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 is invertible, then there exists an open set containing , an open set containing , and a unique continuously differentiable function such that
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 .