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 $(Df)^{-1} = Df^{-1}$?
Thanks