式展開: 8月 2024

original: https://www.youtube.com/watch?v=NEI-U0aF3nY

極座標ラプラシアンの導出

直交座標 $(x, y, z)$ と極座標 $(r, θ, ψ)$ の関係式

\begin{array}{r} {\begin{cases} x & = & r \sin θ \cos ψ \dots a \\ y & = & r \sin θ \sin ψ \dots b \\ z & = & r \cos θ \dots c \end{cases} \end{array}

\begin{array}{r} {\begin{cases} r^{2} & = & x^{2} + y^{2} + z^{2} \dots d \\ \cos θ & = & \frac{z}{r} \dots e \\ \tan ψ & = & \frac{y}{x} \dots f \end{cases} \end{array}

$\frac{\partial r}{\partial x}$ を求める

d

の両辺を

x

の偏微分をとる．

\begin{array}{rcl} \frac{\partial}{\partial x} r^{2} & = & \frac{\partial}{\partial x} (x^{2} + y^{2} + z^{2}) \\ 2 r \frac{\partial r}{\partial x} & = & 2 x \\ \frac{\partial r}{\partial x} & = & \frac{2 x}{2 r} = \frac{x}{r} = \sin θ \cos ψ \dots a よ り \end{array}

$\frac{\partial θ}{\partial x}$ を求める

e

の両辺を

x

の偏微分をとる．

\begin{array}{rcl} \frac{\partial}{\partial x} \cos θ & = & \frac{\partial}{\partial x} \frac{z}{r} \\ - \sin θ \frac{\partial θ}{\partial x} & = & z \frac{\partial}{\partial x} \frac{1}{r} \\ = & z \frac{\partial}{\partial r} (\frac{1}{r}) \frac{\partial r}{\partial x} \\ = & - \frac{z}{r^{2}} \frac{\partial r}{\partial x} \\ = & - \frac{z}{r^{2}} \sin θ \cos ψ \dots 一 つ 前 の \frac{\partial r}{\partial x} の 結 果 \\ \frac{\partial θ}{\partial x} & = & \frac{z \cos ψ}{r^{2}} = \frac{z}{r} \frac{\cos ψ}{r} \\ = & \frac{\cos θ \cos ψ}{r} \dots e よ り \end{array}

$\frac{\partial ψ}{\partial x}$ を求める

f

の両辺を

x

の偏微分をとる．

\begin{array}{rcl} \frac{\partial}{\partial x} \tan ψ & = & \frac{\partial}{\partial x} \frac{y}{x} \\ \frac{1}{\cos^{2} ψ} \frac{\partial ψ}{\partial x} & = & y \frac{\partial}{\partial x} \frac{1}{x} \\ = & y (- \frac{1}{x^{2}}) \\ = & - \frac{y}{x} \frac{1}{x} \\ = & - \tan ψ (\frac{1}{x}) \\ = & - \tan ψ (\frac{1}{r \sin θ \cos ψ}) \\ \frac{\partial ψ}{\partial x} & = & - \tan ψ (\frac{1}{r \sin θ \cos ψ}) \cos^{2} ψ \\ = & - \frac{\sin ψ}{\cos ψ} (\frac{1}{r \sin θ \cos ψ}) \cos^{2} ψ \\ = & - \frac{\sin ψ}{r \sin θ} \end{array}

$\frac{\partial r}{\partial y}$ を求める

d

の両辺を

y

の偏微分をとる． dの両辺をyの偏微分をとる．

\begin{array}{rcl} \frac{\partial}{\partial y} r^{2} & = & \frac{\partial}{\partial y} (x^{2} + y^{2} + z^{2}) \\ 2 r \frac{\partial r}{\partial y} & = & 2 y \\ \frac{\partial r}{\partial y} & = & \frac{2 y}{2 r} = \frac{y}{r} = \sin θ \sin ψ \dots b よ り \end{array}

$\frac{\partial θ}{\partial y}$ を求める

e

の両辺を

y

の偏微分をとる．

\begin{array}{rcl} \frac{\partial}{\partial y} \cos θ & = & \frac{\partial}{\partial y} \frac{z}{r} \\ - \sin θ \frac{\partial θ}{\partial y} & = & z \frac{\partial}{\partial y} \frac{1}{r} \\ = & z \frac{\partial}{\partial r} (\frac{1}{r}) \frac{\partial r}{\partial y} \\ = & - \frac{z}{r^{2}} \frac{\partial r}{\partial y} \\ = & - \frac{z}{r^{2}} \sin θ \sin ψ \dots 一 つ 前 の \frac{\partial r}{\partial y} の 結 果 \\ \frac{\partial θ}{\partial y} & = & (- \frac{1}{\sin θ}) (- \frac{z}{r^{2}} \sin θ \sin ψ) = \frac{z}{r} \frac{\sin ψ}{r} \\ = & \frac{\cos θ \sin ψ}{r} \dots e よ り \end{array}

$\frac{\partial ψ}{\partial y}$ を求める

f

の両辺を

y

の偏微分をとる．

\begin{array}{rcl} \frac{\partial}{\partial y} \tan ψ & = & \frac{\partial}{\partial y} \frac{y}{x} \\ \frac{1}{\cos^{2} ψ} \frac{\partial ψ}{\partial y} & = & \frac{1}{x} \\ \frac{\partial ψ}{\partial y} & = & (\frac{1}{x}) \cos^{2} ψ \\ = & (\frac{1}{r \sin θ \cos ψ}) \cos^{2} ψ \dots a よ り \\ = & \frac{\cos ψ}{r \sin θ} \end{array}

$\frac{\partial r}{\partial z}$ を求める

d

の両辺を

z

の偏微分をとる．

\begin{array}{rcl} \frac{\partial}{\partial z} r^{2} & = & \frac{\partial}{\partial z} (x^{2} + y^{2} + z^{2}) \\ 2 r \frac{\partial r}{\partial z} & = & 2 z \\ \frac{\partial r}{\partial z} & = & \frac{2 z}{2 r} = \frac{z}{r} = \cos θ \dots e よ り \end{array}

$\frac{\partial θ}{\partial z}$ を求める

e

の両辺を

z

の偏微分をとる．

\begin{array}{rcl} \frac{\partial}{\partial z} \cos θ & = & \frac{\partial}{\partial z} \frac{z}{r} \\ - \sin θ \frac{\partial θ}{\partial z} & = & \frac{1}{r} + z \frac{\partial}{\partial z} \frac{1}{r} \\ = & \frac{1}{r} + z \frac{\partial}{\partial r} (\frac{1}{r}) \frac{\partial r}{\partial z} \\ = & \frac{1}{r} - \frac{z}{r^{2}} \frac{\partial r}{\partial z} \\ = & \frac{1}{r} - \frac{z}{r^{2}} \cos θ \dots 一 つ 前 の \frac{\partial r}{\partial z} の 結 果 \\ \frac{\partial θ}{\partial y} & = & (- \frac{1}{\sin θ}) (\frac{1}{r} - \frac{z}{r^{2}} \cos θ) \\ = & - \frac{1}{r \sin θ} + \frac{z}{r} \frac{\cos θ}{r \sin θ} \\ = & - \frac{1}{r \sin θ} + \frac{\cos^{2} θ}{r \sin θ} \dots e よ り \\ = & - \frac{1 - \cos^{2} θ}{r \sin θ} \\ = & - \frac{\sin^{2} θ}{r \sin θ} \\ = & - \frac{\sin θ}{r} \end{array}

$\frac{\partial ψ}{\partial z}$ を求める

f

の両辺を

z

の偏微分をとる．

\begin{array}{rcl} \frac{\partial}{\partial z} \tan ψ & = & \frac{\partial}{\partial z} \frac{y}{x} \\ \frac{1}{\cos^{2} ψ} \frac{\partial ψ}{\partial z} & = & 0 \\ \frac{\partial ψ}{\partial y} & = & 0 \end{array}

$\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z}$ を求める

\begin{array}{rcl} \frac{\partial}{\partial x} & = & \frac{\partial r}{\partial x} \frac{\partial}{\partial r} + \frac{\partial θ}{\partial x} \frac{\partial}{\partial θ} + \frac{\partial ψ}{\partial x} \frac{\partial}{\partial ψ} \\ = & \sin θ \cos ψ \frac{\partial}{\partial r} + \frac{\cos θ \cos ψ}{r} \frac{\partial}{\partial θ} - \frac{\sin ψ}{r \sin θ} \frac{\partial}{\partial ψ} \\ \frac{\partial}{\partial y} & = & \frac{\partial r}{\partial y} \frac{\partial}{\partial r} + \frac{\partial θ}{\partial y} \frac{\partial}{\partial θ} + \frac{\partial ψ}{\partial y} \frac{\partial}{\partial ψ} \\ = & \sin θ \sin ψ \frac{\partial}{\partial r} + \frac{\cos θ \sin ψ}{r} \frac{\partial}{\partial θ} + \frac{\cos ψ}{r \sin θ} \frac{\partial}{\partial ψ} \\ \frac{\partial}{\partial z} & = & \frac{\partial r}{\partial z} \frac{\partial}{\partial r} + \frac{\partial θ}{\partial z} \frac{\partial}{\partial θ} + \frac{\partial ψ}{\partial z} \frac{\partial}{\partial ψ} \\ = & \cos θ \frac{\partial}{\partial r} - \frac{\sin θ}{r} \frac{\partial}{\partial θ} + 0 \frac{\partial}{\partial ψ} \\ = & \cos θ \frac{\partial}{\partial r} - \frac{\sin θ}{r} \frac{\partial}{\partial θ} \end{array}

