I have to present in class "The classical version of the Frobenius theorem". Here is the classic version of the book "Foundations of Differentiable Manifolds and Lie Groups" (Frank W. Warner).
I want more references on this theorem with example and applications of the PDE. Could you suggest me some, please?
You can take any example that uses the Frobenius theorem and rewrite it using classical version instead of the one using vector fields or the one using differential forms.
As Warner says, the Frobenius theorem gives the necessary and sufficient conditions for the existence of a solution to a total system of PDEs. In a total system, there is an equation for each partial of each unknown function. In other words, a total system for an unknown function $$ F: O \rightarrow \mathbb{R}^n, $$ where $O \subset \mathbb{R}^m$ is simply connected and open, is a system of PDEs of the form \begin{align*} \frac{\partial F^b}{\partial x^i} &= G^b_i(x, F(x)), \end{align*} where $G^b_i: O \rightarrow \mathbb{R}$ are given, $1 \le i \le m$, and $1 \le b \le n$. Using differential forms, this is the system $$ dF^b = G^b_i(x, F(x))\,dx^i. $$ The necessary and sufficient conditions that a solution exists are obtained from the fact that partial derivatives commute. For the differential form version, this is equivalent to saying that $d(dF^b) = 0$.
The most basic example of this is the system $$ \frac{\partial f}{\partial x^i} = g_i,\ 1 \le i \le m, $$ and the necessary and sufficient conditions are $$ \frac{\partial g_j}{\partial x^i} =\frac{\partial g_i}{\partial x^j},\ 1 \le i,j \le m. $$ This is sometimes called the Poincaré lemma.
If you're familiar with the differential geometry of a surface in $\mathbb{R}^3$, then an application of the Frobenius theorem is the Fundamental Theorem of Surfaces, which states that the first and second fundamental forms determine the surface up to translation. The easiest way to prove this is by formulating it in terms of moving frames and differential forms. However, a proof using the classical Frobenius theorem can be found in the Appendix to Chapter 4 of Do Carmo's book Differential Geometry of Curves and Surfaces : Revised and Updated Second Edition.