VC3.28

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{\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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