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.