### Commutator of vector fields $x∂_x + y∂_y + z∂_z$ and $∂_z$

The commutator of the vector fields $x∂_x + y∂_y + z∂_z$ and $∂_z$ is given by:$[x∂_x + y∂_y + z∂_z,∂_z] = -∂_z$.I'm having trouble interpreting this result. The commutator is the vector that connects p and p': p being the point one ends in by first following an integral curve of the first vector field over a distance $\delta t$ and then the second, starting in $(x_0,y_0,z_0)$, and p' vice versa, starting from the same point, over the same distance.Here's my problem. I believe the integral curves of the first vector field are straight lines thr...Read more