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Divergence Theorem

  1. Sep 9, 2005 #1
    Let Q denote the unit cube in [tex]\Re^3[/tex] (that is the unite cube with 0<x,y,z<1). Let G(x,y,z) = (y, xe^z+3y, y^3*sinx). Verify the validity of the divergence theorem.

    [tex] \int_{Q} \bigtriangledown} \cdot G dxdydz = \int_{\partial Q} G \cdot n dS [/tex]

    I am not sure how to evaluate the right side. Any help would be good.
     
  2. jcsd
  3. Sep 9, 2005 #2

    arildno

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    Well, you've got six sides on your surface, right?
    Treat the contribution from each side separately.
     
  4. Sep 9, 2005 #3
    What would say..the first integral of the six look like? I just need one example.
     
  5. Sep 9, 2005 #4

    arildno

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    Okay, let's look at the side x=1, 0<=y,z<=1.
    Here, the normal vector is +i.
    G(1,y,z)=(y,e^z+3y,y^3sinx)
    Forming the dot product between G and the normal vector yields the integrand "y".
    This is easy to integrate over the y,z-square (yielding 1/2 in contribution)

    Okay?
     
  6. Sep 9, 2005 #5
    Ok that makes sense. So in the end I add the value of each of the six integrals for the total value of the original integral, [tex]\int_{\partial Q} G \cdot n dS [/tex]?
     
  7. Sep 9, 2005 #6

    Hurkyl

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    By the way, is your book actually using [itex]\partial\Omega[/itex]? I'm more used to seeing [itex]d\Omega[/itex] or [itex]\delta\Omega[/itex].
     
  8. Sep 10, 2005 #7

    arildno

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    That's correct.
    Remember that the integral operation has the additive property; this entails that you may split up the surface in an arbitrary number of sub-surfaces, calculate the individual contributions and then add the individual contributions together to gain the correct value.
     
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