I'm trying to determine something based off of an alternative definition of the Lie Brackect.
It is known that if and are two smooth vector fields, then the Lie bracket is a smooth vector field at a point defined to be
We can express in terms of the local coordinates and of and respectively.
Since and are directional derivatives, it turns out that:
This implies that
By a similar process,
Now, taking the difference between these two values yield the alternative definition of the Lie bracket:
In the textbook I'm using, it says that an easy consequence of this alternative definition is . My question is why is this the case? Given that my professor didn't go over this in class makes it quite challenging for me to grasp this idea.
I'm thinking that it boils down to finding the local coordinates for the basis vectors of , but I'm not sure how to come up with local coordinates for the basis vectors.
I've spent a couple hours on this and really haven't made progress. If anyone could shed some light on this, I would appreciate it!!