How to prove a property of Lie derivatives

Question

I know that there are five properties for Lie derivative. But one of them I don't know how to prove. It is $ L_x[\omega(Y)]=(L_x \omega)(Y)+\omega(L_x Y)$

Note : Here $\omega$ is a covariant vector field while Y is a vector field. Is there anyone has any idea?

score 1 · Answer 1 · answered Oct 17 '13 at 15:44

1

I'll prove this as a collection of exercises. All of the steps should be quite doable from the definition.

Exercise A: prove that $\mathcal{L}_X Y = [X,Y]$

Exercise B: prove that $\mathcal{L}_X\omega = d(X\lrcorner\omega)+X\lrcorner d\omega$

Exercise C: prove that $d\omega(X,Y) = X(\omega(Y))-Y(\omega(X))-\omega([X,Y])$

Now put these all together (remembering the Leibniz property of the Lie derivative).

answered Oct 17 '13 at 15:44

Edward Hughes

5,195

What do you mean by the second exercise? I am not quite sure about the notation.@Edward Hughes – Jack Oct 17 '13 at 21:44
Reading this Wiki article should help. – Edward Hughes Oct 19 '13 at 11:37

How to prove a property of Lie derivatives

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