Inverse function theorem

The inverse function theorem states that a continuously differentiable map with an invertible derivative at a point has a differentiable local inverse near that point.

Jacobian matrix: The matrix of first-order partial derivatives of a differentiable map. Its determinant tests invertibility for maps between equal-dimensional spaces.
Invertible matrix: A square matrix whose associated linear transformation has an inverse. An invertible derivative is the linear-algebraic condition in the theorem.
Coordinate chart: A homeomorphism from part of a space to an open subset of a coordinate space. The theorem validates local coordinates whenever the coordinate map has nonsingular derivative.
Constant rank theorem: A theorem that gives local normal forms for maps whose derivative has constant rank. It describes maps with non-full rank, extending the local picture beyond invertible derivatives.
Local diffeomorphism: A map that is a differentiable bijection with differentiable inverse on suitable neighborhoods. The theorem guarantees this structure near any point with invertible derivative.
Continuous differentiability: The property that a function has first derivatives that exist and vary continuously. Continuity of the derivative is a standard hypothesis ensuring local invertibility.
Change of variables (calculus): A formula for transforming integrals under a differentiable change of coordinates. Local inverses let coordinate transformations be reversed in integration.
Global inverse theorem: A result giving conditions under which a map is invertible over its entire domain. It requires additional global conditions beyond the theorem's pointwise local hypothesis.
Derivative: A linear map giving the first-order approximation of a function near a point. The theorem uses this local linear approximation to establish nonlinear invertibility.
Open set: A subset of a topological space in which every point has a neighborhood contained in the subset. The local inverse is defined on an open neighborhood of the image point.

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