\[\frac{d}{dt} \int \int \int_Vf dV =\int \int int_{V_t} \frac{\partial f}{\partial t} dV + \int \int_{S_t} f \mathbf{v} \cdot d \mathbf{S}\]
where
\[f\]
is a function of \[\mathbf{r}, \:t\]
defined on a volume \[V\]
consisting of a sphere expanding at a constant speed \[v\]
so that \[r=vt\]
Example: If \[f(\mathbf{r} , t) =a \]
The left hand side is
\[\begin{equation} \begin{aligned} \frac{d}{dt} \int \int \int_V a dV &= a \frac{d}{dt} \int \int \int_V dV \\ &= a \frac{d}{dt} (\frac{4}{3} \pi r^3) = 4 \pi a v^3 t^2 \end{aligned} \end{equation}\]
The right hand side is
\[\begin{equation} \begin{aligned} \frac{d}{dt} \int \int \int_{V_t} \frac{\partial a}{\partial t} dV + \int \int_{S_t} a \mathbf{v} \cdot d \mathbf{S} &= \int \int_{S_t} a \mathbf{v} \cdot d \mathbf{S} \\ &= av \int \int_{S_t} \frac{\mathbf{r}}{\left| \mathbf{r} \right| } \cdot \frac{\mathbf{r}}{\left| \mathbf{r} \right|} dS \\ &= av \times (4 \pi r^2) \\ &= av \times (4 \pi v^2 t^2) \\ &= 4 \pi a v^3 t^2 \end{aligned} \end{equation}\]