I would like to prove that the flux volume rate over a manifold is the flux volume rate over a manifold. I mean:
Let $M$ be a Riemannian orientable manifold, let $S$ be a compact oriented submanifold, let $N$ be a normal unit vector field on $S$ and let $X$ be a smooth vector field on $M$. Let $S_+=\{p\in S\mid \left<X_p,N_p\right>>0\}$. On one hand, the flux volume rate of $X$ over $S_+$ is usually defined as $\int_{S_+}\left<X,N\right>\,\omega$, where $\omega$ is the oriented volume form on $S$.
But, on the other hand, we can consider the local one-parameter subgroup determined by $X$, namely, $\Phi:\left]-\delta,\delta\right[\times U\longrightarrow M$, where $U$ is an open subset of $M$ containing $S$. For each $0<h<\delta$ we can define $Q(h)$ as the volume in $M$ of $\Phi[[0,h]\times S_+]$. If we think of $X$ as the velocity field of a fluid, then $Q(h)$ is the volume of fluid thas has crossed $S_+$ in the interval of time $[0,h]$.
Then, what I would like to prove is that $\left.\frac{dQ}{dh}\right|_0=\int_{S_+}\left<X,N\right>\omega$.
In fact, I am able to prove it locally by taking appropriate charts, I mean, I can prove that each point $p\in S_+$ has a neighborhood $S_p$ in $S$ on which the equality is true (replacing $S_+$ with $S_p$ both in the integral and in the definition of $Q$).
However, in order to conclude the global result, I would need to prove that if $W_+$ is partitioned into a countable union of Borel sets, the derivative $dQ/dh|_0$ is the sum of the derivatives corresponding to each set. But I do not know how to show this. In fact, I would only need to show that the volume $Q_B(h)$ for a Borel subset $B$ of $S$ can be bounded as $Q_B(h)\leq K vol(B) h$, for some constant $K$, which seems very intuitive.
Maybe all is simpler than I am figuring out. Maybe a direct global coordinate-free proof is possible. I would be grateful if someone can give me some reference or an indication about how this can be proved.