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Assume $$A^{2}=(x^{2}+y^{2})\cos^{2}\psi+z^{2}\cot^{2}\psi$$ which $A$ is constant. How we can show $\psi(x,y,z)$ satisfies the Laplacian equation $\psi_{xx}+\psi_{yy}+\psi_{zz}=0$ ($\operatorname{div}\nabla\psi=0$) without calculating $\psi(x,y,z)$? I calculate $\psi(x,y,z)$ itself and differentiate, but I'm looking for easier methods, It's not important to use what, only the time that it takes is important.

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Hmmm...WolframAlpha can't simplify the expression, but you could try implicit differentiation and Laplacian in the cylindrical coordinates. May I know the origin of this question? –  Shuhao Cao Jul 28 '13 at 3:17
    
@ShuhaoCao, about the origin of the question, I don't know. A physics student asked it from me and I had no idea so I put it here. –  AmirHosein SadeghiManesh Jul 28 '13 at 7:32
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