I know that
$$\nabla \dot{}\frac{\hat{r}}{r^2}=4\pi \delta(x)\delta(y)\delta(z) $$
However I thing this could work too
$$\nabla \dot{}\frac{\hat{r}}{r^2}=\frac{\delta(r)}{r^2}$$
If I integrate these expressions near the origin, both of them work. Is it correct or am I doing a mistake? Can $\nabla \dot{}\frac{\hat{r}}{r^2}=\frac{\delta(r)}{r^2}$ be used?