$$ \begin{aligned} x &= r \sin \theta \cos \phi\\ y &= r \sin \theta \sin \phi\\ z &= r \cos \theta \end{aligned} $$
$$ \begin{aligned} \begin{pmatrix} \frac{\partial}{\partial r} \\ \frac{\partial}{\partial \theta} \\ \frac{\partial}{\partial \phi} \end{pmatrix} &= \begin{pmatrix} \frac{\partial x}{\partial r} & \frac{\partial y}{\partial r} & \frac{\partial z}{\partial r}\\ \frac{\partial x}{\partial \theta} & \frac{\partial y}{\partial \theta} & \frac{\partial z}{\partial \theta}\\ \frac{\partial x}{\partial \phi} & \frac{\partial y}{\partial \phi} & \frac{\partial z}{\partial \phi} \end{pmatrix} \begin{pmatrix} \frac{\partial}{\partial x} \\ \frac{\partial}{\partial y} \\ \frac{\partial}{\partial z} \end{pmatrix}\\
&= \begin{pmatrix} \sin \theta \cos\phi & \sin \theta \sin \phi & \cos \theta \\ r \cos\theta \cos \phi & r \cos \theta \sin\phi & -r \sin\theta \\ -r \sin \theta \sin \phi & r \sin \theta \cos \phi & 0 \end{pmatrix} \begin{pmatrix} \frac{\partial}{\partial r} \\ \frac{\partial}{\partial \theta} \\ \frac{\partial}{\partial \phi} \end{pmatrix} \end{aligned} $$
余因子行列を行列式で割る方法で逆行列を求める。行列式は:
$$ \cos \theta (r^2 \cos \theta \sin \theta) - r \sin\theta (-r \sin^2 \theta) = r^2 \sin \theta $$
転置前の余因子行列（$(i, j)$成分が$(i, j)$余因子）は:
$$ \begin{pmatrix} r^2 \sin^2 \theta \cos \phi & r^2 \sin^2 \theta \sin \phi & r^2 \cos \theta \sin \theta \\ r \sin \theta \cos \theta \cos \phi & r \sin \theta \cos \theta \sin \phi & -r \sin^2 \theta \\ -r \sin \phi & r \cos \phi & 0 \end{pmatrix} $$
よって、逆行列は:
$$ \begin{pmatrix} \frac{\partial}{\partial x} \\ \frac{\partial}{\partial y} \\ \frac{\partial}{\partial z} \end{pmatrix} = \begin{pmatrix} \sin \theta \cos \phi & \frac{\cos \theta \cos \phi}{r} & -\frac{\sin \phi}{r \sin \theta} \\ \sin \theta \sin \phi & \frac{\cos \theta \sin \phi}{r} & \frac{\cos \phi}{r \sin \theta} \\ \cos\theta& -\frac{\sin \theta}{r} & 0 \end{pmatrix} \begin{pmatrix} \frac{\partial}{\partial r} \\ \frac{\partial}{\partial \theta} \\ \frac{\partial}{\partial \phi} \end{pmatrix} $$
ラプラシアンを計算する:
$$ \begin{aligned} \mathbf{\nabla}^2 = \left(\frac{\partial}{\partial x}\right)^2 + \left(\frac{\partial}{\partial y}\right)^2 + \left(\frac{\partial}{\partial z}\right)^2 \end{aligned} $$
$$ \left(\frac{\partial}{\partial x}\right)^2 = \left( \sin \theta \cos \phi \frac{\partial}{\partial r}
= \sin \theta \cos \phi \left( \sin \theta \cos \phi \frac{\partial^2}{\partial^2 r} + \cos \theta \cos \phi \frac{\partial}{\partial r} \left( \frac{1}{r} \frac{\partial}{\partial \theta} \right) - \frac{\sin \phi}{\sin \theta} \frac{\partial}{\partial r} \left( \frac{1}{r} \frac{\partial}{\partial \phi} \right)\right)\\
$$ \left(\frac{\partial}{\partial y}\right)^2 = \left( \sin \theta \sin\phi \frac{\partial}{\partial r}
