tan(θ)=?

original: https://www.youtube.com/watch?v=rtUBRi2q4mo

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\(\theta=\)?

問い(\(\tan{\left(\theta\right)=}\))

\(\overline{AC}\)に垂直な補助線\(\overline{BD}\)を引く

$$\begin{eqnarray} \angle C &=& \pi - \left(\angle A + \angle B\right) \\ &=& \pi-\left\{\theta+\left(\frac{\pi}{2}+\theta\right)\right\} \\ &=& \frac{\pi}{2}-2\theta \end{eqnarray}$$ $$\begin{eqnarray} \overline{BD}&=&2\sin{\left(\frac{\pi}{2}-2\theta\right)} \\\overline{CD}&=&2\cos{\left(\frac{\pi}{2}-2\theta\right)} \\\overline{AD}&=&\overline{AC}-\overline{CD} \\&=&5-2\cos{\left(\frac{\pi}{2}-2\theta\right)} \end{eqnarray}$$ $$\begin{eqnarray} \tan{\left(\theta\right)}&=&\frac{\overline{BD}}{\overline{AD}} \\&=&\frac{2\sin{\left(\frac{\pi}{2}-2\theta\right)}}{5-2\cos{\left(\frac{\pi}{2}-2\theta\right)}} \\&=&\frac{2\cos{\left(2\theta\right)}}{5-2\sin{\left(2\theta\right)}} \;\cdots\;\cos{\left(\frac{\pi}{2}-x\right)}=\sin{\left(x\right)},\;\sin{\left(\frac{\pi}{2}-x\right)}=\cos{\left(x\right)} \\&=&\frac{2}{5}\;\cdots\;\tan{\left(\theta\right)}=\frac{B\cos{\left(2\theta\right)}}{A-B\sin{\left(2\theta\right)}}ならば\tan{\left(\theta\right)}=\frac{B}{A}(下記) \end{eqnarray}$$

$$\begin{eqnarray} \tan{\left(\theta\right)}&=&\frac{B\cos{\left(2\theta\right)}}{A-B\sin{\left(2\theta\right)}} \\\frac{1}{\tan{\left(\theta\right)}}&=&\frac{A-B\sin{\left(2\theta\right)}}{B\cos{\left(2\theta\right)}} \\&=&\frac{1}{\cos{\left(2\theta\right)}}\left\{\frac{A}{B}-\frac{\cancel{B}}{\cancel{B}}\sin{\left(2\theta\right)}\right\} \\\frac{\cos{\left(2\theta\right)}}{\tan{\left(\theta\right)}}&=&\frac{A}{B}-\sin{\left(2\theta\right)} \\\frac{\cos{\left(2\theta\right)}}{\tan{\left(\theta\right)}}+\sin{\left(2\theta\right)}&=&\frac{A}{B} \\\frac{\cos{\left(2\theta\right)}+\tan{\left(\theta\right)}\sin{\left(2\theta\right)}}{\tan{\left(\theta\right)}}&=&\frac{A}{B} \\\frac{\cos^2{\left(\theta\right)}-\sin^2{\left(\theta\right)}+2\tan{\left(\theta\right)}\sin{\left(\theta\right)}\cos{\left(\theta\right)}}{\tan{\left(\theta\right)}}&=&\frac{A}{B} \;\cdots\;\cos{\left(2\theta\right)}=\cos^2{\left(\theta\right)}-\sin^2{\left(\theta\right)},\;\sin{\left(2\theta\right)}=2\sin{\left(\theta\right)}\cos{\left(\theta\right)} \\\frac{\cos^2{\left(\theta\right)}-\sin^2{\left(\theta\right)}+2\frac{\sin{\left(\theta\right)}}{\cancel{\cos{\left(\theta\right)}}}\sin{\left(\theta\right)}\cancel{\cos{\left(\theta\right)}}}{\tan{\left(\theta\right)}}&=&\frac{A}{B} \;\cdots\;\tan{\left(\theta\right)}=\frac{\sin{\left(\theta\right)}}{\cos{\left(\theta\right)}} \\\frac{\cos^2{\left(\theta\right)}-\sin^2{\left(\theta\right)}+2\sin^2{\left(\theta\right)}}{\tan{\left(\theta\right)}}&=&\frac{A}{B} \\\frac{\cos^2{\left(\theta\right)}+\sin^2{\left(\theta\right)}}{\tan{\left(\theta\right)}}&=&\frac{A}{B} \\\frac{1}{\tan{\left(\theta\right)}}&=&\frac{A}{B}\;\cdots\;\cos^2{\left(\theta\right)}+\sin^2{\left(\theta\right)}=1 \\\tan{\left(\theta\right)}&=&\frac{B}{A} \end{eqnarray}$$

