## 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{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}