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.