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Math Help - diffeomorphism

  1. #1
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    diffeomorphism

    Looking for help with a textbook problem.

    Give an example to demonstrate that a one-to-one and onto mapping need not be a diffeomorphism. (Hint: Take m=n=1)

    I'm assuming that the hint means let F: R^1 \rightarrow R^1

    I'm very stuck, however and having trouble thinking this through. Help and/or a couple of examples would be greatly appreciated.
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  2. #2
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    Take F(x)= x/(x+1) for x in [0,1] and F(x)=x for x>1.
    (reflect this to the negative side).
    I believe it's injective and onto, but F its derivative at x=1 isn't defined.
    ( F'^-(1)=1/2 -1/4=1/4 and F'^+(1)=1).
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  3. #3
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by Zennie View Post
    Give an example to demonstrate that a one-to-one and onto mapping need not be a diffeomorphism. (Hint: Take m=n=1)

    For example, f(x)=x^3


    Fernando Revilla
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