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Homework Help: Verifying difvergence theorem

  1. Jun 4, 2010 #1
    1. The problem statement, all variables and given/known data
    verify that the divergence theorem in 3-d is true for the vector field F(r)=<3x,xy,2xz>
    on the cube bounded by the planes x=0 x=1 y=0 y=1 z=0 z=1


    2. Relevant equations



    3. The attempt at a solution

    so fristly div(F)=d/dx(3x)+d/dy(xy)+d/dz(2xz)=3+3x
    [tex]\int[/tex][tex]\int[/tex][tex]\int[/tex] 3+3xdxdydz=4.5

    now i need to evaluate flux through each faces of the cube seperately so i was just wondering if i am doing this write say i would want to evaluate the top surface of the cube
    then i would have to parametrize it so would the following be corret
    r(x,y,z)=(3x,xy,1)
    dr/dx=(3,y,0)
    dr/dy=(0,x,0)
    (dr/dx) X (dr/dy) = (0,0,3x)
    r(x,y,z).((dr/dx) X(dr/dy))= (3x,xy,1).(0,0,3x) = 3x
    [tex]\int[/tex][tex]\int[/tex] 3x dydx
    =3/2

    and i have to do the same for all other five surfaces so is this the correct way?
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Jun 5, 2010 #2

    vela

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    You're not using the correct r when calculating the normal. You want to describe the surface, not the vector field. For the top face, it would be r=(x, y, 1).
     
  4. Jun 5, 2010 #3
    ok thanks
     
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