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Quick Divergence Theorem question

  1. Mar 11, 2009 #1
    1. The problem statement, all variables and given/known data
    Use the divergence theorem in three dimensions

    [tex]\int\int\int\nabla\bullet V d\tau= \int \int V \bullet n d \sigma[/tex]

    to evaluate the flux of the vector field

    V= (3x-2y)i + x4zj + (1-2z)k

    through the hemisphere bounded by the spherical surface x2+y2+z2=a2 (for z>0) and the x-y plane

    Hint: The direct evauation of the flux may not be the easiest way to proceed

    2. Relevant equations

    3. The attempt at a solution
    I found it pretty simple which means I probably messed up (and I'm not sure what the hint is talking about
    ok so the divergence is
    [tex]\nabla \bullet V = 3-2=1 [/tex]

    and the integral over the volume of the hemisphere (using spherical polar coordinates) is
    [tex] \int_{0}^a r^2 d \tau \int_{0}^{2\pi} d\phi \int_{0}^{\pi/2} sin\theta d\theta = \frac{2\pi a^3}{3} [/tex]

    So am I doing it completely wrong? I don't know the answer but if anyone could look through it and spot anything I would really appreciate it
  2. jcsd
  3. Mar 11, 2009 #2


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    Hi Stef42! :smile:

    (have a del: ∇ and a theta: θ and a phi: φ and a sigma: σ and use \cdot instead of \bullet :wink:)
    Looks fine to me …

    I think the hint just means don't use V.n, which is the flux, use ∇.V … though that's a bit unnecessary since they've already told you to :rolleyes:
  4. Mar 11, 2009 #3
    thanks for the speedy response tiny-tim :-)
    Ok so just to clear up:
    ∇.V= divergence of vector field V
    V.n= flux
    So then the sigma integral is flux density ?
    So what did I calculate? flux density aswell? wasn't I supposed to find the flux? :confused:
  5. Mar 11, 2009 #4


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    oooh, I get confused between flux and flux density …

    anyway, V.n is definitely fluxy, and ∇.V definitely isn't :wink:
  6. Mar 11, 2009 #5
    well, at least I'm not the only one who gets confused, damn vector calculus :frown:
    well from my lectures notes:
    "The divergence represents the flux density of the vector field and, because of the derivative operation, has an associated Fundamental Theorem of Calculus called the divergence theorem : (quotes equation)"

    hmm think I'll go talk to him tomorrow :)
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