Why do we take the dot product of n and f. f is the vector field and n is the unit vector normal to the plane being analyzed.

If you integrate that over the surface, you get the flux of the vector field through that surface.

Oh! Duh, thanks.

If the flow is perpendicular to the surface, then n \cdot f will just be the magnitude of f, since all of the flow is going through the surface. On the other hand, if the flow is parallel, then n \cdot f will be zero–again, expected since none of the flow is going through the surface (it’s all going *along* the surface). And intermediate angles will have intermediate values (proportional to \cos(\theta)).

Nit: The dude’s name was “George Stokes”, so it is “Stokes’ Theorem”, not “Stoke’s Theorem”.