Green's theorem says that (subject to some very reasonable conditions that we need not concern ourselves with here) the counterclockwise line integral of the vector field **F** = [P Q] around the boundary of a region is equal to the double intregral of over the region itself. It's natural to think of it as a special case of Stokes's theorem in the case of a plane. We can also think of the line integral as the integral of the inner product of the vector field with the unit tangent, leading us to write Green's theorem like this:

But some texts (I have Mardsen and Tromba's *Vector Calculus* and Stewart's *Calculus: Early Transcendentals* in my possession; undoubtedly there are others) point out that we can also think of Green's theorem as a special case of the divergence theorem! Suppose we take the integral of the inner product of the vector field with the outward-facing unit normal (instead of the unit tangent)—it turns out that

—which suggests that there's some deep fundamental sense in which Stokes's theorem and the divergence theorem are really just *mere surface manifestations of one and the same underlying idea*! (I'm told that it's called the generalized Stokes's theorem, but regrettably I don't know the details yet.)