What does it mean to evaluate a vector field on a 2-form?

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.

Re: What does it mean to evaluate a vector field on a 2-form?

My guess would ge the following. So, as you've noted a vector field $\displaystyle X$ is just a section (smooth as you please) of the tangent bundle. So, now, if you instead think about the map $\displaystyle Y:X\to TM\oplus T$$ defined by $\displaystyle Y(x)=(X(x),X(x))$. You see then that if $\displaystyle \omega\in \bigwedge^2(T^\ast M)$ then you can think about $\displaystyle \omega(Y)$ defined by $\displaystyle \omega(Y)(x)=\omega(Y(x))=\omega(X(x),X(x))$. That's my best guess without more context.

Re: What does it mean to evaluate a vector field on a 2-form?

Yes a 2-form eats two vectors and outputs a number, but if you only feed one input it outputs a map:

$\displaystyle \omega_X : V \rightarrow V$ defined by $\displaystyle \omega_X=\omega(X, . ) : Y \mapsto \omega(X,Y)$