Over on the Wiki page for interior products: http://en.wikipedia.org/wiki/Interior_product
There is a line that says
$\iota_X \alpha = \alpha(X) = \langle \alpha,X \rangle$
where $\alpha$ is a differential one-form and $X$ is a vector field.
Unfortunately, I'm not entirely sure what any of these mean.
1. How exactly does contraction with a one-form work? I've only ever done it with constant vectors as opposed to vector fields.
2. How do you apply a differential one-form to a vector field? I'm assuming that if you have a parameterization of the vector field then you just plug in the respective coordinate functions. Is this correct?
3. How can you take the inner product of a differential one form and a vector field?
If anyone can answer these questions I'd be grateful. Examples would be very preferable. Thanks!