If S is the surface of a tetrahedron with vertices (0,0,0), (1,0,0), (0,1,0), and (0,0,1), oriented outward, and F is a constant vector field, is why ∫s F · dA = 0?
Use the divergence theorem to change the surface integral into a volume (triple) integral. The divergence of a constant vector field is zero. So the volume integral of the divergence is zero.