I have the integral:
$$ \int_{V} d^{3}\mathbf{r} \; \nabla \cdot \mathbf{F}.$$
The integral is defined such that the volume enclosed is such that an open surface at time $t$, $S(t)$, moves at a constant velocity, generating another open surface at time $t + \delta t$, $S(t + \delta t)$. The integral is evaluated over thae two open surfaces the closed curve(s) bounding them. Here's a picture:
The notation should be clear from the the context.
The textbook argues that since $d^{3}\mathbf{r} = \hat{\mathbf{n}}dS.\mathbf{v}\delta t$, the integral can be written as:
$$ \delta t \int_{S(t)} d\mathbf{S} \cdot \mathbf{v} \; (\nabla \cdot\mathbf{F}).$$
I don't get the line of reasoning involved. Help?