Homework 1.1: if f, g is smooth, gof is also smooth, provided that gf is well-defined.

 

Answer:

 

We suppose f: and g: . We have the following two given condition: for each x

1. there is F defined on open X containing x such that

     2.   there is G defined on open Y containing f(x) such that

We can construct open set U and V such that F map U onto V; and V lies completely inside Y. Therefore, GF defined on U provides a well-defined map satisfying the definition of smooth map. The construction can be as follow:

Let K be an open set lies completely in Y and contains f(x) and define V=. Let U=.                                                                  QED.

 

Homework 1.2: Give an example of differomorphism and a proof.

 

Answer:

 

The simplest  example is the identity map. A homeomorphism is a continuous mapping between two spaces that has an inverse which is also continuous. Identity map does satisfy this definition. It and its inverse are also smooth, obviously.