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inverse function theorem
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(Theorem)
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Let $\mathbf{f}$ be a continuously differentiable, vector-valued function mapping the open set $E \subset \mathbb{R}^{n}$ to $\mathbb{R}^{n}$ and let $S = \mathbf{f}(E)$ . If, for some point $\mathbf{a} \in E$ , the Jacobian, $| J_{\mathbf{f}}(\mathbf{a})
|$ , is non-zero, then there is a uniquely defined function $\mathbf{g}$ and two open sets $X \subset E$ and $Y \subset S$ such that
- $\mathbf{a} \in X$ , $\mathbf{f}(\mathbf{a}) \in Y$ ;
- $Y = \mathbf{f}(X)$ ;
- $\mathbf{f}:X \to Y$ is one-one;
- $\mathbf{g}$ is continuously differentiable on $Y$ and $\mathbf{g}(\mathbf{f}(\mathbf{x})) = \mathbf{x}$ for all $\mathbf{x} \in X$ .
When $n = 1$ , this theorem becomes: Let $f$ be a continuously differentiable, real-valued function defined on the open interval $I$ . If for some point $a \in I$ , $f'(a) \neq 0$ , then there is a neighbourhood $[\alpha, \beta]$ of $a$ in which $f$ is strictly monotonic. Then $y \to f^{-1}(y)$ is a continuously differentiable, strictly monotonic function from
$[f(\alpha), f(\beta)]$ to $[\alpha, \beta]$ . If $f$ is increasing (or decreasing) on $[\alpha, \beta]$ , then so is $f^{-1}$ on $[f(\alpha), f(\beta)]$ .
The inverse function theorem is a special case of the implicit function theorem where the dimension of each variable is the same.
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"inverse function theorem" is owned by azdbacks4234. [ owner history (1) ]
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Cross-references: variable, dimension, implicit function theorem, decreasing, increasing, strictly monotonic, neighbourhood, open interval, theorem, function, Jacobian, point, open set, mapping, vector-valued function, continuously differentiable
There are 6 references to this entry.
This is version 6 of inverse function theorem, born on 2002-08-24, modified 2002-12-28.
Object id is 3346, canonical name is InverseFunctionTheorem.
Accessed 28834 times total.
Classification:
| AMS MSC: | 03E20 (Mathematical logic and foundations :: Set theory :: Other classical set theory ) |
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Pending Errata and Addenda
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