Inverse function theorem

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Statement

Let f:\R^n \to \R^n be a differentiable map such that the determinant of the Jacobian matrix of f at a particular point p \in \R^n, is nonzero (i.e. the Jacobian matrix is nonsingular). Then, there exist open neighbourhoods U \ni p and V \ni f(p) such that the restriction of f to U is a diffeomorphism from p to f(p).

Applications