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Diffeomorphism and Jacobian

  1. Dec 5, 2009 #1
    Why is a non zero jacobian the necessary condition for a diffeomorphism? How to prove it?
  2. jcsd
  3. Dec 5, 2009 #2


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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?
  4. Dec 5, 2009 #3


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    if f is a a diffeo, then f o f^{-1} = id. Differentiate both side, put in matrix form and take the determinant.
  5. Dec 5, 2009 #4
    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?
  6. Dec 5, 2009 #5
    Thanks, now I get it.
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