## Calculating the Lie Derivative

It’s all well and good to define the Lie derivative, but it’s not exactly straightforward to calculate it from the definition. For one thing, it requires knowing the flow of the vector field , which requires solving a differential equation that might be difficult in practice. Luckily, there’s an easier way.

But first, a lemma: if is an interval containing , is an open set, and is a differentiable function with for all , then there is another differentiable function such that

This is basically just like a lemma we proved for functions on star-shaped neighborhoods. Indeed, it suffices to set

Now let , , and is the local flow of a vector field on a region containing . Consider the function , which satisfies the condition of the above lemma. We can thus write

for some . Or if we write we can write

We also can see that . And so we can write

So now we calculate

Last time we wrote the action of on a smooth function as

which pattern we can recognize in our formula. We thus continue

That is, for any two vector fields the Lie derivative is actually the same as the bracket .