# Is the commutator of vector fields an important notion?

• I
Zhang Bei
Hi,

I'm just starting to read Wald and I find the notion of the commutator hard to grasp. Is it a computation device or does it have an intuitive geometric meaning? Can anyone give me an example of two non-commutative vector fields?

Thanks!

## Answers and Replies

Mentor
2022 Award
Hi,

I'm just starting to read Wald and I find the notion of the commutator hard to grasp. Is it a computation device or does it have an intuitive geometric meaning? Can anyone give me an example of two non-commutative vector fields?

Thanks!
Here is an easy example in section 6.2: https://www.physicsforums.com/insights/journey-manifold-su2-part-ii/

As a geometric intuition, you can imagine to start at a certain point ##p## and follow a flow along vector field ##X## for a small distance, from there along vector field ##Y## for a while and reach point ##s##. If you now start again at ##p## but follow first along ##Y## and then ##X##, you will usually end up at a different point ##t \neq s##. That difference is measured by ##[X,Y]=X \circ Y - Y \circ X##. If ##[X,Y]=0## then ##t=s##.

Zhang Bei