Math 3B03 – Assignment 3
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Math 3B03 – Assignment 3
Problems:
1. Fix a, b, c > 0. Consider the ellipsoid
(a) Prove that E is a smooth surface. (Hint: Adapt the surface patches for the sphere which were given in exercise 4.1.2.)
(b) Find one of the (non-trivial) transition maps between allowable patches in your atlas.
(c) Find the tangent plane
2. Find an example of a surface which fails to be a smooth surface. Justify all of your claims.
3. Let
(a) Prove that is diffeomorphic to the xy-plane T in
Hint: Stereographic projection: consider the maps given by and given by
(b) Deduce that has an atlas which consists of just one allowable surface chart.
4. Let be a linear transformation.
(a) If f is invertible and S is a smooth surface, prove that f(S) is a smooth surface.
(b) If f fails to be invertible, must f(S) be a smooth surface? Justify your answer with a proof or counterexample.
2021-11-05