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 apoint in such that 
and the Determinant of the 
Matrix whose elements are theDerivatives 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 
.See also Change of Variables Theorem, Jacobian
References
Munkres, J. R. Analysis on Manifolds. Reading, MA: Addison-Wesley, 1991.
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