# Gradient, divergence and curl - nabla dotted with f(r)r

• Nov 29th 2009, 11:44 AM
chella182
Gradient, divergence and curl - nabla dotted with f(r)r
Sorry for the naff title, couldn't think of anything else. I've ended with a ridiculous answer...

a) Find $\displaystyle \nabla\cdot\left(f(r)\mathbf{r}\right)$, where $\displaystyle r=|\mathbf{r}|$, $\displaystyle \mathbf{r}=(x,y,z)$ and $\displaystyle f$ is a differentiable function.

b) In Part (a), find $\displaystyle f(r)$ such that $\displaystyle \nabla\cdot\left(f(r)\mathbf{r}\right)=0$.

I ended up with something like $\displaystyle \frac{df}{dr}r+3f(r)$ for part a), which I highly doubt is right (Worried) thus I can't really attempt part b)... cheers in advance.
• Dec 1st 2009, 09:53 PM
qmech
So far so good
You're right on part a. Why don't you try this for part (b)?
$\displaystyle f(r) = \frac {1}{r^3}$

This will be very useful when you think about vector fields that go like $\displaystyle \frac {1}{r^2}$ such as electric fields.