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

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    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.
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    MHF Contributor Drexel28's Avatar
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    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 X is just a section (smooth as you please) of the tangent bundle. So, now, if you instead think about the map Y:X\to TM\oplus T$ defined by Y(x)=(X(x),X(x)). You see then that if \omega\in \bigwedge^2(T^\ast M) then you can think about \omega(Y) defined by \omega(Y)(x)=\omega(Y(x))=\omega(X(x),X(x)). That's my best guess without more context.
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    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:
    \omega_X : V \rightarrow V defined by \omega_X=\omega(X, . ) : Y \mapsto \omega(X,Y)
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