## Proof of the Flux Transport Theorem

Theorem
$\frac{d}{dt} \int \int \int_{V_t} f dV =\int \int \int_{V_t} \frac{\partial f}{\partial t} dV + \int \int_{S_t} f \mathbf{v} \cdot d \mathbf{S}$

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{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}