VC3.25

From Example Problems
Jump to: navigation, search

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

Therefore,{\mathrm  {grad}}r^{{-3}}=\nabla r^{{-3}}=i{\frac  {\partial }{\partial x}}(r^{{-3}})+j{\frac  {\partial }{\partial y}}(r^{{-3}})+k{\frac  {\partial }{\partial z}}(r^{{-3}})\,

=-3r^{{-4}}{\frac  {\partial r}{\partial x}}i-3r^{{-4}}{\frac  {\partial r}{\partial y}}j-3r^{{-4}}{\frac  {\partial r}{\partial z}}k\, ----(2)

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}}\, --(3)

Using (3),(2) implies {\mathrm  {grad}}r^{{-3}}=-{\frac  {3r^{{-4}}x}{r}}i-{\frac  {3r^{{-4}}y}{r}}j-{\frac  {3r^{{-4}}z}{r}}k=-3r^{{-5}}(xi+yj+zk)\,

Therefore,r{\mathrm  {grad}}r^{{-3}}=-3r^{{-4}}(xi+yj+zk)=-3r^{{-4}}i-3r^{{-4}}j-3r^{{-4}}k\,

Now,{\mathrm  {div}}(r{\mathrm  {grad}}r^{{-3}})=\nabla \cdot [r\nabla r^{{-3}}]\,

={\frac  {\partial }{\partial x}}(-3r^{{-4}}x)+{\frac  {\partial }{\partial x}}(-3r^{{-4}}y)+{\frac  {\partial }{\partial z}}(-3r^{{-4}}z)\,

=[12r^{{-5}}{\frac  {\partial r}{\partial x}}x-3r^{{-4}}]+[12r^{{-5}}{\frac  {\partial r}{\partial y}}y-3r^{{-4}}]+[12r^{{-5}}{\frac  {\partial r}{\partial z}}z-3r^{{-4}}]\,

={\frac  {12r^{{-5}}x^{2}}{r}}-3r^{{-4}}+{\frac  {12r^{{-5}}y^{2}}{r}}-3r^{{-4}}+{\frac  {12r^{{-5}}z^{2}}{r}}-3r^{{-4}}\,

=12r^{{-6}}(x^{2}+y^{2}+z^{2})-9r^{{-4}}=12r^{{-6}}r^{2}-9r^{{-4}}=3r^{{-4}}\,

Hence the required.

Main Page