Let and let be a Lipschitz continuous map. In order to analyse the solution of the differential equation,

and make the geometry enter into the scene, it is convenient to see as a collection of vector fields , where the ‘s are the columns of the matrix . The differential equation then of course writes

Generally speaking, a vector field on is a map

A vector field can be seen as a differential operator acting on differentiable functions as follows:

We note that is a derivation, that is for ,

For this reason we often use the differential notation for vector fields and write:

Using this action of vector fields on functions, the change of variable formula for solutions of differential equations takes a particularly concise form:

**Proposition:** *Let be a solution of a differential equation that writes
then for any function ,
*

Let be a Lipschitz vector field on . For any , the differential equation

has a unique solution . By time homogeneity of the equation, the flow of this equation satisfies

and therefore is a one parameter group of diffeomorphisms . This group is generated by in the sense that for every ,

For these reasons, we write . Let us now assume that is a Lipschitz vector field on . If is a diffeomorphism, the pull-back of the vector field by the map is the vector field defined by the chain rule,

. In particular, if is another Lipschitz vector field on , then for every , we have a vector field . The Lie bracket between and is then defined as

It is computed that

Observe that the Lie bracket obviously satisfies and the so-called Jacobi identity that is:

What the Lie bracket really quantifies is the lack of commutativity of the respective flows generated by and .

**Lemma:*** Let be two Lipschitz vector fields on . Then, if and only if for every ,
*

**Proof:**This is a classical result in differential geometry, so we only give one part the proof. From the very definition of the Lie bracket and the multiplicativity of the flow, we see that if and only if for every , . Now, suppose that . Let be the solution of the equation

Since , we obtain that is also a solution of the equation. By uniqueness of solutions, we obtain that

As a conclusion,

If we consider a differential equation

as we will see it throughout this class, the Lie brackets play an important role in understanding the geometry of the set of solutions. The easiest result in that direction is the following:

**Proposition:** * Let and let be Lipschitz vector fields on . Assume that for every , , then the solution of the differential equation
can be represented as
*

**Proof:** Let

Since the flows generated by the ‘s are commuting, we get that

The change of variable formula for bounded variation paths implies then that is a solution and we conclude by uniqueness

It seems I can not follow your arugments when you define the pull-back of a vector field. On the RHS you have a differential and some function in the index. Is this well-known notation?

The notation means the following: is the differential of at the point . This is a linear map that we apply then to the vector .

Thank you for the response. As far as I understand $(d \phi^{-1})_{\phi(x)}(V(\phi(x)))=D^{-1}\phi(x) V(x)$, here $D^{-1} \phi(x)$ stands for the inverse of the derivative of $\phi$, in our case this is some matrix. Right?

Yes, this is right.

Thanks for the time spent on my question.