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Let $U$ and $V$ be the open subsets in $\mathbb{R}^n$, $x\in U$ and $f:U\rightarrow V$ is a smooth function. There is an inverse function theorem which states that if the Jacobian determinant at $x$ is nonzero then there exists an open subset $U'\subset U$ such that $f|_{U'}$ is a diffeomorphism on its image.
I read some long and rather difficult proofs of this theorem. Also, I heard recently that there is an easy proof using Banach fixed point theorem. Could you tell me where I can read this? Unfortunately, I didn't manage to prove it myself.

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  • $\begingroup$ Just google and you will find what you want... $\endgroup$ – Lucien Sep 22 '13 at 11:13
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This seems to be what you're looking for: http://www.math.jhu.edu/~jmb/note/invfnthm.pdf

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