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Homework Help: Evaluating a surface integral

  1. Dec 3, 2009 #1
    Find and evaluate numerically

    e6d56a2ad6bccb4e716ccd8157c62f94.jpg x^10 + y^10 + z^10 dS

    x^2 + y^2 + z^2 =4

    It says you're supposed to use gauss' divergence thm to convert surface integral to volume integral, then integrate volume integral by converting to spherical coordinates... I can do the second part but how do i use gauss' thm...? my prof was really bad at explaining this.

  2. jcsd
  3. Dec 3, 2009 #2


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    Science Advisor

    Well, I would think that the first thing you would do is look up "Gauss' theorem" (perhaps better known as the "divergence theorem"). According to Wikipedia, Gauss' theorem says that
    [tex]\int\int\int (\nabla\cdot \vec{F}) dV= \oint\int \vec{F}\cdot\vec{n}dS[/tex]
    where [itex]\vec{n}[/itex] is the normal vector to the surface at each point.

    Here, you are not given a vector function but, fortunately, Wikipedia also notes that "Applying the divergence theorem to the product of a scalar function, f, and a non-zero constant vector, the following theorem can be proven:
    [tex]\int\int\int \nabla f dV= \oint\int f dS[/tex]"

    So, since you are asked to use Gauss' theorem to evaluate a surface integral, you are intended to find [itex]\nabla f[/itex] and integrate that over the region- the ball of radius 2.

    Then- first step- what is [itex]\nabla (x^{10}+ y^{10}+ z^{10})[/itex]?
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