\[\frac{d}{dt} \int \int \int_{V_t} f dV =\int \int \int_{V_t} \frac{\partial f}{\partial t} dV + \int \int_{S_t} f \mathbf{v} \cdot d \mathbf{S}\]
where
\[f\]
is a function of \[\mathbf{r}, \:t\]
defined on a volume of integration \[V_t\]
that varies with time.Proof
Let
\[f= \mathbf{\nabla} \cdot \mathbf{F}\]
then\[\int \int \int_{V_t} f dV = \int \int \int_{V_t} \mathbf{\nabla} \cdot \mathbf{F} dV =\int \int_{S_t} \mathbf{F} \cdot d \mathbf{S} \]
(1)from the Divergence Theorem.
The flux transport theorem for a closed surface
\[S_t\]
of a volume \[V_t\]
states\[\frac{d}{dt} \int \int_{S_t} \mathbf{F} \cdot d \mathbf{S} = \int \int_{S_t} \frac{\partial \mathbf{F}}{\partial t} + (\mathbf{\nabla} \cdot \mathbf{F}) \mathbf{v}) d \mathbf{S}\]
(2)Differentiate (1) and substitute (2) to get
\[\begin{equation} \begin{aligned} \frac{d}{dt} \int \int \int_{V_t} f dV &=
\frac{d}{dt} \int \int_{S_t} \mathbf{F} \cdot d \mathbf{S} \\ &=
\int \int_{S_t} \frac{\partial \mathbf{F}}{\partial t} \cdot d \mathbf{S} + \int \int_{S_t} (\mathbf{\nabla} \cdot \mathbf{F}) \mathbf{v} \cdot d \mathbf{S} \\ &=
\int \int \int_{V_t} \mathbf{\nabla} \cdot \frac{\partial \mathbf{F}}{\partial t} dV + \int \int_{S_t} (\mathbf{\nabla} \cdot \mathbf{F}) \mathbf{v} \cdot d \mathbf{S} \\ &=
\int \int \int_{V_t} \frac{\partial}{\partial t} (\mathbf{\nabla} \cdot \mathbf{F}) dV + \int \int_{S_t} (\mathbf{\nabla} \cdot \mathbf{F}) \mathbf{v} \cdot d \mathbf{S} \\ &=
\int \int \int_{V_t} \frac{\partial f}{\partial t} dV + \int \int_{S_t} f \mathbf{v}) \cdot d \mathbf{S}
\end{aligned} \end{equation}\]