$\frac{\partial^{2}}{\partial x^{2}}$ を求める

\begin{array}{rcl} \frac{\partial^{2}}{\partial x^{2}} & = & \frac{\partial}{\partial x} \frac{\partial}{\partial x} \\ = & (\sin θ \cos ψ \frac{\partial}{\partial r} + \frac{\cos θ \cos ψ}{r} \frac{\partial}{\partial θ} - \frac{\sin ψ}{r \sin θ} \frac{\partial}{\partial ψ}) (\sin θ \cos ψ \frac{\partial}{\partial r} + \frac{\cos θ \cos ψ}{r} \frac{\partial}{\partial θ} - \frac{\sin ψ}{r \sin θ} \frac{\partial}{\partial ψ}) \\ = & \sin θ \cos ψ \frac{\partial}{\partial r} (\sin θ \cos ψ \frac{\partial}{\partial r} + \frac{\cos θ \cos ψ}{r} \frac{\partial}{\partial θ} - \frac{\sin ψ}{r \sin θ} \frac{\partial}{\partial ψ}) \\ + \frac{\cos θ \cos ψ}{r} \frac{\partial}{\partial θ} (\sin θ \cos ψ \frac{\partial}{\partial r} + \frac{\cos θ \cos ψ}{r} \frac{\partial}{\partial θ} - \frac{\sin ψ}{r \sin θ} \frac{\partial}{\partial ψ}) \\ - \frac{\sin ψ}{r \sin θ} \frac{\partial}{\partial ψ} (\sin θ \cos ψ \frac{\partial}{\partial r} + \frac{\cos θ \cos ψ}{r} \frac{\partial}{\partial θ} - \frac{\sin ψ}{r \sin θ} \frac{\partial}{\partial ψ}) \\ = & \sin θ \cos ψ \cdot A_{x} + \frac{\cos θ \cos ψ}{r} \cdot B_{x} - \frac{\sin ψ}{r \sin θ} \cdot C_{x} \end{array}

\begin{array}{rcl} A_{x} & = & \frac{\partial}{\partial r} (\sin θ \cos ψ \frac{\partial}{\partial r}) + \frac{\partial}{\partial r} (\frac{\cos θ \cos ψ}{r} \frac{\partial}{\partial θ}) - \frac{\partial}{\partial r} (\frac{\sin ψ}{r \sin θ} \frac{\partial}{\partial ψ}) \\ = & \frac{\partial}{\partial r} (\sin θ \cos ψ) \cdot \frac{\partial}{\partial r} + \sin θ \cos ψ \cdot \frac{\partial}{\partial r} (\frac{\partial}{\partial r}) \\ + \frac{\partial}{\partial r} (\frac{\cos θ \cos ψ}{r}) \cdot \frac{\partial}{\partial θ} + \frac{\cos θ \cos ψ}{r} \cdot \frac{\partial}{\partial r} (\frac{\partial}{\partial θ}) \\ - \frac{\partial}{\partial r} (\frac{\sin ψ}{r \sin θ}) \cdot \frac{\partial}{\partial ψ} - \frac{\sin ψ}{r \sin θ} \cdot \frac{\partial}{\partial r} (\frac{\partial}{\partial ψ}) \\ = & (0) \frac{\partial}{\partial r} + \sin θ \cos ψ \frac{\partial^{2}}{\partial r^{2}} \\ + (- \frac{\cos θ \cos ψ}{r^{2}}) \frac{\partial}{\partial θ} + \frac{\cos θ \cos ψ}{r} \frac{\partial^{2}}{\partial r \partial θ} \\ - (- \frac{\sin ψ}{r^{2} \sin θ}) \cdot \frac{\partial}{\partial ψ} - \frac{\sin ψ}{r \sin θ} \cdot \frac{\partial}{\partial r \partial ψ} \\ = & \sin θ \cos ψ \frac{\partial^{2}}{\partial r^{2}} - \frac{\cos θ \cos ψ}{r^{2}} \frac{\partial}{\partial θ} + \frac{\cos θ \cos ψ}{r} \frac{\partial^{2}}{\partial r \partial θ} + \frac{\sin ψ}{r^{2} \sin θ} \frac{\partial}{\partial ψ} - \frac{\sin ψ}{r \sin θ} \frac{\partial^{2}}{\partial r \partial ψ} \end{array}

\begin{array}{rcl} \sin θ \cos ψ \cdot A_{x} & = & \sin θ \cos ψ {\sin θ \cos ψ \frac{\partial^{2}}{\partial r^{2}} - \frac{\cos θ \cos ψ}{r^{2}} \frac{\partial}{\partial θ} + \frac{\cos θ \cos ψ}{r} \frac{\partial^{2}}{\partial r \partial θ} + \frac{\sin ψ}{r^{2} \sin θ} \frac{\partial}{\partial ψ} - \frac{\sin ψ}{r \sin θ} \frac{\partial^{2}}{\partial r \partial ψ}} \\ = & \sin^{2} θ \cos^{2} ψ \frac{\partial^{2}}{\partial r^{2}} - \frac{\cos θ \sin θ \cos^{2} ψ}{r^{2}} \frac{\partial}{\partial θ} + \frac{\cos θ \sin θ \cos^{2} ψ}{r} \frac{\partial^{2}}{\partial r \partial θ} + \frac{\sin θ \cos ψ \sin ψ}{r^{2} \sin θ} \frac{\partial}{\partial ψ} - \frac{\sin θ \cos ψ \sin ψ}{r \sin θ} \frac{\partial^{2}}{\partial r \partial ψ} \\ = & \sin^{2} θ \cos^{2} ψ \frac{\partial^{2}}{\partial r^{2}} - \frac{\cos θ \sin θ \cos^{2} ψ}{r^{2}} \frac{\partial}{\partial θ} + \frac{\cos θ \sin θ \cos^{2} ψ}{r} \frac{\partial^{2}}{\partial r \partial θ} + \frac{\cos ψ \sin ψ}{r^{2}} \frac{\partial}{\partial ψ} - \frac{\cos ψ \sin ψ}{r} \frac{\partial^{2}}{\partial r \partial ψ} \end{array}

\begin{array}{rcl} B_{x} & = & \frac{\partial}{\partial θ} (\sin θ \cos ψ \frac{\partial}{\partial r}) + \frac{\partial}{\partial θ} (\frac{\cos θ \cos ψ}{r} \frac{\partial}{\partial θ}) - \frac{\partial}{\partial θ} (\frac{\sin ψ}{r \sin θ} \frac{\partial}{\partial ψ}) \\ = & \frac{\partial}{\partial θ} (\sin θ \cos ψ) \cdot \frac{\partial}{\partial r} + \sin θ \cos ψ \cdot \frac{\partial}{\partial θ} (\frac{\partial}{\partial r}) \\ + \frac{\partial}{\partial θ} (\frac{\cos θ \cos ψ}{r}) \cdot \frac{\partial}{\partial θ} + \frac{\cos θ \cos ψ}{r} \cdot \frac{\partial}{\partial θ} (\frac{\partial}{\partial θ}) \\ - \frac{\partial}{\partial θ} (\frac{\sin ψ}{r \sin θ}) \cdot \frac{\partial}{\partial ψ} - \frac{\sin ψ}{r \sin θ} \cdot \frac{\partial}{\partial θ} (\frac{\partial}{\partial ψ}) \\ = & (\cos θ \cos ψ) \frac{\partial}{\partial r} + \sin θ \cos ψ \frac{\partial^{2}}{\partial θ \partial r} - \frac{\sin θ \cos ψ}{r} \frac{\partial}{\partial θ} + \frac{\cos θ \cos ψ}{r} \frac{\partial^{2}}{\partial θ^{2}} + \frac{\cos θ \sin ψ}{r \sin^{2} θ} \frac{\partial}{\partial ψ} - \frac{\sin ψ}{r \sin θ} \frac{\partial^{2}}{\partial θ \partial ψ} \end{array}

\begin{array}{rcl} \frac{\cos θ \cos ψ}{r} \cdot B_{x} & = & \frac{\cos θ \cos ψ}{r} {(\cos θ \cos ψ) \frac{\partial}{\partial r} + \sin θ \cos ψ \frac{\partial^{2}}{\partial θ \partial r} - \frac{\sin θ \cos ψ}{r} \frac{\partial}{\partial θ} + \frac{\cos θ \cos ψ}{r} \frac{\partial^{2}}{\partial θ^{2}} + \frac{\cos θ \sin ψ}{r \sin^{2} θ} \frac{\partial}{\partial ψ} - \frac{\sin ψ}{r \sin θ} \frac{\partial^{2}}{\partial θ \partial ψ}} \\ = & \frac{\cos^{2} θ \cos^{2} ψ}{r} \frac{\partial}{\partial r} + \frac{\cos θ \sin θ \cos^{2} ψ}{r} \frac{\partial^{2}}{\partial θ \partial r} - \frac{\cos θ \sin θ \cos^{2} ψ}{r^{2}} \frac{\partial}{\partial θ} + \frac{\cos^{2} θ \cos^{2} ψ}{r^{2}} \frac{\partial^{2}}{\partial θ^{2}} + \frac{\cos^{2} θ \cos ψ \sin ψ}{r^{2} \sin^{2} θ} \frac{\partial}{\partial ψ} - \frac{\cos θ \cos ψ \sin ψ}{r^{2} \sin θ} \frac{\partial^{2}}{\partial θ \partial ψ} \end{array}

