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