From: parendt@newshost.nmt.edu (Paul Arendt)
Subject: Re: Formular for Lie derivative
Date: 11 Jan 2000 16:18:07 -0700
Newsgroups: sci.math
Summary: the Cartan identity for differential forms

In article <38766055.3AED3F10@physik.stud.uni-erlangen.de>,
Markus Krapf  <markus.krapf@physik.stud.uni-erlangen.de> wrote:

>Can Someone help me by getting a formular  for L_X_0(\omega(X_1,...,X_p)
>
> whereas X_0,...,X_p  being vectorfields on M and \omega being a p-Form
>on M, L_X denotes the Lie derivative
>in direction X.

(I'm going to write "w" for "omega" here.)
Since Lie derivatives are derivations (first-order derivative
operators), they obey the Leibniz rule ("product rule").  This
means that we can differentiate w(X1,...,Xp) term by term:

L_X0(w(X1,...,Xp)) = (L_X0(w))(X1,...Xp) + w([X0,X1],X2,...,Xp)
                   + w(X1,[X0,X2],...Xp) + ... + w(X1,...,[X0,Xp])

This uses the fact that L_X(Y) = [X,Y] for vector fields X and Y.
It's important to realize that in the first term, we must take the
Lie derivative of w *before* evaluating w on (X1,...,Xp).

I'm not sure if that's what you're looking for -- this can be
rewritten in a number of different ways.  The most obvious
way to make it more complicated is to expand out L_X0(w) by
using the following identity (true for all differential forms):

   L_Y(w) = dw(Y,...) + d(w(Y,...))

This formula is sometimes called the "Cartan identity."
You can use the Cartan identity to expand out the first term
in the above formula, in order that no Lie derivatives appear
explicitly on the right-hand side.