Originally Posted by

**Ackbeet** Hang on a bit, I think I made a mistake earlier. $\displaystyle r$ definitely depends on $\displaystyle x_{i},$ so the partial derivative can't go past that.

Ok, let me make the notation a little clearer:

$\displaystyle \nabla\cdot\mathbf{u}=\dfrac{\partial}{\partial x_{i}}\,u_{i}=\dfrac{\partial}{\partial x_{i}}\,\epsilon_{ijk}\Phi_{j}\dfrac{x_{k}}{r}=\ep silon_{ijk}\Phi_{j}\,\dfrac{\partial}{\partial x_{i}}\,\dfrac{x_{k}}{r}=$ (using the quotient rule)

$\displaystyle \epsilon_{ijk}\Phi_{j}\dfrac{(r)\left(\dfrac{\part ial x_{k}}{\partial x_{i}}\right)-(x_{k})\left(\dfrac{\partial r}{\partial x_{i}}\right)}{r^{2}}.$

Now if $\displaystyle i=k,$ the Levi-Civita symbol cancels out the whole expression. So no need to examine that case. Assume $\displaystyle i\not=k.$ Then how does this expression simplify? Recall that

$\displaystyle r=\sqrt{r_{m}r_{m}}.$