What is the significance of a non-zero Jacobian in proving diffeomorphism?

[kof9595995]

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

 

[Hurkyl]

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?

 

[quasar987]

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

 

[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?

 

[kof9595995]

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

Thanks, now I get it.