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inverse function theorem (Theorem)

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, $ \vert J_{\mathbf{f}}(\mathbf{a}) \vert$, 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

  1. $ \mathbf{a} \in X$, $ \mathbf{f}(\mathbf{a}) \in Y$;
  2. $ Y = \mathbf{f}(X)$;
  3. $ \mathbf{f}:X \to Y$ is one-one;
  4. $ \mathbf{g}$ is continuously differentiable on $ Y$ and $ \mathbf{g}(\mathbf{f}(\mathbf{x})) = \mathbf{x}$ for all $ \mathbf{x} \in X$.

Simplest case

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)]$.

Note

The inverse function theorem is a special case of the implicit function theorem where the dimension of each variable is the same.



"inverse function theorem" is owned by vypertd.
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See Also: derivative of inverse function, Legendre Transform, derivative as parameter for solving differential equations


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proof of inverse function theorem (Proof) by paolini
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Cross-references: variable, dimension, implicit function theorem, decreasing, increasing, strictly monotonic, neighbourhood, open interval, function, Jacobian, point, open set, mapping, vector-valued function, continuously differentiable
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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 24394 times total.

Classification:
AMS MSC03E20 (Mathematical logic and foundations :: Set theory :: Other classical set theory )

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