\begin{array}{rcl} C_{x} & = & \frac{\partial}{\partial ψ} (\sin θ \cos ψ \frac{\partial}{\partial r}) + \frac{\partial}{\partial ψ} (\frac{\cos θ \cos ψ}{r} \frac{\partial}{\partial θ}) - \frac{\partial}{\partial ψ} (\frac{\sin ψ}{r \sin θ} \frac{\partial}{\partial ψ}) \\ = & \frac{\partial}{\partial ψ} (\sin θ \cos ψ) \cdot \frac{\partial}{\partial r} + \sin θ \cos ψ \cdot \frac{\partial}{\partial ψ} (\frac{\partial}{\partial r}) \\ + \frac{\partial}{\partial ψ} (\frac{\cos θ \cos ψ}{r}) \cdot \frac{\partial}{\partial θ} + \frac{\cos θ \cos ψ}{r} \cdot \frac{\partial}{\partial ψ} (\frac{\partial}{\partial θ}) \\ - \frac{\partial}{\partial ψ} (\frac{\sin ψ}{r \sin θ}) \cdot \frac{\partial}{\partial ψ} - \frac{\sin ψ}{r \sin θ} \cdot \frac{\partial}{\partial ψ} (\frac{\partial}{\partial ψ}) \\ = & (- \sin θ \sin ψ) \frac{\partial}{\partial r} + \sin θ \cos ψ \frac{\partial^{2}}{\partial ψ \partial r} \\ + (- \frac{\cos θ \sin ψ}{r}) \frac{\partial}{\partial θ} + \frac{\cos θ \cos ψ}{r} \frac{\partial^{2}}{\partial ψ \partial θ} \\ - (\frac{\cos ψ}{r \sin θ}) \frac{\partial}{\partial ψ} - \frac{\sin ψ}{r \sin θ} \cdot \frac{\partial^{2}}{\partial ψ^{2}} \\ = & - \sin θ \sin ψ \frac{\partial}{\partial r} + \sin θ \cos ψ \frac{\partial^{2}}{\partial ψ \partial r} - \frac{\cos θ \sin ψ}{r} \frac{\partial}{\partial θ} + \frac{\cos θ \cos ψ}{r} \frac{\partial^{2}}{\partial ψ \partial θ} - \frac{\cos ψ}{r \sin θ} \frac{\partial}{\partial ψ} - \frac{\sin ψ}{r \sin θ} \frac{\partial^{2}}{\partial ψ^{2}} \end{array}

\begin{array}{rcl} - \frac{\sin ψ}{r \sin θ} \cdot C_{x} & = & - \frac{\sin ψ}{r \sin θ} {- \sin θ \sin ψ \frac{\partial}{\partial r} + \sin θ \cos ψ \frac{\partial^{2}}{\partial ψ \partial r} - \frac{\cos θ \sin ψ}{r} \frac{\partial}{\partial θ} + \frac{\cos θ \cos ψ}{r} \frac{\partial^{2}}{\partial ψ \partial θ} - \frac{\cos ψ}{r \sin θ} \frac{\partial}{\partial ψ} - \frac{\sin ψ}{r \sin θ} \frac{\partial^{2}}{\partial ψ^{2}}} \\ = & \frac{\sin θ \sin^{2} ψ}{r \sin θ} \frac{\partial}{\partial r} - \frac{\sin θ \cos ψ \sin ψ}{r \sin θ} \frac{\partial^{2}}{\partial ψ \partial r} + \frac{\cos θ \sin^{2} ψ}{r^{2} \sin θ} \frac{\partial}{\partial θ} - \frac{\cos θ \cos ψ \sin ψ}{r^{2} \sin θ} \frac{\partial^{2}}{\partial ψ \partial θ} + \frac{\cos ψ \sin ψ}{r^{2} \sin^{2} θ} \frac{\partial}{\partial ψ} + \frac{\sin^{2} ψ}{r^{2} \sin^{2} θ} \frac{\partial^{2}}{\partial ψ^{2}} \\ = & \frac{\sin^{2} ψ}{r} \frac{\partial}{\partial r} - \frac{\cos ψ \sin ψ}{r} \frac{\partial^{2}}{\partial ψ \partial r} + \frac{\cos θ \sin^{2} ψ}{r^{2} \sin θ} \frac{\partial}{\partial θ} - \frac{\cos θ \cos ψ \sin ψ}{r^{2} \sin θ} \frac{\partial^{2}}{\partial ψ \partial θ} + \frac{\cos ψ \sin ψ}{r^{2} \sin^{2} θ} \frac{\partial}{\partial ψ} + \frac{\sin^{2} ψ}{r^{2} \sin^{2} θ} \frac{\partial^{2}}{\partial ψ^{2}} \end{array}

\begin{array}{rcl} \frac{\partial^{2}}{\partial x^{2}} & = & \sin θ \cos ψ \cdot A_{x} + \frac{\cos θ \cos ψ}{r} \cdot B_{x} - \frac{\sin ψ}{r \sin θ} \cdot C_{x} \\ = & \sin^{2} θ \cos^{2} ψ \frac{\partial^{2}}{\partial r^{2}} - \frac{\cos θ \sin θ \cos^{2} ψ}{r^{2}} \frac{\partial}{\partial θ} + \frac{\cos θ \sin θ \cos^{2} ψ}{r} \frac{\partial^{2}}{\partial r \partial θ} + \frac{\cos ψ \sin ψ}{r^{2}} \frac{\partial}{\partial ψ} - \frac{\cos ψ \sin ψ}{r} \frac{\partial^{2}}{\partial r \partial ψ} \\ + \frac{\cos^{2} θ \cos^{2} ψ}{r} \frac{\partial}{\partial r} + \frac{\cos θ \sin θ \cos^{2} ψ}{r} \frac{\partial^{2}}{\partial θ \partial r} - \frac{\cos θ \sin θ \cos^{2} ψ}{r^{2}} \frac{\partial}{\partial θ} + \frac{\cos^{2} θ \cos^{2} ψ}{r^{2}} \frac{\partial^{2}}{\partial θ^{2}} + \frac{\cos^{2} θ \cos ψ \sin ψ}{r^{2} \sin^{2} θ} \frac{\partial}{\partial ψ} - \frac{\cos θ \cos ψ \sin ψ}{r^{2} \sin θ} \frac{\partial^{2}}{\partial θ \partial ψ} \\ + \frac{\sin^{2} ψ}{r} \frac{\partial}{\partial r} - \frac{\cos ψ \sin ψ}{r} \frac{\partial^{2}}{\partial ψ \partial r} + \frac{\cos θ \sin^{2} ψ}{r^{2} \sin θ} \frac{\partial}{\partial θ} - \frac{\cos θ \cos ψ \sin ψ}{r^{2} \sin θ} \frac{\partial^{2}}{\partial ψ \partial θ} + \frac{\cos ψ \sin ψ}{r^{2} \sin^{2} θ} \frac{\partial}{\partial ψ} + \frac{\sin^{2} ψ}{r^{2} \sin^{2} θ} \frac{\partial^{2}}{\partial ψ^{2}} \\ = & \sin^{2} θ \cos^{2} ψ \frac{\partial^{2}}{\partial r^{2}} + \frac{\cos^{2} θ \cos^{2} ψ}{r^{2}} \frac{\partial^{2}}{\partial θ^{2}} + \frac{\sin^{2} ψ}{r^{2} \sin^{2} θ} \frac{\partial^{2}}{\partial ψ^{2}} \\ + \frac{\cos θ \sin θ \cos^{2} ψ}{r} \frac{\partial^{2}}{\partial r \partial θ} + \frac{\cos θ \sin θ \cos^{2} ψ}{r} \frac{\partial^{2}}{\partial θ \partial r} \\ - \frac{\cos ψ \sin ψ}{r} \frac{\partial^{2}}{\partial r \partial ψ} - \frac{\cos ψ \sin ψ}{r} \frac{\partial^{2}}{\partial ψ \partial r} \\ - \frac{\cos θ \cos ψ \sin ψ}{r^{2} \sin θ} \frac{\partial^{2}}{\partial θ \partial ψ} - \frac{\cos θ \cos ψ \sin ψ}{r^{2} \sin θ} \frac{\partial^{2}}{\partial ψ \partial θ} \\ + \frac{\cos^{2} θ \cos^{2} ψ}{r} \frac{\partial}{\partial r} + \frac{\sin^{2} ψ}{r} \frac{\partial}{\partial r} \\ - \frac{\cos θ \sin θ \cos^{2} ψ}{r^{2}} \frac{\partial}{\partial θ} - \frac{\cos θ \sin θ \cos^{2} ψ}{r^{2}} \frac{\partial}{\partial θ} + \frac{\cos θ \sin^{2} ψ}{r^{2} \sin θ} \frac{\partial}{\partial θ} \\ + \frac{\cos ψ \sin ψ}{r^{2}} \frac{\partial}{\partial ψ} + \frac{\cos^{2} θ \cos ψ \sin ψ}{r^{2} \sin^{2} θ} \frac{\partial}{\partial ψ} + \frac{\cos ψ \sin ψ}{r^{2} \sin^{2} θ} \frac{\partial}{\partial ψ} \\ = & \sin^{2} θ \cos^{2} ψ \frac{\partial^{2}}{\partial r^{2}} + \frac{\cos^{2} θ \cos^{2} ψ}{r^{2}} \frac{\partial^{2}}{\partial θ^{2}} + \frac{\sin^{2} ψ}{r^{2} \sin^{2} θ} \frac{\partial^{2}}{\partial ψ^{2}} \\ + 2 \frac{\cos θ \sin θ \cos^{2} ψ}{r} \frac{\partial^{2}}{\partial r \partial θ} \dots \frac{\partial^{2}}{\partial r \partial θ} = \frac{\partial^{2}}{\partial θ \partial r} を 仮 定 \\ - 2 \frac{\cos ψ \sin ψ}{r} \frac{\partial^{2}}{\partial r \partial ψ} \dots \frac{\partial^{2}}{\partial r \partial ψ} = \frac{\partial^{2}}{\partial ψ \partial r} を 仮 定 \\ - 2 \frac{\cos θ \cos ψ \sin ψ}{r^{2} \sin θ} \frac{\partial^{2}}{\partial θ \partial ψ} \dots \frac{\partial^{2}}{\partial θ \partial ψ} = \frac{\partial^{2}}{\partial ψ \partial θ} を 仮 定 \\ + \frac{\cos^{2} θ \cos^{2} ψ + \sin^{2} ψ}{r} \frac{\partial}{\partial r} \\ + (- 2 \frac{\cos θ \sin θ \cos^{2} ψ}{r^{2}} + \frac{\cos θ \sin^{2} ψ}{r^{2} \sin θ}) \frac{\partial}{\partial θ} \\ + \frac{\cos ψ \sin ψ}{r^{2} \sin^{2} θ} (\sin^{2} θ + \cos^{2} θ + 1) \frac{\partial}{\partial ψ} \dots \sin^{2} θ + \cos^{2} θ = 1 \\ = & \sin^{2} θ \cos^{2} ψ \frac{\partial^{2}}{\partial r^{2}} + \frac{\cos^{2} θ \cos^{2} ψ}{r^{2}} \frac{\partial^{2}}{\partial θ^{2}} + \frac{\sin^{2} ψ}{r^{2} \sin^{2} θ} \frac{\partial^{2}}{\partial ψ^{2}} \\ + 2 \frac{\cos θ \sin θ \cos^{2} ψ}{r} \frac{\partial^{2}}{\partial r \partial θ} - 2 \frac{\cos ψ \sin ψ}{r} \frac{\partial^{2}}{\partial r \partial ψ} - 2 \frac{\cos θ \cos ψ \sin ψ}{r^{2} \sin θ} \frac{\partial^{2}}{\partial θ \partial ψ} \\ + \frac{\cos^{2} θ \cos^{2} ψ + \sin^{2} ψ}{r} \frac{\partial}{\partial r} + (- 2 \frac{\cos θ \sin θ \cos^{2} ψ}{r^{2}} + \frac{\cos θ \sin^{2} ψ}{r^{2} \sin θ}) \frac{\partial}{\partial θ} + 2 \frac{\cos ψ \sin ψ}{r^{2} \sin^{2} θ} \frac{\partial}{\partial ψ} \end{array}