極座標ラプラシアンの導出

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

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極座標ラプラシアンの導出

直交座標\((x,y,z)\)と極座標\((r,\theta,\psi)\)の関係式

$$\begin{eqnarray} \left\{ \begin{array}{l} x&=&r \sin{\theta}\cos{\psi}\;\cdots\;a \\y&=&r \sin{\theta}\sin{\psi}\;\cdots\;b \\z&=&r \cos{\theta}\;\cdots\;c \end{array} \right. \end{eqnarray}$$ $$\begin{eqnarray} \left\{ \begin{array}{l} r^2&=&x^2+y^2+z^2\;\cdots\;d \\\cos{\theta}&=&\frac{z}{r}\;\cdots\;e \\\tan{\psi}&=&\frac{y}{x}\;\cdots\;f \end{array} \right. \end{eqnarray}$$

\(\frac{\partial r}{\partial x}\)を求める

\(d\)の両辺を\(x\)の偏微分をとる． $$\begin{eqnarray} \frac{\partial }{\partial x}r^2&=&\frac{\partial }{\partial x}\left(x^2+y^2+z^2\right) \\2r\frac{\partial r}{\partial x}&=&2x \\\frac{\partial r}{\partial x} &=&\frac{\cancel{2}x}{\cancel{2}r}=\frac{x}{r}=\sin{\theta}\cos{\psi}\;\cdots\;aより \end{eqnarray}$$

\(\frac{\partial \theta}{\partial x}\)を求める

\(e\)の両辺を\(x\)の偏微分をとる． $$\begin{eqnarray} \frac{\partial }{\partial x}\cos{\theta}&=&\frac{\partial }{\partial x}\frac{z}{r} \\-\sin{\theta}\frac{\partial \theta}{\partial x}&=& z\frac{\partial }{\partial x}\frac{1}{r} \\&=&z\frac{\partial }{\partial r}\left(\frac{1}{r}\right)\frac{\partial r}{\partial x} \\&=&-\frac{z}{r^2}\frac{\partial r}{\partial x} \\&=&-\frac{z}{r^2}\;\sin{\theta}\cos{\psi}\;\cdots\;一つ前の\frac{\partial r}{\partial x}の結果 \\\frac{\partial \theta}{\partial x}&=&\frac{z\cos{\psi}}{r^2}=\frac{z}{r}\frac{\cos{\psi}}{r} \\&=&\frac{\cos{\theta}\cos{\psi}}{r}\;\cdots\;eより \end{eqnarray}$$

\(\frac{\partial \psi}{\partial x}\)を求める

\(f\)の両辺を\(x\)の偏微分をとる． $$\begin{eqnarray} \frac{\partial }{\partial x}\tan{\psi}&=&\frac{\partial }{\partial x}\frac{y}{x} \\\frac{1}{\cos^2{\psi}}\frac{\partial \psi}{\partial x} &=&y\frac{\partial }{\partial x}\frac{1}{x} \\&=&y\left(-\frac{1}{x^2}\right) \\&=&-\frac{y}{x}\frac{1}{x} \\&=&-\tan{\psi}\left(\frac{1}{x}\right) \\&=&-\tan{\psi}\left(\frac{1}{r\sin{\theta}\cos{\psi}}\right) \\\frac{\partial \psi}{\partial x} &=&-\tan{\psi}\left(\frac{1}{r\sin{\theta}\cos{\psi}}\right)\cos^2{\psi} \\&=&-\frac{\sin{\psi}}{\cancel{\cos{\psi}}}\left(\frac{1}{r\sin{\theta}\cancel{\cos{\psi}}}\right)\cancel{\cos^2{\psi}} \\&=&-\frac{\sin{\psi}}{r\sin{\theta}} \end{eqnarray}$$

