Implicit function theorem
Implicit function theorem can be considered an extension to the non linear case of a well known problem of solving under determinated system of linear equations having more unknown then equations.
Let a differentiable function of class
Let a point such as and rank .
This means that
Then there are U a neighborhood of a, a neighborhood of b and a function with properties:
(2)for all i.e. for all
(3) f is continuous on U;
Function with these three properties is local unique, any two functions having these properties are identical on the intersection of their domains of definition.
In plus f is differentiable and
By the above relation we mean
After a short calculation or solving a system of linear equations
we have more explicitly