2
2x
,
,
1 + x2 + y 2 1 + x2 + y 2 1 + x2 + y 2
,
which covers all of S except the north pole. Show that th
Math 4061 Homework 7
Due Thursday, 4/22/10
1. Let S be the hyperboloid z = 2xy.
(a) Prove that at the origin, S has mean curvature H = 0 and Gaussian curvature K = 4.
(b) Let T be a surface obtained by turning S about the origin until the principal vector
Math 4061 Homework 4
Due Thursday, 2/25/10
1. Let T be the torus in R3 dened by the equation (r 2)2 + z 2 = 1, in cylindrical coordinates.
One way to parametrize T is via charts of the form
(, ) = (2 + cos ) cos , (2 + cos ) sin , sin ),
for coordinates (
Math 4061 Homework 2
Due Thursday, 2/4/10
1. Let : R R2 be a smooth unit-speed curve in the plane. Let be the same curve
traversed backwards: that is, (t) = (t). How does the signed curvature s at (t) relate to
the signed curvature at (t)?
2. Let : R R2 b