$\frac{\partial^{2}}{\partial y^{2}}$ を求める

\begin{array}{rcl} \frac{\partial^{2}}{\partial y^{2}} & = & \frac{\partial}{\partial y} \frac{\partial}{\partial y} \\ = & (\sin θ \sin ψ \frac{\partial}{\partial r} + \frac{\cos θ \sin ψ}{r} \frac{\partial}{\partial θ} + \frac{\cos ψ}{r \sin θ} \frac{\partial}{\partial ψ}) (\sin θ \sin ψ \frac{\partial}{\partial r} + \frac{\cos θ \sin ψ}{r} \frac{\partial}{\partial θ} + \frac{\cos ψ}{r \sin θ} \frac{\partial}{\partial ψ}) \\ = & \sin θ \sin ψ \frac{\partial}{\partial r} (\sin θ \sin ψ \frac{\partial}{\partial r} + \frac{\cos θ \sin ψ}{r} \frac{\partial}{\partial θ} + \frac{\cos ψ}{r \sin θ} \frac{\partial}{\partial ψ}) \\ + \frac{\cos θ \sin ψ}{r} \frac{\partial}{\partial θ} (\sin θ \sin ψ \frac{\partial}{\partial r} + \frac{\cos θ \sin ψ}{r} \frac{\partial}{\partial θ} + \frac{\cos ψ}{r \sin θ} \frac{\partial}{\partial ψ}) \\ + \frac{\cos ψ}{r \sin θ} \frac{\partial}{\partial ψ} (\sin θ \sin ψ \frac{\partial}{\partial r} + \frac{\cos θ \sin ψ}{r} \frac{\partial}{\partial θ} + \frac{\cos ψ}{r \sin θ} \frac{\partial}{\partial ψ}) \\ = & \sin θ \sin ψ \frac{\partial}{\partial r} A_{y} + \frac{\cos θ \sin ψ}{r} B_{y} + \frac{\cos ψ}{r \sin θ} C_{y} \end{array}

\begin{array}{rcl} A_{y} & = & \frac{\partial}{\partial r} (\sin θ \sin ψ \frac{\partial}{\partial r} + \frac{\cos θ \sin ψ}{r} \frac{\partial}{\partial θ} + \frac{\cos ψ}{r \sin θ} \frac{\partial}{\partial ψ}) \\ = & \frac{\partial}{\partial r} (\sin θ \sin ψ) \cdot \frac{\partial}{\partial r} + \sin θ \sin ψ \cdot \frac{\partial}{\partial r} (\frac{\partial}{\partial r}) \\ + \frac{\partial}{\partial r} (\frac{\cos θ \sin ψ}{r}) \cdot \frac{\partial}{\partial θ} + \frac{\cos θ \sin ψ}{r} \cdot \frac{\partial}{\partial r} (\frac{\partial}{\partial θ}) \\ + \frac{\partial}{\partial r} (\frac{\cos ψ}{r \sin θ}) \cdot \frac{\partial}{\partial ψ} + \frac{\cos ψ}{r \sin θ} \cdot \frac{\partial}{\partial r} (\frac{\partial}{\partial ψ}) \\ = & (0) \frac{\partial}{\partial r} + \sin θ \sin ψ \frac{\partial^{2}}{\partial r^{2}} \\ + (- \frac{\cos θ \sin ψ}{r^{2}}) \frac{\partial}{\partial θ} + \frac{\cos θ \sin ψ}{r} \frac{\partial^{2}}{\partial r \partial θ} \\ + (- \frac{\cos ψ}{r^{2} \sin θ}) \frac{\partial}{\partial ψ} + \frac{\cos ψ}{r \sin θ} \frac{\partial^{2}}{\partial r \partial ψ} \\ = & \sin θ \sin ψ \frac{\partial^{2}}{\partial r^{2}} - \frac{\cos θ \sin ψ}{r^{2}} \frac{\partial}{\partial θ} + \frac{\cos θ \sin ψ}{r} \frac{\partial^{2}}{\partial r \partial θ} - \frac{\cos ψ}{r^{2} \sin θ} \frac{\partial}{\partial ψ} + \frac{\cos ψ}{r \sin θ} \frac{\partial^{2}}{\partial r \partial ψ} \end{array}

\begin{array}{rcl} \sin θ \sin ψ \cdot A_{y} & = & \sin θ \sin ψ (\sin θ \sin ψ \frac{\partial^{2}}{\partial r^{2}} - \frac{\cos θ \sin ψ}{r^{2}} \frac{\partial}{\partial θ} + \frac{\cos θ \sin ψ}{r} \frac{\partial^{2}}{\partial r \partial θ} - \frac{\cos ψ}{r^{2} \sin θ} \frac{\partial}{\partial ψ} + \frac{\cos ψ}{r \sin θ} \frac{\partial^{2}}{\partial r \partial ψ}) \\ = & \sin^{2} θ \sin^{2} ψ \frac{\partial^{2}}{\partial r^{2}} - \frac{\cos θ \sin θ \sin^{2} ψ}{r^{2}} \frac{\partial}{\partial θ} + \frac{\cos θ \sin θ \sin^{2} ψ}{r} \frac{\partial^{2}}{\partial r \partial θ} - \frac{\sin θ \cos ψ \sin ψ}{r^{2} \sin θ} \frac{\partial}{\partial ψ} + \frac{\sin θ \cos ψ \sin ψ}{r \sin θ} \frac{\partial^{2}}{\partial r \partial ψ} \\ = & \sin^{2} θ \sin^{2} ψ \frac{\partial^{2}}{\partial r^{2}} - \frac{\cos θ \sin θ \sin^{2} ψ}{r^{2}} \frac{\partial}{\partial θ} + \frac{\cos θ \sin θ \sin^{2} ψ}{r} \frac{\partial^{2}}{\partial r \partial θ} - \frac{\cos ψ \sin ψ}{r^{2}} \frac{\partial}{\partial ψ} + \frac{\cos ψ \sin ψ}{r} \frac{\partial^{2}}{\partial r \partial ψ} \end{array}

