Reynold's Transport Theorem states that
\[\frac{d}{dt} \int \int \int_{V_t} f dV =\int \int int_{V_t} \frac{\partial f}{\partial t} dV + \int \int_{S_t} f \mathbf{v} \cdot d \mathbf{S}\]
where
\[f\]
is a function of \[\mathbf{r}, \:t\]
defined on a volume \[V\]
consisting of a sphere expanding at a constant speed \[v\]
so that \[r=vt\]
The Continuity Equation states
\[\frac{\partial \rho}{\partial t} + \mathbf{\nabla} (\rho \mathbf{v}) =0\]
Set
\[f = \rho\]
in Reynold's Theorem gives\[\frac{d}{dt} \int \int \int_{V_t} \rho dV =\int \int int_{V_t} \frac{\partial \rho}{\partial t} dV + \int \int_{S_t} \rho \mathbf{v} \cdot d \mathbf{S}\]
Now apply the Divergence Theorem
\[\int \int_S \mathbf{F} \cdot d \mathbf{S} = \int \int \int_V \mathbf{\nabla} \cdot \mathbf{F} dV\]
with
\[\mathbf{F} = \rho \mathbf{v}\]
to get\[\frac{d}{dt} \int \int \int_{V_t} \rho dV =\int \int \int_{V_t} \frac{\partial \rho}{\partial t} dV +\int \int \int_{V_t} \mathbf{\nabla} \cdot (\rho \mathbf{v}) dV\]
Hence
\[\frac{d}{dt} \int \int \int_{V_t} \rho dV =\int \int \int_{V_t}( \frac{\partial \rho}{\partial t} + \mathbf{\nabla} \cdot (\rho \mathbf{v})) dV=0\]
Then the left hand side is zero and
\[ \int \int \int_{V_t} \rho dV =constant\]