Question

Why the picture 18.5.3 is not elementary Jordan region and why we need to break it up to 2 pieces and use green theorem ? It just looks like a Jordan region which can be applied green theorem with out breaking up

961 18.5 GREEN'S THEOREM Figure 18.5.3 showsa Jordan region that is not elementary but can be broken up into two elementary regions. (See Figure 18.5.4.) Green's theorem applied to the elementary parts tells us that Pdx +Qdy àx dy 521 Q aP dx dy bdry of Ω2 Figure 18.5.3 We now add these equations. The sum of the double integrals is, by additivity, the double integral over Ω. The sum of the line integrals is the integral over C (see the figure) plus the integrals over the crosscut. Since the crosscut is traversed twice and in opposite directions, the total contribution of the crosscut is zero and therefore Green's theorem holds ag aP This same argument can be extended to a Jordan region 2 that breaks up into n ele mentary regions S21 integrals over the Ω, add up to the double integral over Ω, and, since the line integrals over the crosscuts cancel, the line integrals over the boundaries of the 2, add up to the line integral over C. (This is as far as we will carry the proof of Green's theorem. It is far enough to cover all the Jordan regions we encounter in practice.) O Ω". (Figure 18.5.5 gives an example with n-4.) The double Figure 18.5.4 C: Figure 18.5.5 xample 1 Use Green's theorem to eval (3x2 + y) dx + (2x + y3)dy. (Figure 18.5.6) where C is the circle x2+ya x2 +y2-a2. With SOLUTION Let be the closed disk 0 Figure 18.5.6 we havc OP -2. and ax ay

Answer 1

Its written in the book just to denote the cases in which its not applicable. You cannot just look at the regions and directly say that green theorem is directly applicable or not ( until and unless any sharp corner or hole is not present in the region). Proper conditions must be checked like :

1) P and Q are functions of (x, y) defined on an open region containing D and have continuous partial derivatives there.

2) Let C be a positively oriented, piecewise smooth, simple closed curve in a plane, and let D be the region bounded by C

Again, I tell you that its just to demonstrate the cases which are not directly applicable to Green's theorem.