## 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:

(1)f(a)=b, i.e.

(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