# Action of Lie Brackets on vector fields multiplied by functions

by tut_einstein
Tags: action, brackets, fields, functions, multiplied, vector
 P: 31 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!
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P: 5,903
 Quote by tut_einstein 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.
 Quote by tut_einstein 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
$$\left[ fX,Y \right]g .$$

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