Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit


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.