Week 16: Stokes' Theorem, The Divergence theorem and a unified theory

Thomas Calculus, 11th edition, pages: 1201 - 1222

Sec 16.7 Stokes' Theorem

In this section we will discuss Stokes’ Theorem.

Sec 16.8 The Divergence Theorem and a Unified Theory

The divergence form of Green’s Theorem in the plane states that the net outward flux of a vector field across a simple closed curve can be calculated by integrating the divergence of the field over the region enclosed by the curve. The corresponding theorem in three dimensions, called the Divergence Theorem, states that the net outward flux of a vector field across a closed surface in space can be calculated by integrating the divergence of the field over the region enclosed by the surface. In this section, we prove the Divergence Theorem and show how it simplifies the calculation of flux. We also derive Gauss’s law for flux in an electric field and the continuity equation of hydrodynamics. Finally, we unify the chapter’s vector integral theorems into a single fundamental theorem.