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For example, I've read the definition of the interior product a dozen times, Interior product - Wikipedia, the free encyclopedia , but my understanding is that a 2-form is something that eats two tangent vectors and spits out a number. A vector field on a manifold is a map v(x) = (x,u) for x in M and u in T_xM, so I don't understand how to plug a map into a form.
My guess would ge the following. So, as you've noted a vector fieldis just a section (smooth as you please) of the tangent bundle. So, now, if you instead think about the map
$ defined by
. You see then that if
then you can think about
defined by
. That's my best guess without more context.
Yes a 2-form eats two vectors and outputs a number, but if you only feed one input it outputs a map:
defined by