VC3.27

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Here r^{2}=x^{2}+y^{2}+z^{2},{\frac  {\partial r}{\partial x}}={\frac  {x}{r}}\, --(1)

Now,\nabla ^{2}[{\frac  {x}{r^{3}}}]=\sum [{\frac  {\partial ^{2}}{\partial x^{2}}}{{\frac  {x}{r^{3}}}}]\, --(2)

But {\frac  {\partial ^{2}}{\partial x^{2}}}[{\frac  {x}{r^{3}}}]={\frac  {\partial }{\partial x}}[{\frac  {\partial }{\partial x}}{{\frac  {x}{r^{3}}}}]={\frac  {\partial }{\partial x}}[{\frac  {1}{r^{3}}}-{\frac  {3}{r^{4}}}{\frac  {\partial r}{\partial x}}]\,

={\frac  {\partial }{\partial x}}[{\frac  {1}{r^{3}}}-{\frac  {3}{r^{4}}}\cdot {\frac  {x}{r}}]\,,by using (1)

={\frac  {\partial }{\partial x}}[{\frac  {1}{r^{3}}}]-3{\frac  {\partial }{\partial x}}[{\frac  {x^{2}}{r^{5}}}]\,

=-{\frac  {3}{r^{4}}}{\frac  {\partial r}{\partial x}}-3[{\frac  {2x}{r^{5}}}-{\frac  {5x^{2}}{r^{6}}}{\frac  {\partial r}{\partial x}}]\,

=-{\frac  {3x}{r^{5}}}-3[{\frac  {2x}{r^{5}}}-{\frac  {5x^{3}}{r^{7}}}]=-{\frac  {9x}{r^{5}}}+{\frac  {15x^{3}}{r^{7}}}\, --(3)

Also,{\frac  {\partial ^{2}}{\partial y^{2}}}[{\frac  {x}{r^{3}}}]=x{\frac  {\partial }{\partial y}}[{\frac  {\partial }{\partial y}}{{\frac  {1}{r^{3}}}}]\,

=x{\frac  {\partial }{\partial y}}[-{\frac  {3}{r^{4}}}{\frac  {\partial r}{\partial y}}]=-3x{\frac  {\partial }{\partial y}}[{\frac  {y}{r^{5}}}]\, by (1)

=-3x[{\frac  {1}{r^{5}}}-{\frac  {5}{r^{6}}}{\frac  {\partial }{\partial y}}y]=-3x[{\frac  {1}{r^{5}}}-{\frac  {5y}{r^{6}}}\cdot {\frac  {y}{r}}]\,

=-3x[{\frac  {1}{r^{5}}}-{\frac  {5y^{2}}{r^{7}}}]=-{\frac  {3x}{r^{5}}}+{\frac  {15xy^{2}}{r^{7}}}\, --(4)

Similarly,{\frac  {\partial ^{2}}{\partial z^{2}}}[{\frac  {x}{r^{3}}}]=-{\frac  {3x}{r^{5}}}+{\frac  {15xz^{2}}{r^{7}}}\, --(5)

Using (3),(4),(5), Equation(2) reduces to

\nabla ^{2}[{\frac  {x}{r^{3}}}]=-{\frac  {9x}{r^{5}}}+{\frac  {15x^{3}}{r^{7}}}-{\frac  {3x}{r^{5}}}+{\frac  {15xy^{2}}{r^{7}}}-{\frac  {3x}{r^{5}}}+{\frac  {15xz^{2}}{r^{7}}}\,

=-{\frac  {15}{r^{5}}}+{\frac  {15x}{r^{7}}}(x^{2}+y^{2}+z^{2})=-{\frac  {15x}{r^{5}}}+{\frac  {15x}{r^{5}}}=0\,, which is to be proved.

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