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 .

**References**

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

© 1996-9

1999-05-26