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

\[\int_S \mathbf{F} \cdot \mathbf{n} dS = \int_{xy} \frac{\mathbf{F} \cdot \mathbf{n}}{| \mathbf{n} \cdot \mathbf{k}|} dxdy \]