Checking the divergence theorem on a sphere

**BERMES39** · January 12th 2014, 05:51 AM

I have not been able to solve this problem. Can someone please help me find the error?

**ebaines** · January 14th 2014, 09:32 AM

I think your problem is that $\bigtriangledown \cdot r^2 \hat r$ is not 2r, but rather is 4r. In spherical coordinates you have:

$\bigtriangledown \cdot \vec v = \frac 1 {r^2} \frac {\partial (r^2 v_r)}{\partial r} + \frac 1 {r \sin \theta} \frac {\partial (v_\theta \sin \theta)}{\partial \theta} + \frac 1 {r \sin \theta} \frac {\partial v_\phi}{\partial \phi}$

For the case of $\vec v=r^2 \hat r$ this becomes:

$\bigtriangledown \cdot \vec v = \frac 1 {r^2} \frac {\partial (r^4)}{\partial r} = 4r$

Use this to determine $\int _{Vol} ( \bigtriangledown \cdot \vec v ) dV$ and it turns out to be $4 \pi R^4$ .

**BERMES39** · January 14th 2014, 10:17 AM

Thank you very much!

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