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Thread: Vector Analysis

  1. #1
    Super Member Aryth's Avatar
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    Vector Analysis

    Evaluate

    $\displaystyle \int\int_S \left[\frac{1}{R}\nabla{\phi} - \phi\nabla{\left(\frac{1}{R}\right)}\right]\cdot d\bold{S}$

    over the surface of the sphere $\displaystyle (x-3)^2 + y^2 + z^2 = 25$, where $\displaystyle \phi = xyz + 5$
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  2. #2
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    I see a vector valued function dotted into the normal of a ball centered at (3,0,0). Can't you use the divergence theorem? So instead of evaluating
    $\displaystyle
    \vec f \cdot n dS
    $
    you instead evaluate
    $\displaystyle
    \nabla \cdot \vec f dV
    $
    It looks like the integrand was chosen so a couple of terms drop out. Then
    $\displaystyle
    \nabla \cdot \nabla \frac{1}{r}
    $
    gives you a constant because the origin is in your ball, and the other
    $\displaystyle
    \nabla \cdot \nabla (xyz+5)
    $
    should give you zero.
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