\begin{array}{rcl} B_{y} & = & \frac{\partial}{\partial θ} (\sin θ \sin ψ \frac{\partial}{\partial r} + \frac{\cos θ \sin ψ}{r} \frac{\partial}{\partial θ} + \frac{\cos ψ}{r \sin θ} \frac{\partial}{\partial ψ}) \\ = & \frac{\partial}{\partial θ} (\sin θ \sin ψ) \cdot \frac{\partial}{\partial r} + \sin θ \sin ψ \cdot \frac{\partial}{\partial θ} (\frac{\partial}{\partial r}) \\ + \frac{\partial}{\partial θ} (\frac{\cos θ \sin ψ}{r}) \cdot \frac{\partial}{\partial θ} + \frac{\cos θ \sin ψ}{r} \cdot \frac{\partial}{\partial θ} (\frac{\partial}{\partial θ}) \\ + \frac{\partial}{\partial θ} (\frac{\cos ψ}{r \sin θ}) \cdot \frac{\partial}{\partial ψ} + \frac{\cos ψ}{r \sin θ} \cdot \frac{\partial}{\partial θ} (\frac{\partial}{\partial ψ}) \\ = & \cos θ \sin ψ \frac{\partial}{\partial r} + \sin θ \sin ψ \frac{\partial^{2}}{\partial θ \partial r} - \frac{\sin θ \sin ψ}{r} \frac{\partial}{\partial θ} + \frac{\cos θ \sin ψ}{r} \frac{\partial^{2}}{\partial θ^{2}} - \frac{\cos θ \cos ψ}{r \sin^{2} θ} \frac{\partial}{\partial ψ} + \frac{\cos ψ}{r \sin θ} \frac{\partial^{2}}{\partial θ \partial ψ} \end{array}

\begin{array}{rcl} \frac{\cos θ \sin ψ}{r} \cdot B_{y} & = & \frac{\cos θ \sin ψ}{r} (\cos θ \sin ψ \frac{\partial}{\partial r} + \sin θ \sin ψ \frac{\partial^{2}}{\partial θ \partial r} - \frac{\sin θ \sin ψ}{r} \frac{\partial}{\partial θ} + \frac{\cos θ \sin ψ}{r} \frac{\partial^{2}}{\partial θ^{2}} - \frac{\cos θ \cos ψ}{r \sin^{2} θ} \frac{\partial}{\partial ψ} + \frac{\cos ψ}{r \sin θ} \frac{\partial^{2}}{\partial θ \partial ψ}) \\ = & \frac{\cos^{2} θ \sin^{2} ψ}{r} \frac{\partial}{\partial r} + \frac{\cos θ \sin θ \sin^{2} ψ}{r} \frac{\partial^{2}}{\partial θ \partial r} - \frac{\cos θ \sin θ \sin^{2} ψ}{r^{2}} \frac{\partial}{\partial θ} + \frac{\cos^{2} θ \sin^{2} ψ}{r^{2}} \frac{\partial^{2}}{\partial θ^{2}} - \frac{\cos^{2} θ \cos ψ \sin ψ}{r^{2} \sin^{2} θ} \frac{\partial}{\partial ψ} + \frac{\cos θ \cos ψ \sin ψ}{r^{2} \sin θ} \frac{\partial^{2}}{\partial θ \partial ψ} \end{array}

\begin{array}{rcl} C_{y} & = & \frac{\partial}{\partial ψ} (\sin θ \sin ψ \frac{\partial}{\partial r} + \frac{\cos θ \sin ψ}{r} \frac{\partial}{\partial θ} + \frac{\cos ψ}{r \sin θ} \frac{\partial}{\partial ψ}) \\ = & \frac{\partial}{\partial ψ} (\sin θ \sin ψ) \cdot \frac{\partial}{\partial r} + \sin θ \sin ψ \cdot \frac{\partial}{\partial ψ} (\frac{\partial}{\partial r}) \\ + \frac{\partial}{\partial ψ} (\frac{\cos θ \sin ψ}{r}) \cdot \frac{\partial}{\partial θ} + \frac{\cos θ \sin ψ}{r} \cdot \frac{\partial}{\partial ψ} (\frac{\partial}{\partial θ}) \\ + \frac{\partial}{\partial ψ} (\frac{\cos ψ}{r \sin θ}) \cdot \frac{\partial}{\partial ψ} + \frac{\cos ψ}{r \sin θ} \cdot \frac{\partial}{\partial ψ} (\frac{\partial}{\partial ψ}) \\ = & (\sin θ \cos ψ) \frac{\partial}{\partial r} + \sin θ \sin ψ \frac{\partial^{2}}{\partial ψ \partial r} \\ + (\frac{\cos θ \cos ψ}{r}) \frac{\partial}{\partial θ} + \frac{\cos θ \sin ψ}{r} \frac{\partial^{2}}{\partial ψ \partial θ} \\ + (- \frac{\sin ψ}{r \sin θ}) \frac{\partial}{\partial ψ} + \frac{\cos ψ}{r \sin θ} \frac{\partial^{2}}{\partial ψ^{2}} \\ = & \sin θ \cos ψ \frac{\partial}{\partial r} + \sin θ \sin ψ \frac{\partial^{2}}{\partial ψ \partial r} + \frac{\cos θ \cos ψ}{r} \frac{\partial}{\partial θ} + \frac{\cos θ \sin ψ}{r} \frac{\partial^{2}}{\partial ψ \partial θ} - \frac{\sin ψ}{r \sin θ} \frac{\partial}{\partial ψ} + \frac{\cos ψ}{r \sin θ} \frac{\partial^{2}}{\partial ψ^{2}} \end{array}

\begin{array}{rcl} \frac{\cos ψ}{r \sin θ} \cdot C_{y} & = & \frac{\cos ψ}{r \sin θ} (\sin θ \cos ψ \frac{\partial}{\partial r} + \sin θ \sin ψ \frac{\partial^{2}}{\partial ψ \partial r} + \frac{\cos θ \cos ψ}{r} \frac{\partial}{\partial θ} + \frac{\cos θ \sin ψ}{r} \frac{\partial^{2}}{\partial ψ \partial θ} - \frac{\sin ψ}{r \sin θ} \frac{\partial}{\partial ψ} + \frac{\cos ψ}{r \sin θ} \frac{\partial^{2}}{\partial ψ^{2}}) \\ = & \frac{\sin θ \cos^{2} ψ}{r \sin θ} \frac{\partial}{\partial r} + \frac{\sin θ \cos ψ \sin ψ}{r \sin θ} \frac{\partial^{2}}{\partial ψ \partial r} + \frac{\cos θ \cos^{2} ψ}{r^{2} \sin θ} \frac{\partial}{\partial θ} + \frac{\cos θ \cos ψ \sin ψ}{r^{2} \sin θ} \frac{\partial^{2}}{\partial ψ \partial θ} - \frac{\cos ψ \sin ψ}{r^{2} \sin^{2} θ} \frac{\partial}{\partial ψ} + \frac{\cos^{2} ψ}{r^{2} \sin^{2} θ} \frac{\partial^{2}}{\partial ψ^{2}} \\ = & \frac{\cos^{2} ψ}{r} \frac{\partial}{\partial r} + \frac{\cos ψ \sin ψ}{r} \frac{\partial^{2}}{\partial ψ \partial r} + \frac{\cos θ \cos^{2} ψ}{r^{2} \sin θ} \frac{\partial}{\partial θ} + \frac{\cos θ \cos ψ \sin ψ}{r^{2} \sin θ} \frac{\partial^{2}}{\partial ψ \partial θ} - \frac{\cos ψ \sin ψ}{r^{2} \sin^{2} θ} \frac{\partial}{\partial ψ} + \frac{\cos^{2} ψ}{r^{2} \sin^{2} θ} \frac{\partial^{2}}{\partial ψ^{2}} \end{array}

