VC3.24

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Here r=xi+yj+zk\, so that r^{2}=|r|^{2}=x^{2}+y^{2}+z^{2}\, --(1)

Now,r^{3}r=r^{3}(xi+yj+zk)=r^{3}xi+r^{3}yj+r^{3}zk\, -----(2)

Therefore,\nabla \cdot (r^{3}r)={\frac  {\partial }{\partial x}}(r^{3}x)+{\frac  {\partial }{\partial y}}(r^{3}y)+{\frac  {\partial }{\partial z}}(r^{3}z)\,

=r^{3}+3r^{2}x{\frac  {\partial r}{\partial x}}+r^{3}+3r^{2}y{\frac  {\partial r}{\partial y}}+r^{3}+3r^{2}z{\frac  {\partial r}{\partial z}}\,

=3r^{3}+3r^{2}[x{\frac  {\partial r}{\partial x}}+y{\frac  {\partial r}{\partial y}}+z{\frac  {\partial r}{\partial z}}]\, -------(3)

From (1),{\frac  {\partial r}{\partial x}}={\frac  {x}{r}},{\frac  {\partial r}{\partial y}}={\frac  {y}{r}},{\frac  {\partial r}{\partial z}}={\frac  {z}{r}}\, --(4).

Using (4)and(5),(3) reduces to

\nabla \cdot (r^{3}r)=3r^{3}+3r^{2}({\frac  {x^{2}}{r}}+{\frac  {y^{2}}{r}}+{\frac  {z^{2}}{r}})\,

=3r^{3}+3r(r^{2})=6r^{3}\,

Hence,\nabla \cdot (r^{3}r)=6r^{3}\,

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