Analysis MOC

Inverse function theorem

Let be open, be differentiable, and . Then if the total derivative is non-singular, there exist open neighbourhoods of in and of in such that

is a diffeomorphism, anal i.e. is locally a diffeomorphism at .

Proof

proof

The constructive proof relates to Newton’s method.

Corollary

The above theorem is easily extended to a differentiable map between differentiable manifolds . If the Tangent map is a Linear isomorphism, then is a local diffeomorphism, as one expects from the Linearization dogma.

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