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Proof of Flux Transport Theorem Using Parametrization of a Surface

Theorem
\[ \frac{d \Phi}{dt} =\int \int_{S} (\frac{\partial \mathbf{F}}{\partial t} + ((\mathbf{\nabla} \cdot \mathbf{F}) \mathbf{v} ) \cdot d \mathbf{S} + \oint_{C_1} (\mathbf{F} \times \mathbf{v}) \cdot d \mathbf{r}\]

Proof
Suppose we have a surface we have a surface  
\[S_0\]
  such that at  
\[t-0\]
 
\[S_0\]
  is parametrized by coordinates  
\[u, \: v\]
  and we can write  
\[\mathbf{r} =r(u,v)\]
.
A point on  
\[S_)\]
  traces out a curve  
\[\mathbf{r}=r(u,v,t)\]
.
We can define the velocity of a point by  
\[\mathbf{v} = \frac{\partial \mathbf{r}}{\partial t}\]
.
The flux out of the surface  
\[S\]
  at time  
\[t\]
  is
\[\Phi_t = \int \int_{S_t} \mathbf{F} d \mathbf{S} = \int \int_{\Omega} \mathbf{F}(r(u,v), t) \cdot (\frac{\partial \mathbf{r}}{\partial u} \times \frac{\partial \mathbf{r}}{\partial v}) du dv\]

Since  
\[\Omega\]
  is fixed and independent of time,
\[\frac{d \Phi}{dt} = \int \int_{\Omega} \frac{d \mathbf{F}}{dt} \cdot (\frac{\partial \mathbf{r}}{\partial u} \times \frac{\partial \mathbf{r}}{\partial v}) dudv + \int \int_{\Omega} \mathbf{F} \cdot \frac{\partial}{\partial t} (\frac{\partial \mathbf{r}}{\partial u} \times \frac{\partial \mathbf{r}}{\partial v}) du dv\]
  (1)
Differentiating  
\[\mathbf{F}\]
  with respect to time gives
\[\frac{d \mathbf{F}}{dt} = \frac{\partial \mathbf{F}}{\partial t} + \frac{\partial u}{\partial t} \frac{\partial \mathbf{F}}{\partial u} + \frac{\partial v}{\partial t} \frac{\partial \mathbf{F}}{\partial v}= \frac{\partial \mathbf{F}}{\partial t} + (\mathbf{v} \cdot \mathbf{\nabla}) \mathbf{F} \]

\[\begin{equation} \begin{aligned} \mathbf{F} \cdot \frac{\partial}{\partial t} (\frac{\partial \mathbf{r}}{\partial u} \times \frac{\partial \mathbf{r}}{\partial v}) &= \mathbf{F} \cdot (\frac{\partial }{\partial u}(\frac{\partial \mathbf{r}}{\partial t} \times \frac{\partial \mathbf{r}}{\partial v}) - \frac{\partial }{\partial v}(\frac{\partial \mathbf{r}}{\partial t} \times \frac{\partial \mathbf{r}}{\partial u}) ) \\ &= \mathbf{F} \cdot \frac{\partial }{\partial u}(\frac{\partial \mathbf{r}}{\partial t} \times \frac{\partial \mathbf{r}}{\partial v}) - \mathbf{F} \cdot \frac{\partial }{\partial v}(\frac{\partial \mathbf{r}}{\partial t} \times \frac{\partial \mathbf{r}}{\partial u}) \\ &= \frac{\partial}{\partial u} (\mathbf{F} \cdot(\mathbf{v} \times \frac{\partial \mathbf{r}}{\partial v} ))- \frac{\partial \mathbf{F}}{\partial u} \cdot (\mathbf{v} \times \frac{\partial \mathbf{r}}{\partial v}) \\ &- \frac{\partial}{\partial v} (\mathbf{F} \cdot(\mathbf{v} \times \frac{\partial \mathbf{r}}{\partial u} ))+ \frac{\partial \mathbf{F}}{\partial v} \cdot (\mathbf{v} \times \frac{\partial \mathbf{r}}{\partial u}) \\ &= \frac{\partial}{\partial u} (\mathbf{F} \cdot (\mathbf{v} \times \frac{\partial \mathbf{r}}{\partial v} ))- \frac{\partial}{\partial v} (\mathbf{F} \cdot(\mathbf{v} \times \frac{\partial \mathbf{r}}{\partial u} )) \\ &+ ((\mathbf{\nabla} \cdot \mathbf{F}) \mathbf{v} - (\mathbf{v} \cdot (\mathbf{\nabla}) \mathbf{F}) \cdot (\frac{\partial \mathbf{r}}{\partial u} \times \frac{\partial \mathbf{r}}{\partial v}) \end{aligned} \end{equation}\]

