S is closed if it has no edge, or boundary, and doesn't go off to infinity in any direction. Any closed surface S encloses a region V , called the interior , similarly to a simple closed curve on the plane. Curiously, though, that is it.
Any closed surface is a sphere with some number of handles attached.
Given a closed surface S , you can choose, consistently, an outward pointing unit normal vector n . The last two lines follow from the definition of surface integrals. The minus sign came from the fact that we want the outward unit normal, but for a graph that formula works for the upward normal, which on the bottom surface is not the outward one from the closed surface. To complete this we really should worry about surfaces that are so complicated that they can't be written as a top and a bottom, and for the other two components. But that works just as it did for Green's theorem.