\begin{array}{rcl} \frac{\partial^{2}}{\partial y^{2}} & = & \sin θ \sin ψ \frac{\partial}{\partial r} A_{y} + \frac{\cos θ \sin ψ}{r} B_{y} + \frac{\cos ψ}{r \sin θ} C_{y} \\ = & \sin^{2} θ \sin^{2} ψ \frac{\partial^{2}}{\partial r^{2}} - \frac{\cos θ \sin θ \sin^{2} ψ}{r^{2}} \frac{\partial}{\partial θ} + \frac{\cos θ \sin θ \sin^{2} ψ}{r} \frac{\partial^{2}}{\partial r \partial θ} - \frac{\cos ψ \sin ψ}{r^{2}} \frac{\partial}{\partial ψ} + \frac{\cos ψ \sin ψ}{r} \frac{\partial^{2}}{\partial r \partial ψ} \\ + \frac{\cos^{2} θ \sin^{2} ψ}{r} \frac{\partial}{\partial r} + \frac{\cos θ \sin θ \sin^{2} ψ}{r} \frac{\partial^{2}}{\partial θ \partial r} - \frac{\cos θ \sin θ \sin^{2} ψ}{r^{2}} \frac{\partial}{\partial θ} + \frac{\cos^{2} θ \sin^{2} ψ}{r^{2}} \frac{\partial^{2}}{\partial θ^{2}} - \frac{\cos^{2} θ \cos ψ \sin ψ}{r^{2} \sin^{2} θ} \frac{\partial}{\partial ψ} + \frac{\cos θ \cos ψ \sin ψ}{r^{2} \sin θ} \frac{\partial^{2}}{\partial θ \partial ψ} \\ + \frac{\cos^{2} ψ}{r} \frac{\partial}{\partial r} + \frac{\cos ψ \sin ψ}{r} \frac{\partial^{2}}{\partial ψ \partial r} + \frac{\cos θ \cos^{2} ψ}{r^{2} \sin θ} \frac{\partial}{\partial θ} + \frac{\cos θ \cos ψ \sin ψ}{r^{2} \sin θ} \frac{\partial^{2}}{\partial ψ \partial θ} - \frac{\cos ψ \sin ψ}{r^{2} \sin^{2} θ} \frac{\partial}{\partial ψ} + \frac{\cos^{2} ψ}{r^{2} \sin^{2} θ} \frac{\partial^{2}}{\partial ψ^{2}} \\ = & \sin^{2} θ \sin^{2} ψ \frac{\partial^{2}}{\partial r^{2}} + \frac{\cos^{2} θ \sin^{2} ψ}{r^{2}} \frac{\partial^{2}}{\partial θ^{2}} + \frac{\cos^{2} ψ}{r^{2} \sin^{2} θ} \frac{\partial^{2}}{\partial ψ^{2}} \\ + \frac{\cos θ \sin θ \sin^{2} ψ}{r} \frac{\partial^{2}}{\partial r \partial θ} + \frac{\cos θ \sin θ \sin^{2} ψ}{r} \frac{\partial^{2}}{\partial θ \partial r} \\ + \frac{\cos ψ \sin ψ}{r} \frac{\partial^{2}}{\partial r \partial ψ} + \frac{\cos ψ \sin ψ}{r} \frac{\partial^{2}}{\partial ψ \partial r} \\ + \frac{\cos θ \cos ψ \sin ψ}{r^{2} \sin θ} \frac{\partial^{2}}{\partial θ \partial ψ} + \frac{\cos θ \cos ψ \sin ψ}{r^{2} \sin θ} \frac{\partial^{2}}{\partial ψ \partial θ} \\ + \frac{\cos^{2} θ \sin^{2} ψ}{r} \frac{\partial}{\partial r} + \frac{\cos^{2} ψ}{r} \frac{\partial}{\partial r} \\ - \frac{\cos θ \sin θ \sin^{2} ψ}{r^{2}} \frac{\partial}{\partial θ} - \frac{\cos θ \sin θ \sin^{2} ψ}{r^{2}} \frac{\partial}{\partial θ} + \frac{\cos θ \cos^{2} ψ}{r^{2} \sin θ} \frac{\partial}{\partial θ} \\ - \frac{\cos ψ \sin ψ}{r^{2}} \frac{\partial}{\partial ψ} - \frac{\cos^{2} θ \cos ψ \sin ψ}{r^{2} \sin^{2} θ} \frac{\partial}{\partial ψ} - \frac{\cos ψ \sin ψ}{r^{2} \sin^{2} θ} \frac{\partial}{\partial ψ} \\ = & \sin^{2} θ \sin^{2} ψ \frac{\partial^{2}}{\partial r^{2}} + \frac{\cos^{2} θ \sin^{2} ψ}{r^{2}} \frac{\partial^{2}}{\partial θ^{2}} + \frac{\cos^{2} ψ}{r^{2} \sin^{2} θ} \frac{\partial^{2}}{\partial ψ^{2}} \\ + 2 \frac{\cos θ \sin θ \sin^{2} ψ}{r} \frac{\partial^{2}}{\partial r \partial θ} \dots \frac{\partial^{2}}{\partial r \partial θ} = \frac{\partial^{2}}{\partial θ \partial r} を 仮 定 \\ + 2 \frac{\cos ψ \sin ψ}{r} \frac{\partial^{2}}{\partial r \partial ψ} \dots \frac{\partial^{2}}{\partial r \partial ψ} = \frac{\partial^{2}}{\partial ψ \partial r} を 仮 定 \\ + 2 \frac{\cos θ \cos ψ \sin ψ}{r^{2} \sin θ} \frac{\partial^{2}}{\partial θ \partial ψ} \dots \frac{\partial^{2}}{\partial θ \partial ψ} = \frac{\partial^{2}}{\partial ψ \partial θ} を 仮 定 \\ + \frac{\cos^{2} θ \sin^{2} ψ + \cos^{2} ψ}{r} \frac{\partial}{\partial r} \\ + (- 2 \frac{\cos θ \sin θ \sin^{2} ψ}{r^{2}} + \frac{\cos θ \cos^{2} ψ}{r^{2} \sin θ}) \frac{\partial}{\partial θ} \\ - \frac{\cos ψ \sin ψ}{r^{2} \sin^{2} θ} (\sin^{2} θ + \cos^{2} θ + 1) \frac{\partial}{\partial ψ} \dots \sin^{2} θ + \cos^{2} θ = 1 \\ = & \sin^{2} θ \sin^{2} ψ \frac{\partial^{2}}{\partial r^{2}} + \frac{\cos^{2} θ \sin^{2} ψ}{r^{2}} \frac{\partial^{2}}{\partial θ^{2}} + \frac{\cos^{2} ψ}{r^{2} \sin^{2} θ} \frac{\partial^{2}}{\partial ψ^{2}} \\ + 2 \frac{\cos θ \sin θ \sin^{2} ψ}{r} \frac{\partial^{2}}{\partial r \partial θ} + 2 \frac{\cos ψ \sin ψ}{r} \frac{\partial^{2}}{\partial r \partial ψ} + 2 \frac{\cos θ \cos ψ \sin ψ}{r^{2} \sin θ} \frac{\partial^{2}}{\partial θ \partial ψ} \\ + \frac{\cos^{2} θ \sin^{2} ψ + \cos^{2} ψ}{r} \frac{\partial}{\partial r} + (- 2 \frac{\cos θ \sin θ \sin^{2} ψ}{r^{2}} + \frac{\cos θ \cos^{2} ψ}{r^{2} \sin θ}) \frac{\partial}{\partial θ} - 2 \frac{\cos ψ \sin ψ}{r^{2} \sin^{2} θ} \frac{\partial}{\partial ψ} \end{array}

$\frac{\partial^{2}}{\partial z^{2}}$ を求める

\begin{array}{rcl} \frac{\partial^{2}}{\partial z^{2}} & = & \frac{\partial}{\partial z} \frac{\partial}{\partial z} \\ = & (\cos θ \frac{\partial}{\partial r} - \frac{\sin θ}{r} \frac{\partial}{\partial θ}) (\cos θ \frac{\partial}{\partial r} - \frac{\sin θ}{r} \frac{\partial}{\partial θ}) \\ = & \cos θ \frac{\partial}{\partial r} (\cos θ \frac{\partial}{\partial r} - \frac{\sin θ}{r} \frac{\partial}{\partial θ}) - \frac{\sin θ}{r} \frac{\partial}{\partial θ} (\cos θ \frac{\partial}{\partial r} - \frac{\sin θ}{r} \frac{\partial}{\partial θ}) \\ = & \cos θ A_{z} - \frac{\sin θ}{r} B_{z} \end{array}

\begin{array}{rcl} A_{z} & = & \frac{\partial}{\partial r} (\cos θ \frac{\partial}{\partial r} - \frac{\sin θ}{r} \frac{\partial}{\partial θ}) \\ = & \frac{\partial}{\partial r} (\cos θ) \cdot \frac{\partial}{\partial r} + \cos θ \cdot \frac{\partial}{\partial r} (\frac{\partial}{\partial r}) \\ + \frac{\partial}{\partial r} (- \frac{\sin θ}{r}) \cdot \frac{\partial}{\partial θ} - \frac{\sin θ}{r} \cdot \frac{\partial}{\partial r} (\frac{\partial}{\partial θ}) \\ = & (0) \frac{\partial}{\partial r} + \cos θ \frac{\partial^{2}}{\partial r^{2}} \\ + (\frac{\sin θ}{r^{2}}) \frac{\partial}{\partial θ} - \frac{\sin θ}{r} \frac{\partial^{2}}{\partial r \partial θ} \\ = & \cos θ \frac{\partial^{2}}{\partial r^{2}} + \frac{\sin θ}{r^{2}} \frac{\partial}{\partial θ} - \frac{\sin θ}{r} \frac{\partial^{2}}{\partial r \partial θ} \end{array}

\begin{array}{rcl} \cos θ \cdot A_{z} & = & \cos θ (\cos θ \frac{\partial^{2}}{\partial r^{2}} + \frac{\sin θ}{r^{2}} \frac{\partial}{\partial θ} - \frac{\sin θ}{r} \frac{\partial^{2}}{\partial r \partial θ}) \\ = & \cos^{2} θ \frac{\partial^{2}}{\partial r^{2}} + \frac{\cos θ \sin θ}{r^{2}} \frac{\partial}{\partial θ} - \frac{\cos θ \sin θ}{r} \frac{\partial^{2}}{\partial r \partial θ} \end{array}