\(\frac{\partial r}{\partial y}\)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． $$\begin{eqnarray} \frac{\partial }{\partial y}r^2&=&\frac{\partial }{\partial y}\left(x^2+y^2+z^2\right) \\2r\frac{\partial r}{\partial y}&=&2y \\\frac{\partial r}{\partial y} &=&\frac{\cancel{2}y}{\cancel{2}r}=\frac{y}{r}=\sin{\theta}\sin{\psi}\;\cdots\;bより \end{eqnarray}$$

\(\frac{\partial \theta}{\partial y}\)を求める

\(e\)の両辺を\(y\)の偏微分をとる． $$\begin{eqnarray} \frac{\partial }{\partial y}\cos{\theta}&=&\frac{\partial }{\partial y}\frac{z}{r} \\-\sin{\theta}\frac{\partial \theta}{\partial y}&=& z\frac{\partial }{\partial y}\frac{1}{r} \\&=&z\frac{\partial }{\partial r}\left(\frac{1}{r}\right)\frac{\partial r}{\partial y} \\&=&-\frac{z}{r^2}\frac{\partial r}{\partial y} \\&=&-\frac{z}{r^2}\;\sin{\theta}\sin{\psi}\;\cdots\;一つ前の\frac{\partial r}{\partial y}の結果 \\\frac{\partial \theta}{\partial y}&=&\left(\cancel{-}\frac{1}{\cancel{\sin{\theta}}}\right)\left(\cancel{-}\frac{z}{r^2}\cancel{\sin{\theta}}\sin{\psi}\right) =\frac{z}{r}\frac{\sin{\psi}}{r} \\&=&\frac{\cos{\theta}\sin{\psi}}{r}\;\cdots\;eより \end{eqnarray}$$

\(\frac{\partial \psi}{\partial y}\)を求める

\(f\)の両辺を\(y\)の偏微分をとる． $$\begin{eqnarray} \frac{\partial }{\partial y}\tan{\psi}&=&\frac{\partial }{\partial y}\frac{y}{x} \\\frac{1}{\cos^2{\psi}}\frac{\partial \psi}{\partial y} &=&\frac{1}{x} \\\frac{\partial \psi}{\partial y} &=&\left(\frac{1}{x}\right)\cos^2{\psi} \\&=&\left(\frac{1}{r\sin{\theta}\cancel{\cos{\psi}}}\right)\cos^\cancel{2}{\psi}\;\cdots\;aより \\&=&\frac{\cos{\psi}}{r\sin{\theta}} \end{eqnarray}$$

\(\frac{\partial r}{\partial z}\)を求める

\(d\)の両辺を\(z\)の偏微分をとる． $$\begin{eqnarray} \frac{\partial }{\partial z}r^2&=&\frac{\partial }{\partial z}\left(x^2+y^2+z^2\right) \\2r\frac{\partial r}{\partial z}&=&2z \\\frac{\partial r}{\partial z} &=&\frac{\cancel{2}z}{\cancel{2}r}=\frac{z}{r}=\cos{\theta}\;\cdots\;eより \end{eqnarray}$$

\(\frac{\partial \theta}{\partial z}\)を求める

\(e\)の両辺を\(z\)の偏微分をとる． $$\begin{eqnarray} \frac{\partial }{\partial z}\cos{\theta}&=&\frac{\partial }{\partial z}\frac{z}{r} \\-\sin{\theta}\frac{\partial \theta}{\partial z}&=& \frac{1}{r}+z\frac{\partial }{\partial z}\frac{1}{r} \\&=&\frac{1}{r}+z\frac{\partial }{\partial r}\left(\frac{1}{r}\right)\frac{\partial r}{\partial z} \\&=&\frac{1}{r}-\frac{z}{r^2}\frac{\partial r}{\partial z} \\&=&\frac{1}{r}-\frac{z}{r^2}\cos{\theta}\;\cdots\;一つ前の\frac{\partial r}{\partial z}の結果 \\\frac{\partial \theta}{\partial y} &=&\left(-\frac{1}{\sin{\theta}}\right)\left(\frac{1}{r}-\frac{z}{r^2}\cos{\theta}\right) \\&=&-\frac{1}{r\sin{\theta}}+\frac{z}{r}\frac{\cos{\theta}}{r\sin{\theta}} \\&=&-\frac{1}{r\sin{\theta}}+\frac{\cos^2{\theta}}{r\sin{\theta}}\;\cdots\;eより \\&=&-\frac{1-\cos^2{\theta}}{r\sin{\theta}} \\&=&-\frac{\sin^\cancel{2}{\theta}}{r\cancel{\sin{\theta}}} \\&=&-\frac{\sin{\theta}}{r} \end{eqnarray}$$

