I am trying to get a better feel for both the exterior derivative of a form and the contraction of a form by a vector field $X$.
Basically, when are these inverses? If I have a one-form $\omega$ and I compute $dw$, getting a 2-form. When can I find a vector field $X$ so that $w = i_Xdw$? Is there a general result describing the relation between forms $w, v$ such that $w=i_Xdv$?
I am aware of the issue of closed and exact forms, but I am having trouble framing this question in this format.
EDIT: Also, is there a "nice" geometric way of seeing the interior product? We can see the exterior derivative in terms of geometric algebra, by creating "parallelepipeds" or their $n$-dimensional equivalent, as volume elements. Is something similarly geometric going on when we contract by a vector field?