The Flux Transport Theorem for a Differentiable velocity Field
The Flux Transport Theorem states\[ \frac{d \Phi}{dt} =\int \int_{S} (\frac{\partial \mathbf{F}}{\partial t} + ((\mathbf{\nabla} \cdot \mathbf{F}) \mathbf{v} ) \cdot d \mathbf{S} + \oint_{C_1} (\mathbf{F} \times \mathbf{v}) \cdot d \mathbf{r}\]
for the flux of a vector field
\[\mathbf{F}\]
through a surface \[S_t\]
with boundary \[C_t\]
at time \[t\]
.If
\[\mathbf{v}\]
is differentiable and defined on a region containing \[S_y\]
we can apply Stoke's Theorem to get\[\oint_{C_1} (\mathbf{F} \times \mathbf{v}) \cdot d \mathbf{r} = \int \int_{S_t} (\mathbf{\nabla} \times (\mathbf{F} \times \mathbf{v}) \cdot d \mathbf{S}\]
The Flux Transport Theorem becomes
\[ \frac{d \Phi}{dt} =\int \int_{S} (\frac{\partial \mathbf{F}}{\partial t} + ((\mathbf{\nabla} \cdot \mathbf{F}) \mathbf{v} + (\mathbf{\nabla} \times (\mathbf{F} \times \mathbf{v})) \cdot d \mathbf{S} \]
Now use the identity
\[\mathbf{\nabla} \times (\mathbf{F} \times \mathbf{v} ) = (\mathbf{v} \cdot \mathbf{\nabla} ) \mathbf{F} - (\mathbf{F} \cdot \mathbf{\nabla} ) \mathbf{v} + (\mathbf{\nabla} \cdot \mathbf{v} ) \mathbf{F} - \mathbf{\nabla} \cdot \mathbf{F} ) \mathbf{v} \]
to get
\[\begin{equation} \begin{aligned} \frac{d \Phi}{dt} &=\int \int_{S} (\frac{\partial \mathbf{F}}{\partial t} + ((\mathbf{\nabla} \cdot \mathbf{F}) \mathbf{v} + (\mathbf{v} \cdot \mathbf{\nabla} ) \mathbf{F} - (\mathbf{F} \cdot \mathbf{\nabla} ) \mathbf{v} + (\mathbf{\nabla} \cdot \mathbf{v} ) \mathbf{F} - (\mathbf{\nabla} \cdot \mathbf{F} ) \mathbf{v}) \cdot d \mathbf{S} \\ &= \int \int_{S} (\frac{\partial \mathbf{F}}{\partial t} + (\mathbf{v} \cdot \mathbf{\nabla} ) \mathbf{F} - (\mathbf{F} \cdot \mathbf{\nabla} ) \mathbf{v} + (\mathbf{\nabla} \cdot \mathbf{v} ) \mathbf{F} ) \cdot d \mathbf{S} \end{aligned} \end{equation}\]