VC3.28

From Exampleproblems

$\mathrm{grad}r^m=\sum i\frac{\partial r^m}{\partial x}\,$ by definition,

= $\sum imr^{m-1}\frac{\partial r}{\partial x}\,$

= $mr^{m-1}\sum i\frac{\partial r}{\partial x}\,$

= $mr^{m-1}[i\frac{x}{r}+j\frac{y}{r}+k\frac{z}{r}]\,$

Thus, $\mathrm{grad}r^m=mr^{m-2}\sum xi\,$ --(1)

Therefore, $\mathrm{div}[\mathrm{grad}r^m]=\mathrm{div}[mr^{m-2}{\sum xi}]\,$

= $\frac{\partial}{\partial x}(mr^{m-2}x)+\frac{\partial}{\partial y}(mr^{m-2}y)+\frac{\partial}{\partial z}(mr^{m-2}z)\,$

= $mr^{m-2}+mx(m-2)r^{m-3}\frac{\partial r}{\partial x}+mr^{m-2}+my(m-2)r^{m-3}\frac{\partial r}{\partial y}+mr^{m-2}+mz(m-2)r^{m-3}\frac{\partial r}{\partial z}\,$

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

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

= $3mr^{m-2}+m(m-2)r^{m-4}(x^2+y^2+z^2)=3mr^{m-2}+m(m-2)r^{m-4}r^2\,$

= $3mr^{m-2}+m(m-2)r^{m-2}=r^{m-2}(3m+m(m-2))=m(m+1)r^{m-2}\,$

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