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.

 

 

        

                                                                  

                                                                  .







let 




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 .

             http://matrix.skku.ac.kr/cal-lab/cal-15-2-3.html 

 

 




 Using Green's Theorem







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

   .

 

 (a)

         .

         ,

         ,

        

            

            

            

         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 .

             http://matrix.skku.ac.kr/cal-lab/cal-15-2-10.html 

 

 




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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