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+nx r^{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)=[i a i]=0\,]

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

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

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