Advanced Engineering Mathematics, 7th Edition

9.12 Green’s Theorem

INTRODUCTION

One of the most important theorems in vector integral calculus relates a line integral around a piecewise-smooth simple closed curve C to a double integral over the region R bounded by the curve.

Line Integrals Along Simple Closed Curves

We say the positive direction around a simple closed curve C is that direction a point on the curve must move, or the direction a person must walk on C, in order to keep the region R bounded by C to the left. See FIGURE 9.12.1(a). Roughly, as shown in Figure 9.12.1(b) and 9.12.1(c), the positive and negative directions correspond to the counterclockwise and clockwise directions, ...