I am trying to understand the interior derivative $\cal{i}$$_X \omega$...
$\omega$ is a 1-form; $X$ is a "vector". I have been looking for worked examples here at Math.SE. What confuses me is that, for instance, the "vector" $X$ is defined as, for example
$$X=a\frac{\partial}{\partial x}+b\frac{\partial}{\partial y}$$.
But this is not a vector, it is the differential operator $\vec{A}\cdot\nabla$ (where say, $\vec{A}$=(a,b,0)). This operator can act on both scalars and vectors.
Where did the vector $\vec{X}$ go? What is the meaning of $X$ in the interior derivative $\cal{i}$$_X \omega$ ?
Can this derivative work with a 3-vector $\vec{F}=(F_x,F_y,F_z)$?