While researching how to derive the divergence theorem from the generalized Stokes' theorem, I ran into the wikipedia page for the volume form, which claimed that the divergence of a vector field $X$ on a manifold $M$ could be defined as the unique scalar function which solves $$(\textrm{div}X)\omega = d(X \;\lrcorner\;\omega) \quad (1)$$ where $\omega$ is the volume form of $M$ and $X \;\lrcorner\;\omega$ is the interior product with $X$. Stokes' theorem then implies that $$ \int_{M} (\textrm{div}X)\omega = \int_{\partial M}X \;\lrcorner\;\omega \quad (2)$$
This is claimed to be a generalization of the divergence theorem.
- Why is $(1)$ a "good" definition for divergence? How does it connect to the usual definition? How should I (intuitively) interpret the relation with the volume form?
- How does $(2)$ restrict to the usual statement of the divergence theorem? I'd assume there's some way to transform $X \;\lrcorner\;\omega$ into $(X\cdot n) \alpha$ where $\alpha$ is the volume form on $\partial M$, but I don't know how to derive it.