Vector Analysis

**Aryth** · December 3rd 2009, 03:55 PM

Evaluate

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

over the surface of the sphere $(x-3)^2 + y^2 + z^2 = 25$ , where $\phi = xyz + 5$

**qmech** · December 10th 2009, 02:41 PM

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
$ \vec f \cdot n dS $
you instead evaluate
$ \nabla \cdot \vec f dV $
It looks like the integrand was chosen so a couple of terms drop out. Then
$ \nabla \cdot \nabla \frac{1}{r} $
gives you a constant because the origin is in your ball, and the other
$ \nabla \cdot \nabla (xyz+5) $
should give you zero.

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