SKKU-Calculus-Sec-15-9 Divergence Theorem 최주영


  15.9    Divergence Theorem    by SGLee, 최주영

Point. Gauss Divergence Theorem는 vector version of Green's Theorem 의 벡터영역을 3차원으로 확장한 것입니다.

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1-4. Using the Divergence Theorem, evaluate the surface integral:

 

 1.  where : .

http://matrix.skku.ac.kr/cal-lab/cal-14-8-1.html 

 Define "Div" function

 

var('x,y,z');   

def Div(F):

    assert(len(F) == 3)

    return (diff(F[0],x)+diff(F[1],y)+diff(F[2],z))

 

 

 

var('x,y,z');

Div([x^3,x^2*y,x*y])

 

  0

 Since divergence is 0, the given integral value is .









 

 2.  where  is a closed surface consisting of the circular cylinder

          and the circular disks  and .

http://matrix.skku.ac.kr/cal-lab/cal-15-6-2.html   

 

 

var('x,y,z');   

def Div(F):

    assert(len(F) == 3)

    return (diff(F[0],x)+diff(F[1],y)+diff(F[2],z))

Div([x^3,x^2*y,x*y])

 

  4*x^2

 

 Parametrized by 

 

var('t')

x=cos(t)

y=sin(t)

z=z

w=integral(4*cos(t)^2,t,0,2*pi)

var('a,b');

w*a*b

 

  4*pi*a*b






 3.  where  is a parallelepiped of ,　,.

http://matrix.skku.ac.kr/cal-lab/cal-15-6-3.html 

 

 

var('x,y,z');

Div([sin(x),(2-cos(x))*y,0])

 

  2

 

 

integral(integral(integral(2,x,0,3),y,0,2),z,0,1)

 

  12






 4.  where : cube of side  and three of whose edges are along the axes.

http://matrix.skku.ac.kr/cal-lab/cal-15-6-4.html 

 

 

var('x,y,z,r,t');

def Div(F):

    assert(len(F) == 3)

    return (diff(F[0],x)+diff(F[1],y)+diff(F[1],z))

Div([x^2-y*z,-2*x^2*y,z])

 

  -2*x^2 + 2*x

 

 Parametrized by 

 

x=cos(t)

y=sin(t)

z=z

integral(integral(integral(-2*cos(t)^2+2*cos(t),t,0,2*pi),r,0,2),z,0,3)

 

  -12*pi






5-13. Verify the Gauss Divergence Theorem for :

 

 5.  taken over the region bounded by ,  and .

http://matrix.skku.ac.kr/cal-lab/cal-14-8-5.html 

 

 

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

p1 = implicit_plot3d(x^2+y^2==4, (x,-2,2), (y, -2,2), (z, -5,5), opacity=0.2, color="red", mesh=True);

p2 = implicit_plot3d(z==0, (x,-2,2), (y, -2,2),(z, -5,5), opacity=0.3, color="blue", mesh=True);

p3 = implicit_plot3d(z==3, (x,-2,2), (y, -2,2), (z, -5,5),   opacity=0.5, color="orange", mesh=True);

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

 

                                                 

 

 

var('x,y,z')

def Div(F):

    assert(len(F) == 3)

    return (diff(F[0],x)+diff(F[1],y)+diff(F[1],z))

Div([4*x,-2*y^2,z^2])

 

  -4*y + 4

 

 

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

integral(integral(2*(-8*sin(t)+2*z+4),t,0,2*pi),z,0,3)

 

  84*pi

 

 , ,  ,

  

  ,

  .

   Therefore .

                                                    









6.  taken over the entire surface of the cube ,,.

  

  ,

  ,

  ,

  ,

   (Since  on ),

  ,

  Therefore 

  ※ div. Hence div.

                                            



7.  taken over the entire surface of the sphere of radius  and centered at the origin.

http://matrix.skku.ac.kr/cal-lab/cal-14-8-7.html

 

 

var('a,b,c,x,y,z')

def Div(F):

    assert(len(F) == 3)

    return (diff(F[0],x)+diff(F[1],y)+diff(F[1],z))

Div([a*x,b*y,c*z])

 

  a+b+c

 

 So, integral of a+b+c over the entire surface of the sphere of radius  is .

 Compute the second part, and show they are equal.



8.  and  is the total surface of the rectangular parallelepiped bounded by the coordinate planes and , , .

 

 

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

p1 = implicit_plot3d(x==1, (x,0,5), (y, 0,5), (z, 0,5), opacity=0.2, color="red", mesh=True);

p2 = implicit_plot3d(y==2, (x,0,5), (y, 0,5), (z, 0,5), opacity=0.3, color="blue", mesh=True);

p3 = implicit_plot3d(z==3, (x,0,5), (y, 0,5),(z, 0,5),   opacity=0.5, color="orange", mesh=True);

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

                                                

 

 

def Div(F):

    assert(len(F) == 3)

    return (diff(F[0],x)+diff(F[1],y)+diff(F[1],z))

d10=Div([2*x*y,y*z^2,x*z])

 

 

 

integral(integral(integral(d10,x,0,1),y,0,2),z,0,3)

 

  48

 

 Compute the second part, and show they are equal.






9.  over the upper half of the sphere .

  

 A parameterization of sphere is

 ,

 ,

 ,

 , .

 Hence  (See the figure).

  

                      ,

  .

  Therefore .

  Compute the second part, and show they are equal.



10.  taken over the rectangular parallelepiped bounded by the coordinate planes and ,  and .

 .



11.  taken over the surface of the ellipsoid .

     Let  be a parameterization of .

     ,

     .

     Hence

     

                          .

  Compute the second part, and show they are equal.



12.  taken over the upper half of the unit sphere .

  

 ,  ,

 ,

  

  .

  Therefore .

  Compute the second part, and show they are equal.

                                        



13.  taken over the closed region of the cylinder , bounded by the planes   and .

 

 ,  ,

 .

  

 ,

 

  Therefore .

  Compute the second part, and show they are equal.

                        



14. (a)  Prove Green's first identity : 

    (b) Let  and . Find .



15. Let  be a solid surrounded by  and  and vector field

    . Find the flux of , that is, 

 

     div,

     div

     



16. Evaluate .

    Here , and  is the boundary of the solid surrounded by , .

      div.

      Let .

      

                      

                      .



17. Let  be a surface  between  and  and . Find a flux .

     div.

     div

     



18. Evaluate the surface integral , where  and  is the part of surface of the paraboloid  above -plane.

 

      Let  and  be the region bounded by  and .

     Then, with the aid of the Divergence Theorem,

      . Since  and  on ,

     we can see that . On the other hand, , and

     therefore, we have  = volume of ,

     which can be computed as follows:

     

              .



19. Let . Evaluate , where  is the part of the sphere

    , .

 

    Note that 

     . Using the Divergence    Theorem and polar coordinates , , where

   ,

   we compute  .

       div 

                               

                               .

       

       

                          

                          .

       : 

       .



20. If , evaluate   over the volume of a cube of side .



21. Evaluate  over the solid region of the sphere  when   where , ,  are constants.



 

                

 

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