inverse function theorem
Let $\mathbf{f}$ be a continuously differentiable, vectorvalued 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 nonzero, 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 oneone;

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, realvalued function defined on the open interval^{} $I$. If for some point $a\in I$, ${f}^{\prime}(a)\ne 0$, 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  20130322 12:58:30 
Last modified on  20130322 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 