SKKU-Calculus-Sec- 15-4 Green’s Theorem in Plane, SGLee+김태현

 15.4    Green’s Theorem in the Plane, SGLee+김태현


1-4. Use Green’s Theorem to evaluate the line integral , when  is


 1. , where  is a circle of , with positive orientation.



2. , where  is the positively oriented curve which is the boundary of the region bounded by .





3. , where  is a circle of , with positive orientation.







4.  where  is a boundary of the rectangle , with positive orientation.




5-8. Verify Green's Theorem by evaluating the line integral  (a) directly, (b) using Green's Theorem, where  is:


5.  with  is a closed curve of the region bounded by  and , with positive orientation.





6.  with  is a boundary of the region defined by  and , with positive orientation.





7.  with  is a boundary of the region enclosed by the circles  and

   , with positive orientation.







8. , where  is a circle of , with positive orientation.






9.  , with positive orientation.



    Let . To evaluate the line integral, we use Green's Theorem as follows


 10.  where  is the arc of the parabola  in the plane  from  to .




 Using Green's Theorem

11. Calculate the area bounded by the ellipse . Hence deduce the area bounded by the circle











         Put , area of circle : .

12. Find the area under arc of the cycloid .

13. Find the area under of one arch of the asteroid .

 14. Find the area of the loop of the folium of Descartes .




15. Let  be smooth functions satisfying the following differential equations:

    . Evaluate the line integral ,

   where  is traversed in the positive direction of the curve .


         Using Green's Theorem, we have


16. 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   for all . Suppose that the solution curve (phase trajectory) is

     a simple closed curve in the phase plane (plane) and  is the region bounded by .

     Prove that .


Proof :  Recalling Green's Theorem,


       Now parametrizing by , the line integral becomes



       Therefore,  we obtain.


17. Evaluate the integral , where  is the boundary of the triangular region 

     with corners  with positive orientation.



18. Let  be the straight lines from  to , from  to , from  to , and from  to . Evaluate .



     Let . Then



19. Let  and  is an any closed curve containing the origin. Find .

 Let’s consider the region  which is the annulus as shown in the figure. Let  and  denote the outer and the inner circles of .


      Then by Green’s Theorem, we have



     Using , we can compute


     Thus the vector field  is not conservative. Note that curl. Here  is not a simply connected domain.



                                                                  The 6th Asian Math Conference (AMC 2013) at Busan


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