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Homework Help: Integral of a closed surface

  1. Jan 6, 2005 #1
    I've been stuck on the following problem: If S is a closed surface that bounds the volume V, prove that: integral over this surface dS = 0.

    I've been reading several textbooks that discuss flux, Stokes' Theorem, Divergence Theorem, but I can't seem to relate them to the problem I'm doing. The examples in the text all have a vector F and present the integral: integral over a surface of F dS, which I understand it as the flux. Is my case a flux problem? There is no vector F given in my problem.

    Should I divide the closed surface into two halves and argue that pairs of normal vectors, one from each half cancel and therefore the integral over this surface dS = 0? What about Stokes' Theorem -- transforming it into a line integral?

  2. jcsd
  3. Jan 6, 2005 #2


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    [tex] \oint\oint_{\partial \Omega} dS=\oint\oint_{\partial\Omega} \vec{n}\cdot \vec{n} dS=\int\int\int_{\Omega} (\nabla\cdot \vec{n}) dV=\int\int\int_{\Omega} 0 dV=0 [/tex]

    Okay??I made use of the fact that
    [tex] \vec{n}\cdot\vec{n}=n^{2}\cos 0=n^{2}=1 [/tex]
    ,as unitvectors of the exterior normal to the surface.
    Because this unit vector is constant (the director cosines are constants),its flux is zero,because its divergence is zero.

  4. Jan 6, 2005 #3
    Thanks. Actually there's a slight problem: I forgot to tell you that the dS is a vector in my problem. Yours is a scalar. I'm not sure what the difference here is between integrating a scalar and integrating a vector. I don't think the first equality holds anymore for vector dS. Sorry for the confusion. Thanks for your help.

    (Show that: [tex] \oint_{S} d\vec{S}=0 [/tex])
    Last edited: Jan 6, 2005
  5. Jan 7, 2005 #4


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    Then perhaps it would be a good idea to tell us what the problem really is! You can integrate dS alone (getting surface area) but you can't integrate [itex]\vec{dS}[/itex] alone over a surface: you integrate the its dot product with some vector function.

    In dextercioby's [itex]\oint\oint_{\partial\Omega} \vec{n}\cdot \vec{n} dS[/itex],
    [itex]\vec{n}dS[/itex] IS the vector [itex]\vec{dS}[/itex].
  6. Jan 7, 2005 #5


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    It seems to me that you want the VECTOR result:
    This is achieved as follows:
    So that we have:
    since the unit vectors [tex]\vec{i},\vec{j},\vec{k}[/tex] are constants you may take out of the integral.

    Use the normal form of the divergence theorem to get your result.
  7. Jan 7, 2005 #6
    Thanks. I was looking for the vector result, but it was also helpful to know how the scalar result is proved. I wasn't paying attention to dS as a vector or a scalar when I was first posting it, so sorry once again for the confusion.
  8. Jan 7, 2005 #7


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    Glad to be of assistance; welcome to PF!
  9. Mar 17, 2010 #8
    how do you prove this???

    Prove that [tex]\int\int_{S} n dS = 0 [/tex]

    for any closed surface S.
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