(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let's define the radial vector [tex]\vec{v}(r) = \hat{r}/r^{2}[/tex] where [tex]\vec{r} = \vec{OP}[/tex] (O being the origin of our coordinate system and P being our observation point at point (x, y, z)). Using spherical coordinates, demonstrate that [tex]\vec{\nabla

} \cdot\vec{v}(r) = 0[/tex] everywhere except at r = 0. At r = 0, demonstrate that [tex]\vec{\nabla}\cdot\vec{v}(r)[/tex] is going to infinity.

I can show the first part but I can't show how it goes to infinity when r = 0.

2. Relevant equations

[tex]\vec{\nabla}\cdot\vec{v} = \frac{1}{r^{2}}\frac{\partial(r^{2} v_{r})}{\partial r} + \frac{1}{r sin\theta}\frac{\partial}{\partial\theta}(sin\theta v_{\theta}) + \frac{1}{r sin\theta}\frac{\partial v_{\phi}}{\partial\phi} [/tex]

3. The attempt at a solution

For the first part, because [tex]\vec{v}[/tex] is only a function of r the two last terms in the divergence formula will equal zero. The first term becomes

[tex]\vec{\nabla}\cdot\vec{v}(r) = \frac{1}{r^{2}}\frac{\partial}{\partial r}(r^{2}\cdot\frac{1}{r^{2}}) = 0[/tex]

zero as well, satisfying part a.

For part b, if we let r = 0 then [tex]\vec{v}(r)[/tex] becomes infinitely large and I am not sure how to take the divergence of that.

Any help will be appreciated.

Thanks in advance,

KEØM

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Divergence of a vector function

**Physics Forums | Science Articles, Homework Help, Discussion**