Inverse function theorem
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In mathematics, the inverse function theorem gives sufficient conditions for a vector-valued function to be invertible on an open region containing a point in its domain.
The theorem states that if at a point p a function f : Rn → Rn has a Jacobian determinant that is nonzero, and f is continuously differentiable near p, then it is an invertible function near p. That is, an inverse function to f exists in some neighborhood of f(p).
The Jacobian matrix of f−1 at f(P) is then the inverse of Jf, evaluated at p.
The inverse function theorem can be generalized to differentiable maps between differentiable manifolds. In this context the theorem states that for a differentiable map f : M → N, if the derivative of f, (Df)p : TpM → Tf(p)N is a linear isomorphism at a point p in M then there exists an open neighborhood U of p such that f|U : U → f(U) is a diffeomorphism. Note that this implies that M and N must have the same dimension.
If the derivative of f is an isomorphism at all points p in M then the map f is a local diffeomorphism.