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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:

[itex]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],[/itex]

where [itex]\hat{k}[/itex] is the orientation of the quadrupoles, and [itex]\hat{r}[/itex] is the direction between the quadrupoles.

If I let [itex]\hat{r}[/itex] be in the [itex]\hat{z}[/itex]-direction, I get

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

My problem now is, that I don't know what to do about the divergence of the [itex]\hat{k}[/itex]-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 [itex]\frac{1}{r}[/itex]-factor, but I can't seem to do it right. I've tried to rewrite [itex]\hat{k}[/itex] 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,

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

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