Advanced Engineering Mathematics, 7th Edition

9.16 Divergence Theorem

INTRODUCTION

In Section 9.14 we saw that Stokes’ theorem was a three-dimensional generalization of a vector form of Green’s theorem. In this section we present a second vector form of Green’s theorem and its three-dimensional analogue.

Another Vector Form of Green’s Theorem

Let F(x, y) = P(x, y)i + Q(x, y)j be a two-dimensional vector field, and let T = (dx/ds)i + (dy/ds)j be a unit tangent to a simple closed plane curve C. In (1) of Section 9.14 we saw that (F · T) ds can be evaluated by a double integral involving curl ...