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Math Help - tangent spaces

  1. #1
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    tangent spaces

    It is an easy problem, but I am not sure that my argument is correct.

    Suppose that f: X ­-> Y is a diffeomorphism, and prove that at each x
    its derivative dfx is an isomorphism of tangent spaces.
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    Let g:Y -> X be the inverse. Then d(f o g)(y) = d(1_Y)(y)=1_{Ty} and similarily d(g o f)(x) = d(1_X)(x)=1_{Tx}. Using the chain rule, you have your result!

    Next time, you can post your solution and we can look at what you did.
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