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

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    Diffeomorphism Jacobian
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Discussion Overview

The discussion revolves around the significance of a non-zero Jacobian in proving that a function is a diffeomorphism. It explores the definitions and implications of differentiability, invertibility, and the relationship between a function and its inverse in the context of differential geometry.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant questions why a non-zero Jacobian is necessary for a diffeomorphism and seeks a proof.
  • Another participant suggests applying definitions and structural theorems to produce equivalent statements related to the condition.
  • A participant states that if a function is a diffeomorphism, then the composition of the function and its inverse equals the identity, and proposes differentiating this relationship.
  • There is a consideration that if a diffeomorphism exists, both the function and its inverse must be differentiable, leading to the idea that the differential of the function's inverse is the inverse of the differential of the function.
  • A participant reiterates the differentiation of the identity relationship and expresses understanding after the discussion.

Areas of Agreement / Disagreement

Participants express varying degrees of understanding regarding the necessity of a non-zero Jacobian, and while some points are clarified, the overall discussion remains unresolved with multiple perspectives presented.

Contextual Notes

There are limitations regarding the assumptions made about differentiability and the specific definitions of diffeomorphism that are not fully explored in the discussion.

kof9595995
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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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