## Proving Euler's Expansion Formula from Reynold's Theorem

Theorem
$\frac{d}{dt} \int \int \int_{V_t} dV = \int \int \int_{V_t} \mathbf{\nabla} \cdot \mathbf{v} dV = \int \int_{S_t} \mathbf{v} d \mathbf{S}$

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
$\frac{d}{dt} \int \int \int_{V_t} dV = \int \int_{S_t} \mathbf{v} \cdot d \mathbf{S}$