Lecture 15: Surface Integrals and Some Related Theorems

Reappraise the simplest integration operation, g(x) = ∫ f(x)dx. Temporarily ignore all the tedious mechanical rules of finding and integral and concentrate on what integration does. Integration replaces a fairly complex process—adding up all the contributions of a function f(x)— with a clever new function g(x) that only needs end-points to return the result of a complicated summation. It is perhaps initially astonishing that this complex operation on the interior of the integration domain can be incorporated merely by the domain’s endpoints. However, careful reflection provides a counterpoint to this marvel. How could it be otherwise? The function f(x) is specified and there are no surprises lurking along the x-axis that will trip up dx as it marches merrily along between the endpoints. All the facts are laid out and they willingly submit to the process their preordination by g(x) by virtue of the endpoints.7 The idea naturally translates to higher dimensional integrals and these are the basis for Green’s theorem in the plane, Stoke’s theorem, and Gauss (divergence) theorem. Here is the idea: