Have to find $\mathrm{div}$ and $\mathrm{curl}$ of $$ \dfrac{\vec{r}}{|\vec{r}|^3}$$
so the definition of curl and div are well known, but how to handle them in light of motion vector in the denominatorof the fraction? if it were constant I would just extract it out of general solution as $\dfrac{1}{r^3}$ and go on with the solution, the main problem is that: if I recall correctly there are no fraction rules either for curl or for div.
In index notation, we want to compute divergence and curl of $$ \frac{x^i}{|x|^3}\,. $$ Divergence $$ \partial_i \frac{x^i}{|x|^3} = \frac{d}{|x|^3}- 3\frac{x_i x^i}{|x|^5} = \frac{d-3}{|x|^3} $$ where $d$ is the number of components of $x$ i.e. the dimension of the space. The curl $$ \partial_i \frac{x_j}{|x|^3} - \partial_j \frac{x_i}{|x|^3} $$ is zero because $$ \frac{x_i}{|x|^3} = -\partial_i \frac{1}{|x|} $$ and partial derivatives commute. All these calculations apply only far away from the origin, where the vector field ceases to be well-defined.