Implicit Function Theorem

(Theory)

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Theory
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What is Implicit function theorem?

In mathematics, more specifically in multivariable calculus, the implicit function theorem is a tool that allows relations to be converted to functions of several real variables. It does so by representing the relation as the graph of a function. There may not be a single function whose graph can represent the entire relation, but there may be such a function on a restriction of the domain of the relation. The implicit function theorem gives a sufficient condition to ensure that there is such a function. More precisely, given a system of m equations fi (x1, ..., xn, y1, ..., ym) = 0, i = 1, ..., m (often abbreviated into F(x, y) = 0), the theorem states that, under a mild condition on the partial derivatives (with respect to the yis) at a point, the m variables yi are differentiable functions of the xj in some neighborhood of the point. As these functions can generally not be expressed in closed form, they are implicitly defined by the equations, and this motivated the name of the theorem. In other words, under a mild condition on the partial derivatives, the set of zeros of a system of equations is locally the graph of a function.

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functionfunctions and mappingmathematical relationmathematical theorempropositionstatementtheoremtheorems in analysistheorems in calculutheorems in real analysistheory

Translations

Implicita funktionssatsen (sv)Satz von der impliziten Funktion (de)Teorema da função implícita (pt)Teorema de la funció implícita (ca)Teorema de la función implícita (es)Teorema delle funzioni implicite (it)Théorème des fonctions implicites (fr)Теорема о неявной функции (ru)Теорема про неявну функцію (uk)음함수 정리 (ko)陰函数定理 (ja)隐函数定理 (zh)

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