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Divergence theorem problem

  1. Nov 29, 2009 #1
    1. The problem statement, all variables and given/known data
    the problem is to calculate
    [tex] \int (\nabla \cdot \vec{F}) d\tau [/tex]
    over the region
    [tex] x^2 + y^2 + x^2 \leq 25 [/tex]
    where
    [tex] \vec{F} = (x^2 + y^2 + x^2)(x\hat{i} +y\hat{j} + z\hat{k}) [/tex]
    in the simplest manner possible.


    2. Relevant equations
    divergence theorem!


    3. The attempt at a solution
    Write
    [tex] \vec{F} = |\vec{r}|^2 \vec{r} = |\vec{r}|^3 \hat{r}, [/tex]
    so
    [tex] \vec{F} \cdot \hat{n} = \vec{F} \cdot \hat{r} = |\vec{r}|^3 \hat{r} \cdot \hat{r} = |\vec{r}|^3 = 125, [/tex]
    since
    [tex] \hat{n} = \hat{r} [/tex]
    and
    [tex] |\vec{r}| = 5 [/tex]
    along the surface of the sphere.
    Then, invoking the divergence theorem, we obtain:
    [tex] \int (\nabla \cdot \vec{F}) d\tau = \oint_{\partial{\tau}} \vec{F} \cdot \hat{n} d\sigma = \oint_{\partial{\tau}} 125 d\sigma = 125 \cdot 4 \cdot \pi \cdot 5^2 [/tex]

    the back of the book gives 100pi as the answer (and i've checked the errata for the book; no correction has been made). am i wrong? or is the book?
     
    Last edited: Nov 29, 2009
  2. jcsd
  3. Nov 29, 2009 #2

    Dick

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    Science Advisor
    Homework Helper

    It's also not very hard to integrate the divergence over the interior of the sphere. I get 4*pi*5^5. It sure looks to me like the book answer is wrong.
     
  4. Nov 29, 2009 #3
    Great, thanks much!
     
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