VC3.30

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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}}}]\,

Now,{\frac  {\partial u}{\partial x}}={\frac  {\partial }{\partial x}}(x^{2}-y^{2}+4z)=2x\, implies {\frac  {\partial ^{2}u}{\partial x^{2}}}=2\,

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

{\frac  {\partial u}{\partial z}}={\frac  {\partial }{\partial z}}(x^{2}-y^{2}+4z)=4\, implies {\frac  {\partial ^{2}u}{\partial x^{2}}}=0\,

Therefore,\nabla ^{2}u=2-2+0=0\,

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