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If $f:M \to N$ is a smooth map between compact closed hypersurfaces $M$, $N \subset \mathbb{R}^n$, does it make sense to write the pushforward as $Df$, the total derivative? Because usually we require $M$ and $N$ to be open sets. If $f$ is a diffeomorphism, does the IFT hold true in the sense that I can write $(Df)^{-1} = Df^{-1}$?

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This is common in many textbooks, for example Guillemin and Pollack's "Differential Topology". Yes, IFT holds. See the text. – Ryan Budney Dec 12 '12 at 0:03

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