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.