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

**chella182** · November 29th 2009, 11:44 AM

Sorry for the naff title, couldn't think of anything else. I've ended with a ridiculous answer...

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

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

I ended up with something like $\frac{df}{dr}r+3f(r)$ for part a), which I highly doubt is right

thus I can't really attempt part b)... cheers in advance.

**qmech** · December 1st 2009, 09:53 PM

You're right on part a. Why don't you try this for part (b)?
$<br /> f(r) = \frac {1}{r^3}<br />$

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

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

So far so good

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