VC3.29

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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  {curl}}[{\mathrm  {grad}}r^{m}]={\begin{vmatrix}i&j&k\\{\frac  {\partial }{\partial x}}&{\frac  {\partial }{\partial y}}&{\frac  {\partial }{\partial z}}\\mr^{{m-2}}x&mr^{{m-2}}y&mr^{{m-2}}z\end{vmatrix}}\,

=\sum [i[{\frac  {\partial }{\partial y}}(mr^{{m-2}}z)-{\frac  {\partial }{\partial z}}(mr^{{m-2}}y)]]\,

=m(m-2)r^{{m-3}}\sum [i[z{\frac  {\partial r}{\partial y}}-y{\frac  {\partial r}{\partial z}}]]\,

=m(m-2)r^{{m-3}}\sum [i[z{\frac  {y}{r}}-y{\frac  {z}{r}}]]\,

=m(m-2)r^{{m-4}}\sum [i(zy-zy)]=i(0)+j(0)+k(0)=0\,

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