# Diffeomorphism and Jacobian

1. Dec 5, 2009

### kof9595995

Why is a non zero jacobian the necessary condition for a diffeomorphism? How to prove it?

2. Dec 5, 2009

### Hurkyl

Staff Emeritus
A necessary condition, you mean.

Have you tried applying the definitions of any of the terms involved, and/or some basic structural theorems? By doing so, what sorts of equivalent statements were you able to produce?

3. Dec 5, 2009

### quasar987

if f is a a diffeo, then f o f^{-1} = id. Differentiate both side, put in matrix form and take the determinant.

4. Dec 5, 2009

### kof9595995

Emm....If I have a diffeomorphism f, that means f and f^-1 are both differentiable. If I can prove it's differential is also invertible, then Jacobian must be non zero. Emm...then what I can think of is: is the differential of f^-1 equals the inverse of the differential of f?

5. Dec 5, 2009

### kof9595995

Thanks, now I get it.