Implicit function theorem

Implicit function theorem: Let $U\subseteq\mathbb R^{n+m}$ be an open set and let $F:U\to\mathbb R^m$ be continuously differentiable. Write points as $(x,y)$ with $x\in\mathbb R^n$ and $y\in\mathbb R^m$. Suppose $(a,b)\in U$ satisfies $F(a,b)=0$ and the Jacobian matrix of $F$ with respect to $y$ at $(a,b)$, denoted $D_yF(a,b)$, is invertible (equivalently, its determinant is nonzero). Then there exist neighborhoods $A$ of $a$ and $B$ of $b$ and a unique continuously differentiable map $g:A\to B$ such that $g(a)=b$ and

Moreover, for each $x\in A$,

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).