The implicit function theorem states that if continuously differentiable functions 
, 
, and 
of six real variables satisfy
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
at a point where the determinant of the Jacobian with respect to 
, 
, and 
is
 |
(4)
|
then 
,
,
and 
can be solved for locally in terms of 
, 
, and 
. Their partial derivatives
can be found by implicit differentiation.
More generally, let 
be an open set in 
and let 
be a Cr
function with 
. Write 
in the form 
, where 
and 
. Suppose that 
is a point in 
such that 
and the determinant
of the 
matrix 
is nonzero. Here, 
is the Jacobian of partial
derivatives with respect to 
. Equivalently,
 |
(5)
|
where rank denotes matrix rank. Then there are open neighborhoods 
of 
and 
of 
, with 
, and a unique Cr
function 
such that 
and, for 
, 
iff 
.
