One direction in the proof is pretty easy since the tangent space of an integral manifold is involutive, so sometimes the theorem is only stated in one direction, that is: If $\Delta$ is involutive then it is completely integrable.

Another way to state the theorem is that if we have $n$ vector fields $\{X_k\}_{k=1}^n$ on a manifold $M$ such that they are linearly independent at every point of the manifold, and furthermore if for any $k,m$ we have $[X_k,X_m] = \sum_{j=1}^n a_j X_j$ for some $C^\infty$ functions $a_j$ , then for any point $x \in N$ , there exists a germ of a submanifold $N \subset M$ , through $x$ , such that $TN$ is spanned by $\{X_k\}_{k=1}^n$ . Note that if we extend $N$ to all of $M$ , it need not be an embedded submanifold anymore, but just an immersed one.

For $n=1$ above, this is just the existence and uniqueness of solution of ordinary differential equations.

Bibliography

1

William M. Boothby. An Introduction to Differentiable Manifolds and Riemannian Geometry, Academic Press, San Diego, California, 2003.

2

Frobenius theorem at Wikipedia: http://en.wikipedia.org/wiki/Frobenius_theorem