Substitute these expressions for  
\[\frac{d \mathbf{F}}{dt}\]
  and  
\[ \mathbf{F} \cdot \frac{\partial}{\partial t} (\frac{\partial \mathbf{r}}{\partial u} \times \frac{\partial \mathbf{r}}{\partial v})\]
  into (1) to get
\[\begin{equation} \begin{aligned} \frac{d \Phi}{dt} &= \int \int_{\Omega} (\frac{\partial \mathbf{F}}{\partial t} + (\mathbf{v} \cdot \mathbf{\nabla}) \mathbf{F} ) \cdot (\frac{\partial \mathbf{r}}{\partial u} \times \frac{\partial \mathbf{r}}{\partial v}) dudv \\ &+ \int \int_{\Omega} \frac{\partial}{\partial u} (\mathbf{F} \cdot (\mathbf{v} \times \frac{\partial \mathbf{r}}{\partial v} ))- \frac{\partial}{\partial v} (\mathbf{F} \cdot(\mathbf{v} \times \frac{\partial \mathbf{r}}{\partial u} )) \\ &+ ((\mathbf{\nabla} \cdot \mathbf{F}) \mathbf{v} - (\mathbf{v} \cdot (\mathbf{\nabla}) \mathbf{F}) \cdot (\frac{\partial \mathbf{r}}{\partial u} \times \frac{\partial \mathbf{r}}{\partial v}) du dv \\ &= \int \int_{\Omega} (\frac{\partial \mathbf{F}}{\partial t} + (\mathbf{v} \cdot \mathbf{\nabla}) \mathbf{F} + ((\mathbf{\nabla} \cdot \mathbf{F}) \mathbf{v} - (\mathbf{v} \cdot (\mathbf{\nabla}) \mathbf{F})) \cdot (\frac{\partial \mathbf{r}}{\partial u} \times \frac{\partial \mathbf{r}}{\partial v}) dudv \\ &+ \int \int_{\Omega} \frac{\partial}{\partial u} (\mathbf{F} \cdot (\mathbf{v} \times \frac{\partial \mathbf{r}}{\partial v} ))- \frac{\partial}{\partial v} (\mathbf{F} \cdot(\mathbf{v} \times \frac{\partial \mathbf{r}}{\partial u} ))du dv \\ &= \int \int_{\Omega} (\frac{\partial \mathbf{F}}{\partial t} + + ((\mathbf{\nabla} \cdot \mathbf{F}) \mathbf{v} ) \cdot (\frac{\partial \mathbf{r}}{\partial u} \times \frac{\partial \mathbf{r}}{\partial v}) dudv \\ &+ \oint_{C_1} \frac{\partial}{\partial u} (\mathbf{F} \cdot (\mathbf{v} \times \frac{\partial \mathbf{r}}{\partial v} ))- \frac{\partial}{\partial v} (\mathbf{F} \cdot(\mathbf{v} \times \frac{\partial \mathbf{r}}{\partial u} ))du dv \\ &= \int \int_{\Omega} (\frac{\partial \mathbf{F}}{\partial t} + + ((\mathbf{\nabla} \cdot \mathbf{F}) \mathbf{v} ) \cdot (\frac{\partial \mathbf{r}}{\partial u} \times \frac{\partial \mathbf{r}}{\partial v}) dudv \\ &+ \int \int_{\Omega} \frac{\partial}{\partial u}( (\mathbf{F} \times \mathbf{v}) \cdot \frac{\partial \mathbf{r}}{\partial v} )- \frac{\partial}{\partial v} ((\mathbf{F} \times \mathbf{v}) \cdot \frac{\partial \mathbf{r}}{\partial u} )du dv \\ &= \int \int_{S} (\frac{\partial \mathbf{F}}{\partial t} + + ((\mathbf{\nabla} \cdot \mathbf{F}) \mathbf{v} ) \cdot d \mathbf{S} + \oint_{C_1} (\mathbf{F} \times \mathbf{v}) \cdot d \mathbf{r} \end{aligned} \end{equation}\]

Using Green's Theorem.

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