Given
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
if the determinantof the Jacobian
 |
(4)
|
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
 |
(5)
|
Then there exists a neighborhood 
of 
in 
and a unique 
function 
such that 
and 
for all 
.
