## The Lie Derivative on Forms

We’ve defined the Lie derivative of one vector field by another, . This worked by using the flow of to compare nearby points, and used the derivative of the flow to translate vectors.

Well now we know how to translate -forms by pulling back, and thus we can define another Lie derivative:

What happens if is a -form — a function ? We check

That is, the Lie derivative by acts on exactly the same as the vector field does itself.

I also say that the Lie derivative by is a degree-zero derivation of the algebra . That is, it’s a real-linear transformation, and it satisfies the Leibniz rule:

for any -form and -form . Linearity is straightforward, and given linearity the Leibniz rule follows if we can show

for -forms . Indeed, we can write and as linear combinations of such - and -fold wedges, and then the Leibniz rule is obvious.

So, let us calculate:

So we see how we can peel off one of the -forms. A simple induction gives us the general case.

[...] Lie derivative looks sort of familiar as a derivative, but we have another sort of derivative on the algebra of [...]

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[...] . But since it takes -forms and sends them to -forms, it has degree one instead of zero like the Lie derivative. As a consequence, the Leibniz rule looks a little different. If is a -form and is an -form, I [...]

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[...] Cartan’s formula in hand we can show that the Lie derivative is a chain map . That is, it commutes with the exterior derivative. And indeed, it’s easy to [...]

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[...] Armstrong: The algebra of differential forms, Pulling back forms, The Lie derivative on forms, The exterior derivative is a derivative, The exterior derivative is nilpotent, De Rham Cohomology, [...]

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