In Stewart's Calculus book, an argument is given to intuitively explain why the theorem for change of variables, using Jacobian matrix, is valid without giving the full proof.
However, there are parts that I have little intuition over and would like to understand more coherently. I would appreciate it if someone explains them to me and preferably, if it is not too hard, by giving the exact proof.
- What is it about $C^1$ Transformation that is of interest to us?
- Why should $T$ (the $C^1$ transformation) map one region to another region (and not to multiple one)? Is this because of $C^1$ transformation?
- In Example 1: "The transformation maps the boundary of S into the boundary of the image" - I know this is not generally true. But where is it true? And how do we know that it is in this particular example?
- By moving around the boundary in the counterclockwise direction, we also move around the boundary of the image in the counterclockwise direction. Why is this true?
Thank you in advance.