VC3.38

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Note that in the xy-plane,r^{2}=x^{2}+y^{2}\, so that

{\frac  {\partial r}{\partial x}}={\frac  {x}{r}}\, --(1)

Also,for xy-plane \nabla ^{2}={\frac  {\partial ^{2}}{\partial x^{2}}}+{\frac  {\partial ^{2}}{\partial y^{2}}}\, --(2)

Since u=\log r\,,using (2),we have

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

Now,{\frac  {\partial }{\partial }}\log r={\frac  {1}{r}}{\frac  {\partial r}{\partial x}}={\frac  {1}{r}}{\frac  {x}{r}}\,,by using(1)

Therefore, {\frac  {\partial ^{2}}{\partial x^{2}}}\log r={\frac  {\partial }{\partial x}}[{\frac  {\partial }{\partial x}}[{\frac  {x}{r^{2}}}]={\frac  {1}{r^{2}}}-{\frac  {2x}{r^{4}}}\, --(4)

Similarly,{\frac  {\partial ^{2}}{\partial y^{2}}}\log r={\frac  {1}{r^{2}}}-{\frac  {2y^{2}}{r^{4}}}\, --(5)

Adding (4) and (5), we have

{\frac  {\partial ^{2}}{\partial x^{2}}}\log r+{\frac  {\partial ^{2}}{\partial y^{2}}}\log r={\frac  {2}{r^{2}}}-{\frac  {2}{r^{4}}}(x^{2}+y^{2})={\frac  {2}{r^{2}}}-{\frac  {2}{r^{4}}}(r^{2})={\frac  {2}{r^{2}}}-{\frac  {2}{r^{2}}}=0\,

or \nabla ^{2}u=0\, by using (3)

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