Straightening theorem for vector fields

In differential calculus, the domain-straightening theorem states that, given a vector field on a manifold, there exist local coordinates such that in a neighborhood of a point where is nonzero. The theorem is also known as straightening out of a vector field.
The Frobenius theorem in differential geometry can be considered as a higher-dimensional generalization of this theorem.

Proof

It is clear that we only have to find such coordinates at 0 in. First we write where is some coordinate system at and are the component function of relative to Let. By linear change of coordinates, we can assume Let be the solution of the initial value problem and let
is smooth by smooth dependence on initial conditions in ordinary [differential equations]. It follows that
and, since, the differential is the identity at. Thus, is a coordinate system at. Finally, since, we have: and so
as required.