## Alternative Expression for Flux Out of a Suface - Projection onto xy Plane

The projection of an element of surface
$dS$
onto the
$xy$
plane is
$dxdy = | \mathbf{n} \cdot \mathbf{k}| dS$

We can write this as
$dS=\frac{dxdy}{| \mathbf{n} \cdot \mathbf{k}|}$

Hence the flux out of the surface die to a vector field
$\mathbf{F}$
is
$\int_S \mathbf{F} \cdot \mathbf{n} dS = \int_{xy} \frac{\mathbf{F} \cdot \mathbf{n}}{| \mathbf{n} \cdot \mathbf{k}|} dxdy$