Implicit Function Theorem

Given

if the Jacobian

then , , and can be solved for in terms of , , and and Partial Derivatives of , , with respect to , , and can be found by differentiating implicitly.

More generally, let be an Open Set in and let be a Function. Write in the form , where and are elements of and . Suppose that (, ) is a point in such that and the Determinant of the Matrix whose elements are the Derivatives of the component Functions of with respect to the variables, written as , evaluated at , is not equal to zero. The latter may be rewritten as

Then there exists a Neighborhood of in and a unique Function such that and for all .