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