Implicit Function Theorem

Given

$$F_1(x,y,z,u,v,w) = 0$$

$$F_2(x,y,z,u,v,w) = 0$$

$$F_3(x,y,z,u,v,w) = 0,$$

if the Jacobian

$$JF(u,v,w) = {\partial(F_1,F_2,F_3)\over\partial(u,v,w)}\not= 0,$$

then $u$, $v$, and $w$ can be solved for in terms of $x$, $y$, and $z$ and Partial Derivatives of $u$, $v$, $w$ with respect to $x$, $y$, and $z$ can be found by differentiating implicitly.

More generally, let $A$ be an Open Set in ${\Bbb{R}}^{n+k}$ and let $f:A\to\Bbb{R}^n$ be a $C^r$ Function. Write $f$ in the form $f(x,y)$, where $x$ and $y$ are elements of $\Bbb{R}^k$ and $\Bbb{R}^n$. Suppose that ($a$, $b$) is a point in $A$ such that $f(a,b)=0$ and the Determinant of the $n\times n$ Matrix whose elements are the Derivatives of the $n$ component Functions of $f$ with respect to the $n$ variables, written as $y$, evaluated at $(a,b)$, is not equal to zero. The latter may be rewritten as

$$\mathop{\rm rank}(Df(a,b)) = n.$$

Then there exists a Neighborhood $B$ of $a$ in $\Bbb{R}^k$ and a unique $C^r$ Function $g:B\to \Bbb{R}^n$ such that $g(a)=b$ and $f(x,g(x))=0$ for all $x\in B$.

See also Change of Variables Theorem, Jacobian

References

Munkres, J. R. Analysis on Manifolds. Reading, MA: Addison-Wesley, 1991.