## 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 |

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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 |