inverse function 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, $|J_{\mathbf{f}}(\mathbf{a})|$ , 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

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 $\mathbf{x}\in X$ .

0.0.1 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 $a\in I$ , $f^{\prime}(a)\neq 0$ , 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 $[f(\alpha),f(\beta)]$ .

0.0.2 Note

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

Title	inverse function theorem
Canonical name	InverseFunctionTheorem
Date of creation	2013-03-22 12:58:30
Last modified on	2013-03-22 12:58:30
Owner	azdbacks4234 (14155)
Last modified by	azdbacks4234 (14155)
Numerical id	9
Author	azdbacks4234 (14155)
Entry type	Theorem
Classification	msc 03E20
Related topic	DerivativeOfInverseFunction
Related topic	LegendreTransform
Related topic	DerivativeAsParameterForSolvingDifferentialEquations
Related topic	TheoryForSeparationOfVariables