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Let $f\colon M \rightarrow N$ be a smooth map. If $f$ is a diffeomorphism I am trying to show that the linear map $f_*$ : $T_pM \rightarrow T_{f(p)}M$ is an isomorphism for all $p \in M$. I know the derivative map $T_pf\colon T_pM \rightarrow T_{f(p)}N$ is an isomorphism, but I have trouble to proving when the map defines $T_pM \rightarrow T_{f(p)}M$.

Thank you for your help.

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    Another redundant answer, just to drive home what every else has said already. The mapping you talk about, called the push-forward of $ f $, takes a tangent vector at $ p $ to $ M $ and associates it with that at $ f(p) $ to $ N $. So the $ T_p(f) $ you speak about doesn't make sense as that point $ f(p) $ cannot be in $ M $ unless $ f: M \rightarrow M $. If that is the case then probably your question would make sense.2012-10-12

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