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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク inverse function theorem ： ウィキペディア英語版 inverse function theorem In mathematics, specifically differential calculus, the inverse function theorem gives sufficient conditions for a function to be invertible in a neighborhood of a point in its domain. 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. In this case, the theorem gives 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. ==Statement of the theorem== For functions of a single variable, the theorem states that if $f$ is a continuously differentiable function with nonzero derivative at the point $a$, then $f$ is invertible in a neighborhood of $a$, the inverse is continuously differentiable, and :$\backslash bigl(f^\backslash bigr)\text{'}(f(a))\; =\; \backslash frac,$ where notationally the left side refers to the derivative of the inverse function evaluated at its value ''f''(''a''). For functions of more than one variable, the theorem states that if the total derivative of a continuously differentiable function $F$ defined from an open set of $\backslash mathbb^n$ into $\backslash mathbb^n$ is invertible at a point $p$ (i.e., the Jacobian determinant of $F$ at $p$ is non-zero), then $F$ is an invertible function near $p$. That is, an inverse function to $F$ exists in some neighborhood of $F(p)$. Moreover, the inverse function $F^$ is also continuously differentiable. In the infinite dimensional case it is required that the Fréchet derivative have a bounded inverse at $p$. Finally, the theorem says that :$J\_,$ where $()^$ denotes matrix inverse and $J\_F(p)$ is the Jacobian matrix of the function $F$ at the point $p$. This formula can also be derived from the chain rule. The chain rule states that for functions $G$ and $H$ which have total derivatives at $H(p)$ and $p$ respectively, :$J\_\; (p)\; =\; J\_G\; (H(p))\; \backslash cdot\; J\_H\; (p).$ Letting $G$ be $F^$ and $H$ be $F$, $G\; \backslash circ\; H$ is the identity function, whose Jacobian matrix is also the identity. In this special case, the formula above can be solved for $J\_$ has a total derivative at $p$. The existence of an inverse function to $F$ is equivalent to saying that the system of $n$ equations $y\_i\; =\; F\_i(x\_1,\; \backslash dots,\; x\_n)$ can be solved for $x\_1,\; \backslash dots,\; x\_n$ in terms of $y\_1,\; \backslash dots,\; y\_n$ if we restrict $x$ and $y$ to small enough neighborhoods of $p$ and $F(p)$, respectively. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「inverse function theorem」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.