## 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.

## The Interior Product

We have yet another operation on the algebra of differential forms: the “interior product”. Given a vector field and a -form , the interior product is the -form defined by

That is, we just take the vector field and stick it into the first “slot” of a -form. We extend this to functions by just defining .

Two interior products anticommute: , which follows easily from the antisymmetry of differential forms. Each one is also clearly linear, and in fact is also a graded derivation of with degree -1:

where is the degree of . This can be shown by reducing to the case where and are wedge products of -forms, but rather than go through all that tedious calculation we can think about it like this: sticking into a slot of means either sticking it into a slot of or into one of . In the first case we just get , while in the second we have to “move the slot” through all of , which incurs a sign of , as asserted.