implicit function theorem

Theorem.

Let $\Omega$ be an open subset of $\mathbb{R}^{n}\times\mathbb{R}^{m}$ and let $f\in C^{1}(\Omega,\mathbb{R}^{m})$. Let $(x_{0},y_{0})\in\Omega\subset\mathbb{R}^{n}\times\mathbb{R}^{m}$. If the matrix $D_{y}f(x_{0},y_{0})$ defined by

 $D_{y}f(x_{0},y_{0})=\left(\frac{\partial f_{j}}{\partial y_{k}}(x_{0},y_{0})% \right)_{j,k}\quad j=1,\ldots,m\quad k=1,\ldots,m$

is invertible, then there exists a neighborhood $U\subset\mathbb{R}^{n}$ of $x_{0}$ and a function $g\in C^{1}(U,\mathbb{R}^{m})$ such that

 $f(x,g(x))=f(x_{0},y_{0})\qquad\forall x\in U.$

Moreover

 $Dg(x)=-(D_{y}f(x,g(x)))^{-1}D_{x}f(x,g(x)).$
Title implicit function theorem ImplicitFunctionTheorem 2013-03-22 12:58:33 2013-03-22 12:58:33 azdbacks4234 (14155) azdbacks4234 (14155) 12 azdbacks4234 (14155) Theorem msc 26B10 FlowBoxTheorem DerivativeAsParameterForSolvingDifferentialEquations