We end this section by using Green's Theorem to discuss a result that was stat preceding section. SKETCH OF PROOF OF THEOREM 16.3.6 We're assuming that F = Pi + Qj is field on an open simply-connected region D, that P and Q have continuous firs partial derivatives, and that ap aQ ay ax throughout D If C is any simple closed path in D and R is the region that C encloses, then G Theorem gives ap { F . dr - 8 Pdx + 0 dy - ]) ax ay dA - fodA = 0 R A curve that is not simple crosses itself at one or more points and can be broke into a number of simple curves. We have shown that the line integrals of F aro simple curves are all 0 and, adding these integrals, we see that Jc F . dr = Of closed curve C. Therefore Jc F . dr is independent of path in D by Theorem 1 follows that F is a conservative vector field. 16.4 EXERCISES 1-4 Evaluate the line integral by two methods: (a) directly and 2. bcy dx - x dy, C is the circle with center the origin and radius 4 (b) using Green's Theorem. 3. Doxy dx + xy' dy, 1. gcy? dx + x2 y dy, C is the triangle with vertices (0, 0), (1, 0), and (1, 2 C is the rectangle with vertices (0, 0), (5, 0), (5, 4), and (0, 4)