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

${\displaystyle \frac{{\partial }^2\psi }{\partial y^2}}$ ${\displaystyle =\frac{{\partial }^2\psi }{\partial r^2}{\left(\frac{\partial r}{\partial y}\right)}^2+}$ ${\displaystyle \frac{{\partial }^2\psi }{\partial \theta \partial r}\frac{\partial \theta }{\partial y}\frac{\partial r}{\partial y}}$${\displaystyle +\frac{\partial \psi }{\partial r}\frac{{\partial }^{2}r}{\partial y^2}}$${\displaystyle +\frac{{\partial }^2\psi }{\partial r\partial \theta }\frac{\partial r}{\partial y}\frac{\partial \theta }{\partial y}}$${\displaystyle +\frac{{\partial }^2\psi }{\partial {\theta }^2}{\left(\frac{\partial \theta }{\partial y}\right)}^2}$${\displaystyle +\frac{\partial \psi }{\partial \theta }\frac{{\partial }^2\theta }{\partial y^2}}$

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

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

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

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

${\displaystyle =-\frac{\cos \theta \sin \phi }{r^2}\frac{\partial r}{\partial y}-\frac{\sin \theta \sin \phi }{r}\frac{\partial \theta }{\partial y}-\frac{\cos \theta \cos \phi }{r}\frac{\partial \phi }{\partial y}}$

(13)､(15)､(14)を代入して

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

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

${\displaystyle \frac{{\partial }^2\phi }{\partial y^2}}$${\displaystyle =\frac{\partial }{\partial y}\left(\frac{\partial \phi }{\partial y}\right)}$

(3)を代入して

${\displaystyle =\frac{\partial }{\partial y}\left(\frac{\cos \phi }{r\sin \theta }\right)}$

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

${\displaystyle =-\frac{\cos \phi }{r^2\sin \theta }\frac{\partial r}{\partial y}-\frac{\cos \theta \cos \phi }{r{\sin }^2\theta }\frac{\partial \theta }{\partial y}-\frac{\sin \phi }{r\sin \theta }\frac{\partial \phi }{\partial y}}$

(2)，(4)，(3)を代入して

${\displaystyle =-\frac{\cos \phi }{r^2\sin \theta }\sin \theta \sin \phi -\frac{\cos \theta \cos \phi }{r{\sin }^2\theta }\frac{\cos \theta \sin \phi }{r}-\frac{\sin \phi }{r\sin \theta }\frac{\cos \phi }{r\sin \theta }}$

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

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

${\displaystyle =-\frac{2\sin \phi \cos \phi }{r^2{\sin }^2\theta }}$　･･････(7)

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

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

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

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

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

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