\begin{array}{rcl} B_{z} & = & \frac{\partial}{\partial θ} (\cos θ \frac{\partial}{\partial r} - \frac{\sin θ}{r} \frac{\partial}{\partial θ}) \\ = & \frac{\partial}{\partial θ} (\cos θ) \cdot \frac{\partial}{\partial r} + \cos θ \cdot \frac{\partial}{\partial θ} (\frac{\partial}{\partial r}) \\ + \frac{\partial}{\partial θ} (- \frac{\sin θ}{r}) \cdot \frac{\partial}{\partial θ} - \frac{\sin θ}{r} \cdot \frac{\partial}{\partial θ} (\frac{\partial}{\partial θ}) \\ = & (- \sin θ) \frac{\partial}{\partial r} + \cos θ \frac{\partial^{2}}{\partial θ \partial r} \\ + (- \frac{\cos θ}{r}) \frac{\partial}{\partial θ} - \frac{\sin θ}{r} \frac{\partial^{2}}{\partial θ^{2}} \\ = & - \sin θ \frac{\partial}{\partial r} + \cos θ \frac{\partial^{2}}{\partial θ \partial r} - \frac{\cos θ}{r} \frac{\partial}{\partial θ} - \frac{\sin θ}{r} \frac{\partial^{2}}{\partial θ^{2}} \end{array}

\begin{array}{rcl} - \frac{\sin θ}{r} B_{z} & = & - \frac{\sin θ}{r} (- \sin θ \frac{\partial}{\partial r} + \cos θ \frac{\partial^{2}}{\partial θ \partial r} - \frac{\cos θ}{r} \frac{\partial}{\partial θ} - \frac{\sin θ}{r} \frac{\partial^{2}}{\partial θ^{2}}) \\ = & \frac{\sin^{2} θ}{r} \frac{\partial}{\partial r} - \frac{\cos θ \sin θ}{r} \frac{\partial^{2}}{\partial θ \partial r} + \frac{\cos θ \sin θ}{r^{2}} \frac{\partial}{\partial θ} + \frac{\sin^{2} θ}{r^{2}} \frac{\partial^{2}}{\partial θ^{2}} \end{array}

\begin{array}{rcl} \frac{\partial^{2}}{\partial z^{2}} & = & \cos θ \cdot A_{z} - \frac{\sin θ}{r} \cdot B_{z} \\ = & \cos^{2} θ \frac{\partial^{2}}{\partial r^{2}} + \frac{\cos θ \sin θ}{r^{2}} \frac{\partial}{\partial θ} - \frac{\cos θ \sin θ}{r} \frac{\partial^{2}}{\partial r \partial θ} \\ + \frac{\sin^{2} θ}{r} \frac{\partial}{\partial r} - \frac{\cos θ \sin θ}{r} \frac{\partial^{2}}{\partial θ \partial r} + \frac{\cos θ \sin θ}{r^{2}} \frac{\partial}{\partial θ} + \frac{\sin^{2} θ}{r^{2}} \frac{\partial^{2}}{\partial θ^{2}} \\ = & \cos^{2} θ \frac{\partial^{2}}{\partial r^{2}} + \frac{\sin^{2} θ}{r^{2}} \frac{\partial^{2}}{\partial θ^{2}} \\ - \frac{\cos θ \sin θ}{r} \frac{\partial^{2}}{\partial r \partial θ} - \frac{\cos θ \sin θ}{r} \frac{\partial^{2}}{\partial θ \partial r} \\ + \frac{\sin^{2} θ}{r} \frac{\partial}{\partial r} \\ + \frac{\cos θ \sin θ}{r^{2}} \frac{\partial}{\partial θ} + \frac{\cos θ \sin θ}{r^{2}} \frac{\partial}{\partial θ} \\ = & \cos^{2} θ \frac{\partial^{2}}{\partial r^{2}} + \frac{\sin^{2} θ}{r^{2}} \frac{\partial^{2}}{\partial θ^{2}} \\ - 2 \frac{\cos θ \sin θ}{r} \frac{\partial^{2}}{\partial r \partial θ} \dots \frac{\partial^{2}}{\partial r \partial θ} = \frac{\partial^{2}}{\partial θ \partial r} を 仮 定 \\ + \frac{\sin^{2} θ}{r} \frac{\partial}{\partial r} \\ + 2 \frac{\cos θ \sin θ}{r^{2}} \frac{\partial}{\partial θ} \\ = & \cos^{2} θ \frac{\partial^{2}}{\partial r^{2}} + \frac{\sin^{2} θ}{r^{2}} \frac{\partial^{2}}{\partial θ^{2}} - 2 \frac{\cos θ \sin θ}{r} \frac{\partial^{2}}{\partial r \partial θ} + \frac{\sin^{2} θ}{r} \frac{\partial}{\partial r} + 2 \frac{\cos θ \sin θ}{r^{2}} \frac{\partial}{\partial θ} \end{array}

$Δ (= \nabla^{2})$ を求める

\begin{array}{rcl} Δ & = & \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} + \frac{\partial^{2}}{\partial z^{2}} \\ = & \sin^{2} θ \cos^{2} ψ \frac{\partial^{2}}{\partial r^{2}} + \frac{\cos^{2} θ \cos^{2} ψ}{r^{2}} \frac{\partial^{2}}{\partial θ^{2}} + \frac{\sin^{2} ψ}{r^{2} \sin^{2} θ} \frac{\partial^{2}}{\partial ψ^{2}} + 2 \frac{\cos θ \sin θ \cos^{2} ψ}{r} \frac{\partial^{2}}{\partial r \partial θ} - 2 \frac{\cos ψ \sin ψ}{r} \frac{\partial^{2}}{\partial r \partial ψ} - 2 \frac{\cos θ \cos ψ \sin ψ}{r^{2} \sin θ} \frac{\partial^{2}}{\partial θ \partial ψ} + \frac{\cos^{2} θ \cos^{2} ψ + \sin^{2} ψ}{r} \frac{\partial}{\partial r} + (- 2 \frac{\cos θ \sin θ \cos^{2} ψ}{r^{2}} + \frac{\cos θ \sin^{2} ψ}{r^{2} \sin θ}) \frac{\partial}{\partial θ} + 2 \frac{\cos ψ \sin ψ}{r^{2} \sin^{2} θ} \frac{\partial}{\partial ψ} \\ + \sin^{2} θ \sin^{2} ψ \frac{\partial^{2}}{\partial r^{2}} + \frac{\cos^{2} θ \sin^{2} ψ}{r^{2}} \frac{\partial^{2}}{\partial θ^{2}} + \frac{\cos^{2} ψ}{r^{2} \sin^{2} θ} \frac{\partial^{2}}{\partial ψ^{2}} + 2 \frac{\cos θ \sin θ \sin^{2} ψ}{r} \frac{\partial^{2}}{\partial r \partial θ} + 2 \frac{\cos ψ \sin ψ}{r} \frac{\partial^{2}}{\partial r \partial ψ} + 2 \frac{\cos θ \cos ψ \sin ψ}{r^{2} \sin θ} \frac{\partial^{2}}{\partial θ \partial ψ} + \frac{\cos^{2} θ \sin^{2} ψ + \cos^{2} ψ}{r} \frac{\partial}{\partial r} + (- 2 \frac{\cos θ \sin θ \sin^{2} ψ}{r^{2}} + \frac{\cos θ \cos^{2} ψ}{r^{2} \sin θ}) \frac{\partial}{\partial θ} - 2 \frac{\cos ψ \sin ψ}{r^{2} \sin^{2} θ} \frac{\partial}{\partial ψ} \\ + \cos^{2} θ \frac{\partial^{2}}{\partial r^{2}} + \frac{\sin^{2} θ}{r^{2}} \frac{\partial^{2}}{\partial θ^{2}} - 2 \frac{\cos θ \sin θ}{r} \frac{\partial^{2}}{\partial r \partial θ} + \frac{\sin^{2} θ}{r} \frac{\partial}{\partial r} + 2 \frac{\cos θ \sin θ}{r^{2}} \frac{\partial}{\partial θ} \\ = & \sin^{2} θ \cos^{2} ψ \frac{\partial^{2}}{\partial r^{2}} + \sin^{2} θ \sin^{2} ψ \frac{\partial^{2}}{\partial r^{2}} + \cos^{2} θ \frac{\partial^{2}}{\partial r^{2}} \\ + \frac{\cos^{2} θ \cos^{2} ψ}{r^{2}} \frac{\partial^{2}}{\partial θ^{2}} + \frac{\cos^{2} θ \sin^{2} ψ}{r^{2}} \frac{\partial^{2}}{\partial θ^{2}} + \frac{\sin^{2} θ}{r^{2}} \frac{\partial^{2}}{\partial θ^{2}} \\ + \frac{\sin^{2} ψ}{r^{2} \sin^{2} θ} \frac{\partial^{2}}{\partial ψ^{2}} + \frac{\cos^{2} ψ}{r^{2} \sin^{2} θ} \frac{\partial^{2}}{\partial ψ^{2}} \\ + 2 \frac{\cos θ \sin θ \cos^{2} ψ}{r} \frac{\partial^{2}}{\partial r \partial θ} + 2 \frac{\cos θ \sin θ \sin^{2} ψ}{r} \frac{\partial^{2}}{\partial r \partial θ} - 2 \frac{\cos θ \sin θ}{r} \frac{\partial^{2}}{\partial r \partial θ} \\ - 2 \frac{\cos ψ \sin ψ}{r} \frac{\partial^{2}}{\partial r \partial ψ} + 2 \frac{\cos ψ \sin ψ}{r} \frac{\partial^{2}}{\partial r \partial ψ} \\ - 2 \frac{\cos θ \cos ψ \sin ψ}{r^{2} \sin θ} \frac{\partial^{2}}{\partial θ \partial ψ} + 2 \frac{\cos θ \cos ψ \sin ψ}{r^{2} \sin θ} \frac{\partial^{2}}{\partial θ \partial ψ} \\ + \frac{\cos^{2} θ \cos^{2} ψ + \sin^{2} ψ}{r} \frac{\partial}{\partial r} + \frac{\cos^{2} θ \sin^{2} ψ + \cos^{2} ψ}{r} \frac{\partial}{\partial r} + \frac{\sin^{2} θ}{r} \frac{\partial}{\partial r} \\ + (- 2 \frac{\cos θ \sin θ \cos^{2} ψ}{r^{2}} + \frac{\cos θ \sin^{2} ψ}{r^{2} \sin θ}) \frac{\partial}{\partial θ} + (- 2 \frac{\cos θ \sin θ \sin^{2} ψ}{r^{2}} + \frac{\cos θ \cos^{2} ψ}{r^{2} \sin θ}) \frac{\partial}{\partial θ} + 2 \frac{\cos θ \sin θ}{r^{2}} \frac{\partial}{\partial θ} \\ + 2 \frac{\cos ψ \sin ψ}{r^{2} \sin^{2} θ} \frac{\partial}{\partial ψ} - 2 \frac{\cos ψ \sin ψ}{r^{2} \sin^{2} θ} \frac{\partial}{\partial ψ} \\ = & \sin^{2} θ (\cos^{2} ψ + \sin^{2} ψ) \frac{\partial^{2}}{\partial r^{2}} + \cos^{2} θ \frac{\partial^{2}}{\partial r^{2}} \dots \cos^{2} ψ + \sin^{2} ψ = 1 \\ + \frac{\cos^{2} θ}{r^{2}} (\cos^{2} ψ + \sin^{2} ψ) \frac{\partial^{2}}{\partial θ^{2}} + \frac{\sin^{2} θ}{r^{2}} \frac{\partial^{2}}{\partial θ^{2}} \dots \cos^{2} ψ + \sin^{2} ψ = 1 \\ + \frac{1}{r^{2} \sin^{2} θ} (\sin^{2} ψ + \cos^{2} ψ) \frac{\partial^{2}}{\partial ψ^{2}} \dots \cos^{2} ψ + \sin^{2} ψ = 1 \\ + 2 \frac{\cos θ \sin θ}{r} (\cos^{2} ψ + \sin^{2} ψ) \frac{\partial^{2}}{\partial r \partial θ} - 2 \frac{\cos θ \sin θ}{r} \frac{\partial^{2}}{\partial r \partial θ} \dots \cos^{2} ψ + \sin^{2} ψ = 1 \\ + \frac{\cos^{2} θ (\cos^{2} ψ + \sin^{2} ψ) + (\cos^{2} ψ + \sin^{2} ψ) + \sin^{2} θ}{r} \frac{\partial}{\partial r} \dots \cos^{2} ψ + \sin^{2} ψ = 1 \\ + {- 2 \frac{\cos θ \sin θ}{r^{2}} (\cos^{2} ψ + \sin^{2} ψ) + 2 \frac{\cos θ \sin θ}{r^{2}} + \frac{\cos θ}{r^{2} \sin θ} (\cos^{2} ψ + \sin^{2} ψ)} \frac{\partial}{\partial θ} \dots \cos^{2} ψ + \sin^{2} ψ = 1 \\ = & (\cos^{2} θ + \sin^{2} θ) \frac{\partial^{2}}{\partial r^{2}} \\ + \frac{1}{r^{2}} (\cos^{2} θ + \sin^{2} θ) \frac{\partial^{2}}{\partial θ^{2}} \dots \cos^{2} θ + \sin^{2} θ = 1 \\ + \frac{1}{r^{2} \sin^{2} θ} \frac{\partial^{2}}{\partial ψ^{2}} \\ + 2 \frac{\cos θ \sin θ}{r} \frac{\partial^{2}}{\partial r \partial θ} - 2 \frac{\cos θ \sin θ}{r} \frac{\partial^{2}}{\partial r \partial θ} \\ + \frac{\cos^{2} θ + \sin^{2} θ + 1}{r} \frac{\partial}{\partial r} \dots \cos^{2} θ + \sin^{2} θ = 1 \\ + {- 2 \frac{\cos θ \sin θ}{r^{2}} + 2 \frac{\cos θ \sin θ}{r^{2}} + \frac{\cos θ}{r^{2} \sin θ}} \frac{\partial}{\partial θ} \\ = & \frac{\partial^{2}}{\partial r^{2}} + \frac{1}{r^{2}} \frac{\partial^{2}}{\partial θ^{2}} + \frac{1}{r^{2} \sin^{2} θ} \frac{\partial^{2}}{\partial ψ^{2}} + \frac{2}{r} \frac{\partial}{\partial r} + \frac{\cos θ}{r^{2} \sin θ} \frac{\partial}{\partial θ} \end{array}

