Inverse Function Theorem

(Theory)

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Theory
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What is Inverse function theorem?

In mathematics, specifically differential calculus, the inverse function theorem gives a sufficient condition for a function to be invertible in a neighborhood of a point in its domain: namely, that its derivative is continuous and non-zero at the point. The theorem also gives a formula for the derivative of the inverse function.In multivariable calculus, this theorem can be generalized to any continuously differentiable, vector-valued function whose Jacobian determinant is nonzero at a point in its domain, giving a formula for the Jacobian matrix of the inverse. There are also versions of the inverse function theorem for complex holomorphic functions, for differentiable maps between manifolds, for differentiable functions between Banach spaces, and so forth.

Technology Types

differential topologyfunctioninverse functionmathematical relationmultivariable calculupropositionstatementtheoremtheorems in analysistheorems in calculutheorems in real analysistheorems in topologytheory

Synonyms

Constant rank theoremDerivative rule for inversesInverse transformation theoremInversion theorem

Translations

Inversa funktionssatsen (sv)Teorema da função inversa (pt)Teorema de la funció inversa (ca)Teorema de la función inversa (es)Teorema della funzione inversa (it)Théorème d'inversion locale (fr)Теорема про обернену функцію (uk)역함수 정리 (ko)反函数定理 (zh)逆函数定理 (ja)

Tech Info


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