VC3.24

Here $r=xi+yj+zk\,$ so that $r^2=|r|^2=x^2+y^2+z^2\,$ --(1)

Now, $r^3r=r^3(xi+yj+zk)=r^3xi+r^3yj+r^3zk\,$ -----(2)

Therefore, $\nabla\cdot(r^3r)=\frac{\partial}{\partial x}(r^3x)+\frac{\partial}{\partial y}(r^3y)+\frac{\partial}{\partial z}(r^3z)\,$

= $r^3+3r^2x\frac{\partial r}{\partial x}+r^3+3r^2y\frac{\partial r}{\partial y}+r^3+3r^2z\frac{\partial r}{\partial z}\,$

= $3r^3+3r^2[x\frac{\partial r}{\partial x}+y\frac{\partial r}{\partial y}+z\frac{\partial r}{\partial z}]\,$ -------(3)

From (1), $\frac{\partial r}{\partial x}=\frac{x}{r},\frac{\partial r}{\partial y}=\frac{y}{r},\frac{\partial r}{\partial z}=\frac{z}{r}\,$ --(4).

Using (4)and(5),(3) reduces to

$\nabla\cdot(r^3r)=3r^3+3r^2(\frac{x^2}{r}+\frac{y^2}{r}+\frac{z^2}{r})\,$

= $3r^3+3r(r^2)=6r^3\,$

Hence, $\nabla\cdot(r^3r)=6r^3\,$