VC3.31

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\nabla ^{2}u=[{\frac  {\partial ^{2}}{\partial x^{2}}}+{\frac  {\partial ^{2}}{\partial y^{2}}}+{\frac  {\partial ^{2}}{\partial z^{2}}}]u\,

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

Now {\frac  {\partial u}{\partial x}}={\frac  {\partial }{\partial x}}(ax^{2}+by^{2}+cz^{2})=2ax\,

{\frac  {\partial ^{2}u}{\partial x^{2}}}={\frac  {\partial }{\partial x}}(2ax)=2a\,

Similarly,{\frac  {\partial ^{2}u}{\partial y^{2}}}=2b,{\frac  {\partial ^{2}u}{\partial z^{2}}}=2c\,

So,(1)=\nabla ^{2}u=2a+2b+2c=2(a+b+c)\,

Hence \nabla ^{2}u=0\, provided a+b+c=0\,

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