15.5 Stokes’ Theorem: Transformation between line integral and surface integral

1.Evaluate using Stokes’ theorem, given that is the circle: that lies inside the cylinder and above the -plane.

http://math2.skku.ac.kr/home/pub/57 


 Sol)

 

x,y,z,t=var('x,y,z,t')

p1 = implicit_plot3d(x^2+y^2+z^2==4, (x,-2,2), (y, -2,2), (z, 0,2), opacity=0.2, color="red", mesh=True);

p2 = implicit_plot3d(x^2+y^2==1, (x,-2,2), (y, -2,2), (z, 0,sqrt(3)), opacity=0.5, color="blue", mesh=True);

p3 = plot3d(0, (x,-2,2), (y, -2,2), opacity=0.3, color="orange", mesh=True);

show(p1+p2+p3, aspect_ratio=1)


 


 

i,j,k=var('i,j,k')

r_t=vector(SR,[cos(t),sin(t),sqrt(3)]);

r_t

  (cos(t), sin(t), sqrt(3))


dr_t=diff(r_t,t);

dr_t

  (-sin(t), cos(t), 0)


F=vector(SR, [x,y,x*y]);

F

  (x, y, x*y)


F_r_t=vector(SR, [cos(t),sin(t),cos(t)*sin(t)]);

F_r_t

  (cos(t), sin(t), sin(t)*cos(t))


integral(F_r_t*dr_t,t,0,2*pi())

  0


2. Evaluate (a) directly (b) using Stoke's theorem where is the ellipse  .

 Sol) (Using Stoke's theorem) Let .

      So and

            

                               


Verify Stoke's theorem in the following problems:


3. where :upper half surface of the sphere

   [Hint] : square .

 Sol) Since curl, we have curl.


4. where is square   in the .

        [Hint] : square .

http://matrix.skku.ac.kr/cal-lab/cal-14-7-4.html

 Sol)

 

@interact

def _(a=(1,(-5,5))):

 var('x,y,z')

 p1 = implicit_plot3d(x==0, (x,-a,a), (y, -a,a), (z, -a,a), opacity=0.2, color="red", mesh=True);

 p2 = implicit_plot3d(y==0, (x,-a,a), (y, -a,a), (z, -a,a), opacity=0.5, color="blue", mesh=True);

 show(p1+p2, aspect_ratio=1)

 var('u,v,w')

 u=curl([x^2,x*y,0])

 v=vector([0,0,1])

 w=u.dot_product(v)

 ANSWER=integral(integral(w,x,0,a),y,0,a)

 print ANSWER


 


5. taken around the rectangle bounded by .

http://matrix.skku.ac.kr/cal-lab/cal-15-5-5.html

 Sol)

 

@interact

def _(a=(1,(-5,5)),b=(1,(-5,5))):

 l1=line([(-a,0), (a,0)],color='black');

 l2=line([(a,0), (a,b)],color='red');

 l3=line([(a,b), (-a,b)],color='green');

 l4=line([(-a,b), (-a,0)]);

 show(l1+l2+l3+l4)

 var('u,v,w')

 u=curl([x^2,x*y,0])

 v=vector([0,0,1])

 w=u.dot_product(v)

 ANSWER=integral(integral(w,x,0,a),y,0,b)

 print ANSWER


 


6. Evaluate , where and is the curve which is the intersection of and . ( is upward anticlockwise).

 Sol) curl and the curve is a boundary of on .

     (Using Strokes' Theorem)

           curl.


7. We consider the vector field and the curve which is the boundary of the triangle with vertices . Compute the work done by the force field in moving a particle along the curve . (First, the particle starts from to , and starts from to , finally starts from back to ). 

 Sol) Consider the triangle . Then its unit normal vector is

     .

     (Orientation is considered) Using Stokes' Theorem, we have

          

     The area of is .

         


8. Evaluate . Here and is a triangle with vertices  .

 Sol) curl.

      Here we have .


      ,

      ,

      curl

                


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

 Sol) Let . Then

     


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

 Sol) (Using Green's Theorem)

        

        

        .

     Using , we can compute