= \sin \theta \sin \phi \left( \sin \theta \sin \phi \frac{\partial^2}{\partial^2 r} + \cos \theta \sin \phi \frac{\partial}{\partial r} \left( \frac{1}{r} \frac{\partial}{\partial \theta} \right) + \frac{\cos \phi}{\sin \theta} \frac{\partial}{\partial r} \left( \frac{1}{r} \frac{\partial}{\partial \phi} \right)\right)\\
\frac{\cos \theta \sin \phi}{r} \left( \sin \phi \frac{\partial}{\partial \theta} \left( \sin \theta \frac{\partial}{\partial r} \right) + \frac{\sin \phi}{r} \frac{\partial}{\partial \theta} \left( \cos \theta \frac{\partial}{\partial \theta} \right) + \frac{\cos \phi}{r} \frac{\partial}{\partial \theta} \left( \frac{1}{\sin \theta} \frac{\partial}{\partial \phi} \right) \right)\\
\frac{\cos \phi}{r \sin \theta} \left( \sin \theta \frac{\partial}{\partial \phi} \left( \sin \phi \frac{\partial}{\partial r} \right) + \frac{\cos \theta}{r} \frac{\partial}{\partial \phi} \left( \sin \phi \frac{\partial}{\partial \theta} \right) + \frac{1}{r \sin \theta} \frac{\partial}{\partial \phi} \left( \cos \phi \frac{\partial}{\partial \phi} \right) \right) $$
$$ \left(\frac{\partial}{\partial x}\right)^2 + \left(\frac{\partial}{\partial y}\right)^2 \\
= \sin^2 \theta \frac{\partial^2}{\partial^2 r} + \sin \theta \cos \theta \frac{\partial}{\partial r} \left( \frac{1}{r} \frac{\partial}{\partial \theta} \right)\\
\frac{\sin \phi}{r} \frac{\partial}{\partial \phi} \left( \cos \phi \frac{\partial}{\partial r} \right) + \frac{\cos \phi}{r} \frac{\partial}{\partial \phi} \left( \sin \phi \frac{\partial}{\partial r} \right) \\
\frac{\cos \theta \sin \phi}{r^2 \sin \theta} \frac{\partial}{\partial \phi} \left( \cos \phi \frac{\partial}{\partial \theta} \right) + \frac{\cos \theta \cos \phi}{r^2 \sin \theta} \frac{\partial}{\partial \phi} \left( \sin \phi \frac{\partial}{\partial \theta} \right) \\
最後の6項は:
$$
\frac{1}{r} \frac{\partial}{\partial r}
よって:
$$ \left(\frac{\partial}{\partial x}\right)^2 + \left(\frac{\partial}{\partial y}\right)^2 \\
= \sin^2 \theta \frac{\partial^2}{\partial^2 r} + \sin \theta \cos \theta \frac{\partial}{\partial r} \left( \frac{1}{r} \frac{\partial}{\partial \theta} \right) \\
\frac{\cos\theta}{r} \frac{\partial}{\partial \theta} \left( \sin \theta \frac{\partial}{\partial r} \right) + \frac{\cos \theta}{r^2} \frac{\partial}{\partial \theta} \left( \cos \theta \frac{\partial}{\partial \theta} \right)\\
\frac{1}{r} \frac{\partial}{\partial r}
\frac{\cos \theta}{r^2 \sin \theta} \frac{\partial}{\partial \theta}
\frac{1}{r^2 \sin^2 \theta} \frac{\partial^2}{\partial^2 \phi} $$
$$ \left(\frac{\partial}{\partial z}\right)^2 = \left( \cos \theta \frac{\partial}{\partial r}
= \cos^2 \theta \frac{\partial^2}{\partial^2 r}
\sin \theta \cos \theta \frac{\partial}{\partial r} \left( \frac{1}{r} \frac{\partial}{\partial \theta} \right)\\
\frac{\sin \theta}{r} \frac{\partial}{\partial \theta} \left( \cos \theta \frac{\partial}{\partial r} \right)
上記2式より: