Find the $\nabla(\nabla \cdot \vec{F})$ of $\vec{F}(x,y,z)=\frac{\vec{r}}{r}$, where $\vec{r}=(x,y,z)$ and $r=\sqrt{x^2+y^2+z^2}$
I've done quite easily $\nabla\cdot \vec{F}$: $$\nabla\cdot \vec{F}=\frac{\partial F_1}{\partial x}+\frac{\partial F_2}{\partial y}+\frac{\partial F_3}{\partial z}=\dots=\frac{2}{r}$$ but then what do I do to find the nabla operator on a scalar $\nabla(\frac{2}{r})$? I've never even thought that the nabla could be used on an scalar, so I'm pretty lost there
$\nabla(\frac{2}{r})$ is the gradient of $\frac{2}{r}$
$$\nabla\left(\frac{2}{r}\right)=\left(\frac{\partial}{\partial x} \frac{2}{r}, ~~\frac{\partial}{\partial y} \frac{2}{r}, ~~\frac{\partial}{\partial z} \frac{2}{r}\right)$$
For example,
$$\frac{\partial}{\partial x} \frac{2}{r}=-\frac{2}{r^2}\frac{\partial r}{\partial x}=-\frac{2}{r^2}\cdot\frac{x}{r}=-\frac{2x}{r^3}$$