\begin{array}{rcl} \frac{1}{r^{2}} \frac{\partial}{\partial r} (r^{2} \frac{\partial}{\partial r}) & = & \frac{1}{r^{2}} {\frac{\partial}{\partial r} (r^{2}) \cdot \frac{\partial}{\partial r} + r^{2} \cdot \frac{\partial}{\partial r} (\frac{\partial}{\partial r})} \\ = & \frac{1}{r^{2}} {2 r \frac{\partial}{\partial r} + r^{2} \frac{\partial^{2}}{\partial r^{2}}} \\ = & \frac{2}{r} \frac{\partial}{\partial r} + \frac{r^{2}}{r^{2}} \frac{\partial^{2}}{\partial r^{2}} \\ = & \frac{\partial^{2}}{\partial r^{2}} + \frac{2}{r} \frac{\partial}{\partial r} \end{array}

\begin{array}{rcl} \frac{1}{r^{2} \sin θ} \frac{\partial}{\partial θ} (\sin θ \frac{\partial}{\partial θ}) & = & \frac{1}{r^{2} \sin θ} {\frac{\partial}{\partial θ} (\sin θ) \cdot \frac{\partial}{\partial θ} + \sin θ \cdot \frac{\partial}{\partial θ} (\frac{\partial}{\partial θ})} \\ = & \frac{1}{r^{2} \sin θ} {\cos θ \frac{\partial}{\partial θ} + \sin θ \frac{\partial^{2}}{\partial θ^{2}}} \\ = & \frac{\cos θ}{r^{2} \sin θ} \frac{\partial}{\partial θ} + \frac{\sin θ}{r^{2} \sin θ} \frac{\partial^{2}}{\partial θ^{2}} \\ = & \frac{1}{r^{2}} \frac{\partial^{2}}{\partial θ^{2}} + \frac{\cos θ}{r^{2} \sin θ} \frac{\partial}{\partial θ} \end{array}

\begin{array}{rcl} Δ & = & \frac{\partial^{2}}{\partial r^{2}} + \frac{1}{r^{2}} \frac{\partial^{2}}{\partial θ^{2}} + \frac{1}{r^{2} \sin^{2} θ} \frac{\partial^{2}}{\partial ψ^{2}} + \frac{2}{r} \frac{\partial}{\partial r} + \frac{\cos θ}{r^{2} \sin θ} \frac{\partial}{\partial θ} \\ = & {\frac{\partial^{2}}{\partial r^{2}} + \frac{2}{r} \frac{\partial}{\partial r}} + {\frac{1}{r^{2}} \frac{\partial^{2}}{\partial θ^{2}} + \frac{\cos θ}{r^{2} \sin θ} \frac{\partial}{\partial θ}} + \frac{1}{r^{2} \sin^{2} θ} \frac{\partial^{2}}{\partial ψ^{2}} \\ = & \frac{1}{r^{2}} \frac{\partial}{\partial r} (r^{2} \frac{\partial}{\partial r}) + \frac{1}{r^{2} \sin θ} \frac{\partial}{\partial θ} (\sin θ \frac{\partial}{\partial θ}) + \frac{1}{r^{2} \sin^{2} θ} \frac{\partial^{2}}{\partial ψ^{2}} \end{array}

式展開

極座標ラプラシアンの導出

極座標ラプラシアンの導出

直交座標(x,y,z)と極座標(r,θ,ψ)の関係式

∂r∂xを求める

∂θ∂xを求める

∂ψ∂xを求める

∂r∂yを求める

∂θ∂yを求める

∂ψ∂yを求める

∂r∂zを求める

∂θ∂zを求める

∂ψ∂zを求める

∂∂x,∂∂y,∂∂zを求める

∂2∂x2を求める

∂2∂y2を求める

∂2∂z2を求める

Δ(=∇2)を求める

直交座標 $(x, y, z)$ と極座標 $(r, θ, ψ)$ の関係式

$\frac{\partial r}{\partial x}$ を求める

$\frac{\partial θ}{\partial x}$ を求める

$\frac{\partial ψ}{\partial x}$ を求める

$\frac{\partial r}{\partial y}$ を求める

$\frac{\partial θ}{\partial y}$ を求める

$\frac{\partial ψ}{\partial y}$ を求める

$\frac{\partial r}{\partial z}$ を求める

$\frac{\partial θ}{\partial z}$ を求める

$\frac{\partial ψ}{\partial z}$ を求める

$\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z}$ を求める

$\frac{\partial^{2}}{\partial x^{2}}$ を求める

$\frac{\partial^{2}}{\partial y^{2}}$ を求める

$\frac{\partial^{2}}{\partial z^{2}}$ を求める

$Δ (= \nabla^{2})$ を求める