\(\frac{\partial \psi}{\partial z}\)を求める

\(f\)の両辺を\(z\)の偏微分をとる． $$\begin{eqnarray} \frac{\partial }{\partial z}\tan{\psi}&=&\frac{\partial }{\partial z}\frac{y}{x} \\\frac{1}{\cos^2{\psi}}\frac{\partial \psi}{\partial z}&=&0 \\\frac{\partial \psi}{\partial y}&=&0 \end{eqnarray}$$

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

$$\begin{eqnarray} \frac{\partial }{\partial x}&=& \frac{\partial r}{\partial x}\frac{\partial }{\partial r} +\frac{\partial \theta}{\partial x}\frac{\partial }{\partial \theta} +\frac{\partial \psi}{\partial x}\frac{\partial }{\partial \psi} \\&=&\sin{\theta}\cos{\psi}\frac{\partial }{\partial r} +\frac{\cos{\theta}\cos{\psi}}{r}\frac{\partial }{\partial \theta} -\frac{\sin{\psi}}{r\sin{\theta}}\frac{\partial }{\partial \psi} \\\; \frac{\partial }{\partial y}&=& \frac{\partial r}{\partial y}\frac{\partial }{\partial r} +\frac{\partial \theta}{\partial y}\frac{\partial }{\partial \theta} +\frac{\partial \psi}{\partial y}\frac{\partial }{\partial \psi} \\&=&\sin{\theta}\sin{\psi}\frac{\partial }{\partial r} +\frac{\cos{\theta}\sin{\psi}}{r}\frac{\partial }{\partial \theta} +\frac{\cos{\psi}}{r\sin{\theta}}\frac{\partial }{\partial \psi} \\\; \\\frac{\partial }{\partial z}&=& \frac{\partial r}{\partial z}\frac{\partial }{\partial r} +\frac{\partial \theta}{\partial z}\frac{\partial }{\partial \theta} +\frac{\partial \psi}{\partial z}\frac{\partial }{\partial \psi} \\&=&\cos{\theta}\frac{\partial }{\partial r} -\frac{\sin{\theta}}{r}\frac{\partial }{\partial \theta} +0\frac{\partial }{\partial \psi} \\&=&\cos{\theta}\frac{\partial }{\partial r} -\frac{\sin{\theta}}{r}\frac{\partial }{\partial \theta} \end{eqnarray}$$

\(\frac{\partial^2 }{\partial x^2}\)を求める

$$\begin{eqnarray} \frac{\partial^2 }{\partial x^2}&=& \frac{\partial }{\partial x}\frac{\partial }{\partial x} \\&=& \left( \sin{\theta}\cos{\psi}\frac{\partial }{\partial r} +\frac{\cos{\theta}\cos{\psi}}{r}\frac{\partial }{\partial \theta} -\frac{\sin{\psi}}{r\sin{\theta}}\frac{\partial }{\partial \psi} \right) \left( \sin{\theta}\cos{\psi}\frac{\partial }{\partial r} +\frac{\cos{\theta}\cos{\psi}}{r}\frac{\partial }{\partial \theta} -\frac{\sin{\psi}}{r\sin{\theta}}\frac{\partial }{\partial \psi} \right) \\&=&\sin{\theta}\cos{\psi} \color{red}{ \frac{\partial }{\partial r}\left( \sin{\theta}\cos{\psi}\frac{\partial }{\partial r} +\frac{\cos{\theta}\cos{\psi}}{r}\frac{\partial }{\partial \theta} -\frac{\sin{\psi}}{r\sin{\theta}}\frac{\partial }{\partial \psi} \right)} \\&&+\frac{\cos{\theta}\cos{\psi}}{r} \color{green}{ \frac{\partial }{\partial \theta}\left( \sin{\theta}\cos{\psi}\frac{\partial }{\partial r} +\frac{\cos{\theta}\cos{\psi}}{r}\frac{\partial }{\partial \theta} -\frac{\sin{\psi}}{r\sin{\theta}}\frac{\partial }{\partial \psi} \right) } \\&&-\frac{\sin{\psi}}{r\sin{\theta}} \color{blue}{ \frac{\partial }{\partial \psi}\left( \sin{\theta}\cos{\psi}\frac{\partial }{\partial r} +\frac{\cos{\theta}\cos{\psi}}{r}\frac{\partial }{\partial \theta} -\frac{\sin{\psi}}{r\sin{\theta}}\frac{\partial }{\partial \psi} \right)} \\&=&\sin{\theta}\cos{\psi}\cdot \color{red}{A_x}\color{black}{} +\frac{\cos{\theta}\cos{\psi}}{r}\cdot \color{green}{B_x}\color{black}{} -\frac{\sin{\psi}}{r\sin{\theta}}\cdot \color{blue}{C_x}\color{black}{} \end{eqnarray}$$

