VC3.34

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

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

Now,{\frac  {\partial }{\partial x}}{(x^{2}+y^{2}+z^{2})^{{{\frac  {-1}{2}}}}}=-{\frac  {1}{2}}(x^{2}+y^{2}+z^{2})^{{{\frac  {-3}{2}}}}\cdot 2x\, --(2)

Therefore,{\frac  {\partial ^{2}}{\partial x^{2}}}{(x^{2}+y^{2}+z^{2})^{{{\frac  {-1}{2}}}}}={\frac  {\partial }{\partial x}}[{\frac  {\partial }{\partial x}}{(x^{2}+y^{2}+z^{2})^{{{\frac  {-1}{2}}}}}]={\frac  {\partial }{\partial }}{-x(x^{2}+y^{2}+z^{2})^{{{\frac  {-3}{2}}}}}\,,using (2)

=(-1)(x^{2}+y^{2}+z^{2})^{{{\frac  {-3}{2}}}}+(-x)({\frac  {-3}{2}})(x^{2}+y^{2}+z^{2})^{{{\frac  {-5}{2}}}}\cdot (2x)\,

=(x^{2}+y^{2}+z^{2})^{{{\frac  {-5}{2}}}}[3x^{2}-(x^{2}+y^{2}+z^{2})]\,

=(2x^{2}-y^{2}-z^{2})(x^{2}+y^{2}+z^{2})^{{{\frac  {-5}{2}}}}\, --(3)

Proceeding similarly,we have

{\frac  {\partial ^{2}}{\partial y^{2}}}{(x^{2}+y^{2}+z^{2})^{{{\frac  {-1}{2}}}}}=(2y^{2}-x^{2}-z^{2})(x^{2}+y^{2}+z^{2})^{{{\frac  {-5}{2}}}}\, --(4)

And {\frac  {\partial ^{2}}{\partial z^{2}}}{(x^{2}+y^{2}+z^{2})^{{{\frac  {-1}{2}}}}}=(2z^{2}-x^{2}-y^{2})(x^{2}+y^{2}+z^{2})^{{{\frac  {-5}{2}}}}\, --(5)

Adding (3),(4),(5),we obtain

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

=(x^{2}+y^{2}+z^{2})^{{{\frac  {-5}{2}}}}[2x^{2}-y^{2}-z^{2}+2y^{2}-x^{2}-z^{2}+2z^{2}-x^{2}-y^{2}]=0\,

or \nabla ^{2}({\frac  {1}{r}})=0\, using (1)

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