VC3.25

From Exampleproblems

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

Therefore, $\mathrm{grad}r^{-3}=\nabla r^{-3}=i\frac{\partial}{\partial x}(r^{-3})+j\frac{\partial}{\partial y}(r^{-3})+k\frac{\partial}{\partial z}(r^{-3})\,$

= $-3r^{-4}\frac{\partial r}{\partial x}i-3r^{-4}\frac{\partial r}{\partial y}j-3r^{-4}\frac{\partial r}{\partial z}k\,$ ----(2)

From (1), $\frac{\partial r}{\partial x}=\frac{x}{r},\frac{\partial r}{\partial y}=\frac{y}{r},\frac{\partial r}{\partial z}=\frac{z}{r}\,$ --(3)

Using (3),(2) implies $\mathrm{grad}r^{-3}=-\frac{3r^{-4}x}{r}i-\frac{3r^{-4}y}{r}j-\frac{3r^{-4}z}{r}k=-3r^{-5}(xi+yj+zk)\,$

Therefore, $r\mathrm{grad}r^{-3}=-3r^{-4}(xi+yj+zk)=-3r^{-4}i-3r^{-4}j-3r^{-4}k\,$

Now, $\mathrm{div}(r\mathrm{grad}r^{-3})=\nabla\cdot[r\nabla r^{-3}]\,$

= $\frac{\partial}{\partial x}(-3r^{-4}x)+\frac{\partial}{\partial x}(-3r^{-4}y)+\frac{\partial}{\partial z}(-3r^{-4}z)\,$

= $[12r^{-5}\frac{\partial r}{\partial x}x-3r^{-4}]+[12r^{-5}\frac{\partial r}{\partial y}y-3r^{-4}]+[12r^{-5}\frac{\partial r}{\partial z}z-3r^{-4}]\,$