\[\mathbf{F}\]
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\[\int_{uv} \mathbf{F} \cdot (\frac{\partial \mathbf{r}}{\partial u} \times \frac{\partial \mathbf{r}}{\partial v})dudv\]