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position vector
In the space $\mathbb{R}^{3}$, the vector
$\vec{r}\;:=\;(x,\,y,\,z)\;=\;x\vec{i}+y\vec{j}+z\vec{k}$ 
directed from the origin to a point $(x,\,y,\,z)$ is the position vector of this point. When the point is variable, $\vec{r}$ represents a vector field and its length
$r\;:=\;\sqrt{x^{2}\!+\!y^{2}\!+\!z^{2}}$ 
The simple formulae

$\nabla\!\cdot\vec{r}\;=\;3$

$\nabla\!\times\!\vec{r}\;=\;\vec{0}$

$\displaystyle\nabla r\;=\;\frac{\vec{r}}{r}\;=\;\vec{r}^{{\,0}}$

$\displaystyle\nabla\frac{1}{r}\;=\;\frac{\vec{r}}{r^{3}}\;=\;\frac{\vec{r}^{% {\,0}}}{r^{2}}$

$\displaystyle\nabla^{2}\frac{1}{r}\;=\;0$
are valid, where $\vec{r}^{{\,0}}$ is the unit vector having the direction of $\vec{r}$.
If $\vec{c}$ is a constant vector, $\vec{U}\!\!:\mathbb{R}^{3}\to\mathbb{R}^{3}$ a vector function and $f\!\!:\mathbb{R}\to\mathbb{R}$ is a twice differentiable function, then the formulae

$\nabla(\vec{c}\cdot\!\vec{r})\;\;=\vec{c}$

$\nabla\cdot(\vec{c}\times\vec{r})\;=\;0$

$(\vec{U}\!\cdot\!\nabla)\vec{r}\;=\;\vec{U}$

$(\vec{U}\!\times\!\nabla)\!\cdot\!\vec{r}\;=\;0$

$(\vec{U}\!\times\!\nabla)\!\times\!\vec{r}\;=\;2\vec{U}$

$\nabla f(r)\;=\;f^{{\prime}}(r)\,\vec{r}^{{\,0}}$

$\displaystyle\nabla^{2}f(r)\;=\;f^{{\prime\prime}}(r)\!+\frac{2}{r}f^{{\prime}% }(r)$
hold.
References
 1 K. Väisälä: Vektorianalyysi. Werner Söderström Osakeyhtiö, Helsinki (1961).
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Comments
Browsing in PM is very slow
Since I can to browse fine in another sites, I assume that issue is in PM. I can't look TeX source entries because of excesive delay.
Also I got either SERVER ERROR or the page can't be displayed.
Connection transfer=768Kbps.
perucho
Re: Browsing in PM is very slow
I have had the same problem. Many pages come back as "Possibly out of date" or "too busy to view", so I suspect for some reason there is an abnormal amount of traffice to the site?
Re: Browsing in PM is very slow
> I have had the same problem. Many pages come back as
> "Possibly out of date" or "too busy to view", so I suspect
> for some reason there is an abnormal amount of traffice to
> the site?
If you ever tried to run Noosphere (the PM software)
on your computer, you'd appreciate the reason:
running a TeX + ghostscript conversion process
for every[*] math image in an entry takes a lot of resources.
A page that is "possibly out of date" or "too busy to view"
means that the author has changed it recently, and
the page has yet to be regenerated
(TeX + ghostscript and all).
// Steve
Re: Browsing in PM is very slow
Yes Steve, that was my problem. Thank you!
Re: Browsing in PM is very slow
Good point. My appologies to anyone who thought I was complaining. I guess I was trying to understand if this was normal (I'm still a little new to the working of the site.) Your reasons make sense.
Does PlanetMath/Noosphere support some type of distributed computing? If so I would be happy to hear about how to join.
Re: Browsing in PM is very slow
Actually, the site should never be that slow. It got out of control since last time I looked at it. But normally, only a few rendering processes are allowed to happen concurrently, so the system should always be fast.
It turns out that this time an entry (one of yours in fact, SteveCommonFourierSeries), was locking up in the rendering process, and that was causing CPUeating latex instances to pile up (dozens of them, each of which would consume as much CPU as was available to it).
I cleaned up the damage and marked the entry as "bad" (so it wont be allowed to render in l2h mode) for now. Since I can't figure out what is wrong with it, I've contacted Ross Moore for help.
Things should be snappy again now.
apk
Re: Browsing in PM is very slow
Did my entry cause that problem before?
My last change to that article was simply to reupload
the lost EPS diagrams there because it was broken
(the cached HTML rendering of that entry expired, I guess)
// Steve
Re: Browsing in PM is very slow
Not that I know of. I'm not entirely clear of the timeline. The problem could purely be some interaction with the new server.
apk