Let F(x, y, z) be a continuous vector field whose components have continuous partial derivatives on an open region which contains the surface of the earth. Since the northern hemisphere DN, oriented so that its normal is point out to space, and the southern hemisphere Ds, oriented so that its normal is point out to space, both have the equator oriented west-to-east (i.e. counterclockwise) as their boundary, we know by Stokes' theorem. JJDN (V x F). dS = J/ (V x F) .ds s