The last equality on page $398$ of Jeffrey Lee's Manifold and Differential Geometry textbook states that
$$\int_{M} div(X) \omega = \int_{\partial M} i_{X} \omega,$$ where $M$ is an oriented manifold with boundary $\partial M$, $\omega$ is a volume form on $M$, and $i_{X}$ is the interior derivative(definition) with respect to a given smooth vector field $X.$
Note that its proof is a direct consequence of the definition of Lie derivative(definition), Cartan's magic formula, and "generalized" Stoke's theorem on forms (statement can be found here). I have a clear understanding of the proof. However, the author comments that this formula helps give a geometric meaning to $div(X)$, which I don't fully understand: How does this equation help us to give a geometric meaning of $div (X)$?
We can give an interpretation of divergence from the classic divergence theorem/Gauss theorem from multivariable calculus, which relates the integral of divergence and flux of a vector field. Does the interior derivative generalize/formalize the notion of flux? More precisely, how can we think about interior derivatives geometrically?
When we define the interior derivative of an $n$ form $\omega,$ we basically form an $n-1$ form $w(X, ......)$ by fixing a vector field slot. Can this contraction by a vector field be interpreted as a"projection of the vector field" along the "surface area element $\partial \omega$"? I know what I am saying is very vague. I would appreciate it if you can help me to understand the notion of interior derivative more concretely/geometrically/intuitively. Thanks so much.
Thank you for your time.
Suppose $M$ and $\partial M$ are compact and oriented. Equip $M$ with a Riemannian metric, and let $N:\partial M \to TM$ be a unit vector field which is normal to $\partial M$ and inward(or outward?) pointing.
Then the Riemannian volume form of $ M$ over the boundary is equal to $$ \operatorname{vol}_M = N^\# \wedge \operatorname{vol}_{\partial M}$$ where $N^\# = \langle N, \cdot\rangle $ is the 1-form associated to $N$. Note that we can extend $N$ to a vector field over the whole $M$ if needed.
What happens when you take the interior product? Over the boundary we have (denoting by $\iota:\partial M \to M$ the inclusion)
$$\iota^*(i_X \operatorname{vol}_M) = \langle N,X\rangle \operatorname{vol}_{\partial M} \in \Omega^{\dim M -1}(\partial M),$$ note that in general $i_X \operatorname{vol}_M = \langle N,X\rangle \operatorname{vol}_{\partial M} + rest$ but when you pull it back to $\partial M$ the rest dies since $\Lambda^{\dim M-1}\partial M $ is spanned by $\operatorname{vol}_{\partial M} $.
From this we see that when we compute $\int_{\partial M} i_X\operatorname{vol}_M = \int_{\partial M} \langle N,X\rangle \operatorname{vol}_{\partial M}$ we are really computing the flow of $X$ along $\partial M$.