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

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