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

Yes.
Which book?

Let's go back a couple of steps.

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