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.
Why is this post in the “point-set topology” category?
Comment by Andrei | July 26, 2011 |
An errant click on a very tightly-spaced list.
Comment by John Armstrong | July 26, 2011 |
[…] turns out that there is a fantastic relationship between the interior product, the exterior derivative, and the Lie […]
Pingback by Cartan’s Formula « The Unapologetic Mathematician | July 26, 2011 |
[…] along the projection to get a “time” vector field on the cylinder. Then we use the interior product to assert that and […]
Pingback by The Poincaré Lemma (proof) « The Unapologetic Mathematician | December 3, 2011 |
[…] and applying it to the vector itself: . We also take the canonical volume form , and we use the interior product to define the -form […]
Pingback by A Family of Nontrivial Homology Classes (part 1) « The Unapologetic Mathematician | December 20, 2011 |
hi thank you for this post.
when we insert X into a slot of say a, then don’t we need to move X so that it will be on the first slot of a? And this will require multiplying by (-1) times the number of swaps.
Comment by circa1687 | November 15, 2015 |
That’s true, circa1687. Think of it like this: because of antisymmetry, the interior product doesn’t really care which slot you insert into, since it just picks up a sign as you move it to the front. Inserting into the second slot of is the same as inserting it into the second slot of , but inserting into the second slot of is like inserting it into the th slot of , so we always pick up an extra signs.
Comment by John Armstrong | November 15, 2015 |