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Differential Geometry : Inverse Function Theorem, Isomorphisms and Diffeomorphisms

Inverse Function Theorem. Let M and N be differentiable manifolds, φ : M -> N a differentiable mapping and p Є M such that dφp : TM ?> T(p)N is an isomorphism. Prove that is a local diffeomorphism at p; that is, there are neighborhoods U C M of p and V C N of p(p) such that φ : U ?> V is a diffeomorphism.
Hint: Coordinatize!

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Inverse Function Theorem, Isomorphisms and Diffeomorphisms are investigated.

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