Cartan’s Formula
It turns out that there is a fantastic relationship between the interior product, the exterior derivative, and the Lie derivative.
It starts with the observation that for a function and a vector field
, the Lie derivative is
and the exterior derivative evaluated at
is
. That is,
on functions.
Next we consider the differential of a function. If we apply
to it, the nilpotency of the exterior derivative tells us that we automatically get zero. On the other hand, if we apply
, we get
, which it turns out is
. To see this, we calculate
just as if we took and applied it to
.
So on exact -forms,
gives zero while
gives
. And on functions
gives
, while
gives zero. In both cases we find that
and in fact this holds for all differential forms, which follows from these two base cases by a straightforward induction. This is Cartan’s formula, and it’s the natural extension to all differential forms of the basic identity on functions.
[…] Cartan’s formula in hand we can show that the Lie derivative is a chain map . That is, it commutes with the exterior […]
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