Lie Derivative - The Lie Derivative of Differential Forms

The Lie Derivative of Differential Forms

The Lie derivative can also be defined on differential forms. In this context, it is closely related to the exterior derivative. Both the Lie derivative and the exterior derivative attempt to capture the idea of a derivative in different ways. These differences can be bridged by introducing the idea of an antiderivation or equivalently an interior product, after which the relationships fall out as a set of identities.

Let M be a manifold and X a vector field on M. Let be a k+1-form. The interior product of X and ω is

Note that

and that is a -antiderivation. That is, is R-linear, and

i_X (\omega \wedge \eta) = (i_X \omega) \wedge \eta + (-1)^k \omega \wedge (i_X \eta)

for and η another differential form. Also, for a function, that is a real or complex-valued function on M, one has

The relationship between exterior derivatives and Lie derivatives can then be summarized as follows. For an ordinary function f, the Lie derivative is just the contraction of the exterior derivative with the vector field X:

For a general differential form, the Lie derivative is likewise a contraction, taking into account the variation in X:

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This identity is known variously as "Cartan's formula" or "Cartan's magic formula," and shows in particular that:

.

The derivative of products is distributed:

\mathcal{L}_{fX}\omega = f\mathcal{L}_X\omega + df \wedge i_X \omega

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