# What is the divergence of a unit vector not in the r direction?

1. Feb 1, 2013

### SiggyYo

Hi guys,

I've run across a problem. In finding the potential energy between two electrical quadrupoles, I've come across the expression for the energy as follows:

$U_{Q}=\frac{3Q_{0}}{4r^{4}}\left[(\hat{k}\cdot \nabla)(5(\hat{k}\cdot \hat{r})^3-2(\hat{k}\cdot \hat{r})^2-(\hat{k}\cdot \hat{r}))\right],$

where $\hat{k}$ is the orientation of the quadrupoles, and $\hat{r}$ is the direction between the quadrupoles.

If I let $\hat{r}$ be in the $\hat{z}$-direction, I get

$U_{Q}=\frac{3Q_{0}}{4r^{4}}\left[(\hat{k}\cdot \nabla)(5(\cos{\theta})^3-2(\cos{\theta})^2-(\cos{\theta}))\right].$

My problem now is, that I don't know what to do about the divergence of the $\hat{k}$-vector. I would like to do the differentiation in cartesian coordinates, but have them translated into spherical polar coordinates. I know, that the result should probably involve a $\frac{1}{r}$-factor, but I can't seem to do it right. I've tried to rewrite $\hat{k}$ in polar coordinates and tried using the chain rule on the derivative, but I get 3 as an answer. So I don't know if the initial expression is wrong, or I just dont know how to take the derivative. Can anyone please help?

Thanks,

2. Feb 1, 2013

### chiro

Hey SiggyYo.

Have you tried representing a transformation between spherical and cartesian?

(Example: for (r,theta) -> (x,y) we have y/x = arctan(theta) and x^2 + y^2 = r^2 which can be used to get (x,y)).

3. Feb 2, 2013

### SiggyYo

Thank you chiro for the quick response.

I am afraid I don't know what you mean. Wouldn't I just obtain the usual
$x=r\sin{\theta}\cos{\phi}$
$y=r\sin{\theta}\sin{\phi}$
$z=r\cos{\theta}$?

Also, I want $\hat{k}$ to be a unit vector, which gives me $r=1$. How do I take this into account, when trying to get a result with a factor of $\frac{1}{r}$? I am really lost on this one :P

4. Feb 2, 2013

### chiro

If k is a unit vector, then I don't think you will have any extra terms.

I'm not really sure what you are doing or trying to say: you have a conversion from polar to R^3 and provided the formula is correct, you should be able to plug these definitions in.

Also is the r term in your equation related to some vector in polar or is it some other variable?