\[\frac{d}{dt} \int \int \int_{V_t} dV = \int \int \int_{V_t} \mathbf{\nabla} \cdot \mathbf{v} dV = \int \int_{S_t} \mathbf{v} d \mathbf{S} \]
where
\[\mathbf{v} = \frac{d \mathbf{r}}{dt}\]
on a region \[V_t\]
which is a function of time.Proof
\[\frac{d}{dt} \int \int \int_{V_t} f dV =\int \int \int_{V_t} \frac{\partial f}{\partial t} dV + \int \int_{S_t} f \mathbf{v} \cdot d \mathbf{S}\]
where
\[f\]
is a function of \[\mathbf{r}, \:t\]
defined on a volume \[V_t\]
which is a function of time.Set
\[f=1\]
then\[\frac{d}{dt} \int \int \int_{V_t} dV = \int \int_{S_t} \mathbf{v} \cdot d \mathbf{S}\]