${\displaystyle z}$による関数 ${\displaystyle \psi }$ の2階偏微分

${\displaystyle \frac{{\partial }^2\psi }{\partial z^2}}$ ${\displaystyle =\frac{{\partial }^2\psi }{\partial r^2}{\left(\frac{\partial r}{\partial z}\right)}^2}$${\displaystyle +\frac{{\partial }^2\psi }{\partial \theta \partial r}\frac{\partial \theta }{\partial z}\frac{\partial r}{\partial z}}$${\displaystyle +\frac{\partial \psi }{\partial r}\frac{{\partial }^{2}r}{\partial z^2}}$${\displaystyle +\frac{{\partial }^2\psi }{\partial r\partial \theta }\frac{\partial r}{\partial z}\frac{\partial \theta }{\partial z}}$${\displaystyle +\frac{{\partial }^2\psi }{\partial {\theta }^2}{\left(\frac{\partial \theta }{\partial z}\right)}^2+\frac{\partial \psi }{\partial \theta }\frac{{\partial }^2\theta }{\partial z^2}}$ ${\displaystyle +\frac{{\partial }^2\psi }{\partial r\partial \phi }\frac{\partial r}{\partial z}\frac{\partial \phi }{\partial z}}$${\displaystyle +\frac{{\partial }^2\psi }{\partial {\phi }^2}{\left(\frac{\partial \phi }{\partial z}\right)}^2}$${\displaystyle +\frac{\partial \psi }{\partial \phi }\frac{{\partial }^2\phi }{\partial z^2}}$　･･････(1)

${\displaystyle =\frac{\partial }{\partial r}\left(-\frac{\sin \theta }{r}\right)\frac{\partial r}{\partial z}+\frac{\partial }{\partial \theta }\left(-\frac{\sin \theta }{r}\right)\frac{\partial \theta }{\partial z}+\frac{\partial }{\partial \phi }\left(-\frac{\sin \theta }{r}\right)\frac{\partial \phi }{\partial z}}$

${\displaystyle \frac{{\partial }^2\psi }{\partial z^2}}$ ${\displaystyle =\frac{{\partial }^2\psi }{\partial r^2}{\left(\cos \theta \right)}^2+\frac{{\partial }^2\psi }{\partial \theta \partial r}{\displaystyle \left(-\frac{\sin \theta }{r}\right)\cos \theta +\frac{\partial \psi }{\partial r}}\frac{{\sin }^2\theta }{r}}$

${\displaystyle +\frac{{\partial }^2\psi }{\partial r\partial \theta }\cos \theta \left(-\frac{\sin \theta }{r}\right)+\frac{{\partial }^2\psi }{\partial {\theta }^2}{\left(-\frac{\sin \theta }{r}\right)}^2+\frac{\partial \psi }{\partial \theta }\frac{2\sin \theta \cos \theta }{r^2}}$

${\displaystyle +0+0+0}$

${\displaystyle =\frac{{\partial }^2\psi }{\partial r^2}{\cos }^2\theta -\frac{{\partial }^2\psi }{\partial \theta \partial r}{\displaystyle \frac{\sin \theta \cos \theta }{r}+\frac{\partial \psi }{\partial r}}\frac{{\sin }^2\theta }{r}}$

${\displaystyle -\frac{{\partial }^2\psi }{\partial r\partial \theta }\frac{\sin \theta \cos \theta }{r}+\frac{{\partial }^2\psi }{\partial {\theta }^2}\frac{{\sin }^2\theta }{r^2}+\frac{\partial \psi }{\partial \theta }\frac{2\sin \theta \cos \theta }{r^2}}$