implicit function theorem
ImplicitFunctionTheorem
2002-08-24 06:21:57
2008-12-13 15:01:05
Theorem
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\begin{thm*}
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
\[
Dg(x) = - (D_y f(x,g(x))) ^ {-1} D_x f(x,g(x)).
\]
\end{thm*}