The Transport Theorem

Theorem
The rate of change of flux from a vector field  
\[\mathbf{F}\]
  through a surface  
\[S\]
  both time dependent i
\[\frac{d \Phi}{dt} = \int \int_{S_t}( \frac{\partial \mathbf{F}}{\partial t} + \mathbf{v} \mathbf{\nabla} \cdot \mathbf{F}) \cdot d \mathbf{n} dS + \oint_{C_1} (\mathbf{F} \times \mathbf{v}) \cdot d \mathbf{r} \]

Proof
At any time  
\[t_1\]
  the vector field is  
\[\mathbf{F} (\mathbf{r}, t_1)\]
. Let the surface at  
\[t=t_1\]
  be  
\[S_{t_1}\]
  and let the boundary of  
\[S_{t_1}\]
  be  
\[C_{t_1}\]
.
The flux out of  
\[S_{t_1}\]
  is  
\[\Phi = \int_{S_{t_1}} \mathbf{F} (\mathbf{r}, t_1) \cdot \mathbf{n}_{t_1} dS_{t_1}\]
.
\[\frac{d \Phi}{dt} = lim_{t_2 \rightarrow t_1} \frac{\Phi_{t_2} - \Phi_{t_1}}{t_2 -t_1} =lim_{t_2 \rightarrow t_1} \frac{\int \int_{S_{t_2}} \mathbf{F} (\mathbf{r}, t_2) \cdot \mathbf{n}_{t_2} dS_{t_2} - \int \int_{S_{t_1}} \mathbf{F} (\mathbf{r}, t_1) \cdot \mathbf{n}_{t_1} dS_{t_1}}{t_2 -t_1}\]

Apply the Divergence Theorem to the volume traced out by  
\[S\]
  between  
\[t_2\]
  and  
\[t_1\]
  gives
\[ \begin{equation} \begin{aligned} \int \int \int_{V_{t_1}} \mathbf{\nabla} \cdot \mathbf{F}(\mathbf{r}, t_{t_1}) dV_{t_1} &= lim_{t_2 \rightarrow t_1} ( \int \int_{S_{t_2}} \mathbf{F} (\mathbf{r},t_2) \cdot \mathbf{n}_{t_2} dS_{t_2} \\ &- \int \int_{S_{t_1}} \mathbf{F} (\mathbf{r},t_1) \cdot \mathbf{n}_{t_1} dS_{t_1} + \int \int_{Sides} \mathbf{F} (\mathbf{r},t_2) \cdot \mathbf{n}_{t_2} dS_{t_2}) \end{aligned} \end{equation} \]

\[\mathbf{F}(\mathbf{r}, t_2) \simeq \mathbf{F}(\mathbf{r}, t_1) + \frac{\partial \mathbf{F}}{\partial t} dt \]

so
where  
\[dt=t_2 -t_1\]
.
\[\begin{equation} \begin{aligned} \int \int_{S_{t_2}} \mathbf{F} (\mathbf{r},t_2) \cdot \mathbf{n}_{t_2} dS_{t_2} - \int \int_{S_{t_1}} \mathbf{F} (\mathbf{r},t_1) \cdot \mathbf{n}_{t_1} dS_{t_1} &\simeq= \int \int_{S_{t_2}} \mathbf{F}(\mathbf{r}, t_1) dS_{t_2} \\ &+ \int \int_{S_{t_2}} \frac{\partial \mathbf{F}}{\partial t} dt dS_{t_2} - \int \int_{S_{t_1}} \mathbf{F} (\mathbf{r},t_1) \cdot \mathbf{n}_{t_1} dS_{t_1} \\ &+ \int \int \int_{V_{t_1}} \mathbf{\nabla} \cdot \mathbf{F}(\mathbf{r}, t_{t_1}) dV_{t_1} - \int \int_{Sides} \mathbf{F} (\mathbf{r},t_2) \cdot \mathbf{n}_{t_2} dS_{t_2} \end{aligned} \end{equation}\]

\[dV \simeq v dS dt\]

Hence
\[\begin{equation} \begin{aligned} \int \int_{S_{t_2}} \mathbf{F} (\mathbf{r},t_2) \cdot \mathbf{n}_{t_2} dS_{t_2} - \int \int_{S_{t_1}} \mathbf{F} (\mathbf{r},t_1) \cdot \mathbf{n}_{t_1} dS_{t_1} &\simeq= \int \int_{S_{t_2}} \mathbf{F}(\mathbf{r}, t_1) dS_{t_2} \\ &+ \int \int_{S_{t_2}} \frac{\partial \mathbf{F}}{\partial t} dt dS_{t_2} - \int \int_{S_{t_1}} \mathbf{F} (\mathbf{r},t_1) \cdot \mathbf{n}_{t_1} dS_{t_1} \\ &+ \int \int \int_{V_{t_1}} \mathbf{\nabla} \cdot \mathbf{F}(\mathbf{r}, t_{t_1}) v dS_{t_1} dt - \int \int_{Sides} \mathbf{F} (\mathbf{r},t_2) \cdot \mathbf{n}_{t_2} dS_{t_2} \end{aligned} \end{equation}\]

Now divide by  
\[dt=t_2 -t_1\]
  to get
\[\frac{d \Phi}{dt} = \int \int_{S_t}( \frac{\partial \mathbf{F}}{\partial t} + \mathbf{v} \mathbf{\nabla} \cdot \mathbf{F}) \cdot d \mathbf{n} dS + \oint_{C_1} (\mathbf{F} \times \mathbf{v}) \cdot d \mathbf{r} \]