If a continuously differentiable map has invertible derivative at a point, it is a local continuously differentiable diffeomorphism there.
A local diffeomorphism restricts near every point to a diffeomorphism onto an open subset. In particular, every local diffeomorphism is an open map.
The displayed smooth map from to the punctured plane has Jacobian determinant one: in polar coordinates its radius is and its angle is , so the determinant is . Nevertheless for every integer . Thus a local volume-preserving map need not be injective or have bounded weighted inverse multiplicity. Compactly supported data distributed over many inverse branches disprove a determinant-only dispersion with a nonlinear velocity map estimate.
If and the partial derivative with respect to is invertible at , then the nearby zero set is the graph of a continuously differentiable function.
If a holomorphic function satisfies and , then near its zero set is the graph of a unique holomorphic function. The analogous statement with uses as the local coordinate.
Implicit differentiation differentiates an identity and uses the chain rule to solve for derivatives of the implicitly defined function. In one dimension, when .
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The Inverse Function Theorem is a fundamental result in differential calculus that provides conditions under which a function has a differentiable inverse. It states the following: ### Statement of the Theorem: Let \( f: U \to \mathbb{R}^n \) be a function defined on an open set \( U \subseteq \mathbb{R}^n \). If: 1. \( f \) is continuously differentiable in \( U \), 2.