15.4 Green’s Theorem in Plane: Transformation between line integral and double integral 1-4. Use Green’s theorem to evaluate the line integral when equals to: 1. where circle .

http://math2.skku.ac.kr/home/pub/56 Sol)

 x,y,r,t=var('x,y,r,t'); p = implicit_plot(x^2+y^2==9, (x,-3,3), (y, -3,3),cmap=["red"]); p1 = implicit_plot(x==0, (x,-4,4), (y, -4,4));  p2 = implicit_plot(y==0, (x,-4,4), (y, -4,4));  show(p+p1+p2, aspect_ratio=1) f1=-y; f2=x; integral(integral((diff(f2,x)-diff(f1,y))*r, r, 0, 3), t, 0, 2*pi)

18*pi

2. where : the boundary of the region bounded by , .

Sol)  .

3. where : circle .

Sol)   .

4. where : the boundary of the rectangle , .

Sol)   .

5-8. Verify Green's theorem or evaluate the line integral (a) directly (b) using Green's theorem, where is:

5. with : closed curve of the region bounded by and .

Sol)   .

6. with : boundary of the region defined by and .

Sol)   .

7. with : boundary of the region enclosed by the circles and .

Sol)     .

8. with : .

Sol)   .

9. Calculate the area bounded by the ellipse . Deduce the area bounded by the circle .

Sol) (a) . , ,    Put , area of circle : .

10. Let be smooth functions satisfying the following differential equations: , . Evaluate the line integral , where is the positively oriented curve .

Sol) Using the Green's Theorem, we have  .

11. Let be a solution of a system of differential equations , where are smooth functions in variables.
Assume further that is periodic with a periodicity , namely  . Suppose that solution curve(phase trajectory) is a simple closed curve in the phase plane ( plane) and is the region bounded by .

Prove that .

Proof) Recalling the Green's theorem, .

Now parametrizing in -variable, the line integral becomes  .

Therefore,  we obtain .