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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