VC3.14

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\nabla \cdot (a\times r)r^{n}=\sum i\cdot {\frac  {\partial }{\partial x}}{(a\times r)r^{n}}\,,by definition of convergence.

=\sum i\cdot [{{\frac  {\partial }{\partial x}}(a\times r)}r^{n}+(a\times r){\frac  {\partial r^{n}}{\partial x}}]\,

=\sum i\cdot {({\frac  {\partial a}{\partial x}}\times r+a\times {\frac  {\partial r}{\partial x}})r^{n}+(a\times r)nr^{{n-1}}{\frac  {\partial r}{\partial x}}}\,

=\sum i\cdot [(a\times r)r^{n}+(a\times r)nr^{{n-1}}({\frac  {x}{r}})]\, [Since a is a constant vector,{\frac  {\partial a}{\partial x}}=0\,,Again r=xi+yj+zk\, and r^{2}=x^{2}+y^{2}+z^{2}\,,then {\frac  {\partial r}{\partial x}}=i,{\frac  {\partial r}{\partial x}}={\frac  {x}{r}}\,]

=\sum i\cdot [(a\times i)r^{n}+nxr^{{n-2}}(a\times r)]\,

=\sum [(i\cdot a\times i)r^{n}+nr^{{n-2}}xi\cdot (a\times r)]\,

=\sum nr^{{n-2}}(a\times r)xi\cdot (a\times r)\, [As i\cdot (a\times i)=[iai]=0\,]

=nr^{{n-2}}(a\times r).(\sum xi)=nr^{{n-2}}(a\times r)\cdot r=0\, [As

(a\times r)\cdot r=[arr]=0\,]

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