The implicit function theorem is a strong theorem that allows us to express a variable as a function of another variable. For instance, if 
, can we make 
the subject, i.e. write as a function of 
? The implicit function theorem allows us to answer such questions, though like most Pure Math theorems, it only guarantees existence, the theorem does not explicitly tell us how to write out such a function.
The below material are taken from Wikipedia.
Implicit function theorem
Let 
be a continuously differentiable function, and let 
have coordinates 
. Fix a point 
with 
, where 
. If the matrix ![\displaystyle [(\partial f_i/\partial y_j)(\mathbf{a},\mathbf{b})]](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5B%28%5Cpartial+f_i%2F%5Cpartial+y_j%29%28%5Cmathbf%7Ba%7D%2C%5Cmathbf%7Bb%7D%29%5D&bg=ffffff&fg=333333&s=0 "\displaystyle [(\partial f_i/\partial y_j)(\mathbf{a},\mathbf{b})]")
is invertible, then there exists an open set 
containing 
, an open set 
containing 
, and a unique continuously differentiable function 
such that 
Elaboration:
Abbreviating 
to 
, the Jacobian matrix is
where 
is the matrix of partial derivatives in the variables 
and 
is the matrix of partial derivatives in the variables 
.
The implicit function theorem says that if is an invertible matrix, then there are , , and 
as desired.
Example (Unit circle)
In this case 
and 
.
Note that 
is invertible iff 
. By the implicit function theorem, we see that we can locally write the circle in the form 
for all points where 
.
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