Implicit function theorem

Implicit function theorem. Let $U\subseteq\mathbb R^{n+m}$ be open, and let $F:U\to\mathbb R^m$ be continuously differentiable. Write points as $(x,y)$ , where $x\in\mathbb R^n$ and $y\in\mathbb R^m$ . Suppose $(a,b)\in U$ , $F(a,b)=0$ , and the partial Jacobian $D_yF(a,b)$ is invertible. Then there are neighborhoods $A$ of $a$ and $B$ of $b$ , with $A\times B\subseteq U$ , and a unique continuously differentiable map $g:A\to B$ such that $g(a)=b$ and

F(x,y)=0\quad\Longleftrightarrow\quad y=g(x)

for all $(x,y)\in A\times B$ .

Moreover, for each $x\in A$ ,

Dg(x)=-(D_yF(x,g(x)))^{-1}D_xF(x,g(x)).

Remarks

This theorem produces an implicitly defined function from an equation, and it is tightly connected to the inverse function theorem (which can be recovered as a special case).

Higher regularity

If $F$ is $C^k$ for $1\le k\le\infty$ , the local solution $g$ is $C^k$ . Starting from the displayed derivative formula, repeated differentiation proves this by induction, since matrix inversion is smooth on the open set of invertible matrices. In particular, a smooth equation with an invertible partial Jacobian defines a smooth local solution.