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

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SUMMARY

A non-zero Jacobian is a necessary condition for proving a diffeomorphism, as it indicates that the differential of the function is invertible. When a function \( f \) is a diffeomorphism, both \( f \) and its inverse \( f^{-1} \) are differentiable. By differentiating the identity \( f \circ f^{-1} = id \) and analyzing the determinant in matrix form, one can establish that the Jacobian must be non-zero to satisfy the invertibility of the differential.

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Why is a non zero jacobian the necessary condition for a diffeomorphism? How to prove it?
 
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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?
 
if f is a a diffeo, then f o f^{-1} = id. Differentiate both side, put in matrix form and take the determinant.
 
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?
 
quasar987 said:
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.
 

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