A time dependent point charge $q\left ( t \right )$ at the origin $\rho\left ( \vec{r},t \right )=q\left ( t \right )\delta ^{3}\left ( \vec{r} \right ) $, is fed by a current $\vec{J}\left ( \vec{r},t \right )=-\left ( \frac{1}{4 \pi} \right )\left ( \frac{\dot{q}}{r^{2}} \right )\hat{r}$ where $\dot{q}=\frac{\mathrm{dq} }{\mathrm{d} t}$
I want to determine that $\bigtriangledown \cdot \vec{J}=\frac{\partial \rho}{\partial t}$ but the integrals are giving me a terribly hard time.
Here's my attempt: $\vec{\bigtriangledown} \cdot \vec{J}=0$
$\frac{\partial \rho}{\partial t}$, as I've worked, is $\dot{q}\delta ^{3}\left ( \vec{r} \right )$
but it doesn't seem to satisfy the continuity equation.
Any help is appreciated.
Thanks in advance.
The divergence of $\hat{r}/r^2$ is not zero, in fact it is equal to $4\pi\delta^{(3)}(\vec{r})$. This fact can be found in the early chapters of most electrodynamics texts.