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I am not able to solve this problem:

Prove that if $f:M\rightarrow N$ is $C^{\infty}$, one-to-one, onto, and everywhere non-singular, then $f$ is a diffeomorphism.

This $f$ is a diffeomorphism $\iff$ $df$ is surjective everywhere, right? Then how to proceed?

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    My guess is that he's referring to this particular $f$, in the last line.2012-06-28
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    Ah, good point Dylan.2012-06-28
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    @Zev But it's a good thing to point out! I was typing out a comment to the same effect until I saw yours. I'll try to make it clearer in the edit. To Mex: If you are truly under the impression that for a general smooth map $M \to N$, surjectivity of the derivative at all points means that the map is a diffeomorphism, then you should tell us.2012-06-28
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    Since $f$ is a bijective smooth map it is a diffeomorphism iff $f^{-1}$ is smooth. This is guaranteed by the non-singularity condition , which means that the differential $T_mf:T_mM\to T_{f(m)}N$ is invertible at all $m\in M$.2012-06-28
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    Exactly. As Georges points out this is an application of the Inverse Function Theorem and doesn't require you to worry about $df$ being surjective everywhere.2012-06-28
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    Can I ask whether or not these problems are homework? In the past day you've asked questions ranging from confusion about the definition of differentiable structure on $\mathbb{R}$ to flows on manifolds. Are these review questions for a final or qual or something? Are you self-studying? It would be easier to know how to answer this string of questions if we knew more background.2012-06-28
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    Matt, I am self studying the book by FW Warner, and I am in first Chapter Now, solving problems also.2012-06-29
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    @Mex I loved that book once I had thoroughly gone through Lee's Introduction to Smooth Manifolds because of how succinctly the material was presented. Can I recommend supplementing Warner with Lee because Lee is incredibly thorough presenting lots of examples, motivation, and intuition.2012-06-29
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    Dear Matt, Iam simply pleased and delighted at your suggestion.Collecting Lee and starting reading it :)2012-06-29

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