Action of Lie Brackets on vector fields multiplied by functions

[tut_einstein]

Hi,

Is there a specific product rule or something one must follow when applying the lie bracket/ commutator to two vector fields such that one of them is multiplied by a function and added to another vector field? This is the expression given in my textbook but I don't see how:

[fX+Z,Y] = f[X,Y] + [Z,Y] - (Yf)Xg

I don't see where the third term on the right hand side comes from.

I'd really appreciate some help on this because I'm self-learning differential geometry for a research project and almost all my doubts revolve around my not understanding how lie brackets work. So any help will be appreciated.

Thanks!

 

[George Jones]

Is there a specific product rule or something one must follow when applying the lie bracket/ commutator to two vector fields such that one of them is multiplied by a function

Yes.

This is the expression given in my textbook

Which book?

Let's go back a couple of steps.

If $X$ is a vector field and $f$ and $g$ are smooth functions, then both $Xf$ and $Xg$ are functions. Because $f$ and $g$ are both functions, the product $fg$ is also a function on which $X$ can act. Consquently, $X \left(fg\right)$ is a function. $X$ acts like a derivative (is a derivation) on the set (ring) of smooth functions, i.e.,
$$X \left(fg\right) = g Xf + f Xg.$$
Now, use the above and expand
$$\left[ fX,Y \right]g .$$