## Alternative Expression for the Flux out of a Surface

The flux of a vector field
$\mathbf{F}$
out of a surface
$S$
is
$\int_S \mathbf{F} \cdot d \mathbf{S}$

If the surface is parametrized with parameters
$u,v$
we can write
$d \mathbf{S} = \frac{\partial \mathbf{r}}{\partial u} \times \frac{\partial \mathbf{r}}{\partial v}du dv$
hence the flux out of the surface is
$\int_{uv} \mathbf{F} \cdot (\frac{\partial \mathbf{r}}{\partial u} \times \frac{\partial \mathbf{r}}{\partial v})dudv$