$$\begin{eqnarray} \\A_x&=& \frac{\partial }{\partial r}\left(\sin{\theta}\cos{\psi}\frac{\partial }{\partial r}\right) +\frac{\partial }{\partial r}\left(\frac{\cos{\theta}\cos{\psi}}{r}\frac{\partial }{\partial \theta}\right) -\frac{\partial }{\partial r}\left(\frac{\sin{\psi}}{r\sin{\theta}}\frac{\partial }{\partial \psi}\right) \\&=& \frac{\partial }{\partial r}\left(\sin{\theta}\cos{\psi}\right)\cdot\frac{\partial }{\partial r} +\sin{\theta}\cos{\psi}\cdot\frac{\partial }{\partial r}\left(\frac{\partial }{\partial r}\right) \\&&+\frac{\partial }{\partial r}\left(\frac{\cos{\theta}\cos{\psi}}{r}\right)\cdot\frac{\partial }{\partial \theta} +\frac{\cos{\theta}\cos{\psi}}{r}\cdot\frac{\partial }{\partial r}\left(\frac{\partial }{\partial \theta}\right) \\&&-\frac{\partial }{\partial r}\left(\frac{\sin{\psi}}{r\sin{\theta}}\right)\cdot\frac{\partial }{\partial \psi} -\frac{\sin{\psi}}{r\sin{\theta}}\cdot\frac{\partial }{\partial r}\left(\frac{\partial }{\partial \psi}\right) \\&=& \left(0\right)\frac{\partial }{\partial r} +\sin{\theta}\cos{\psi}\frac{\partial^2 }{\partial r^2} \\&&+\left(-\frac{\cos{\theta}\cos{\psi}}{r^2}\right)\frac{\partial }{\partial\theta} +\frac{\cos{\theta}\cos{\psi}}{r}\frac{\partial^2 }{\partial r \partial \theta} \\&&-\left(-\frac{\sin{\psi}}{r^2\sin{\theta}}\right)\cdot\frac{\partial }{\partial \psi} -\frac{\sin{\psi}}{r\sin{\theta}}\cdot\frac{\partial }{\partial r \partial \psi} \\&=& \sin{\theta}\cos{\psi}\frac{\partial^2 }{\partial r^2} -\frac{\cos{\theta}\cos{\psi}}{r^2}\frac{\partial }{\partial\theta} +\frac{\cos{\theta}\cos{\psi}}{r}\frac{\partial^2 }{\partial r \partial \theta} +\frac{\sin{\psi}}{r^2\sin{\theta}}\frac{\partial }{\partial \psi} -\frac{\sin{\psi}}{r\sin{\theta}}\frac{\partial^2 }{\partial r \partial \psi} \end{eqnarray}$$ $$\begin{eqnarray} \sin{\theta}\cos{\psi}\cdot A_x &=&\sin{\theta}\cos{\psi} \left\{ \sin{\theta}\cos{\psi}\frac{\partial^2 }{\partial r^2} -\frac{\cos{\theta}\cos{\psi}}{r^2}\frac{\partial }{\partial\theta} +\frac{\cos{\theta}\cos{\psi}}{r}\frac{\partial^2 }{\partial r \partial \theta} +\frac{\sin{\psi}}{r^2\sin{\theta}}\frac{\partial }{\partial \psi} -\frac{\sin{\psi}}{r\sin{\theta}}\frac{\partial^2 }{\partial r \partial \psi} \right\} \\&=& \sin^2{\theta}\cos^2{\psi}\frac{\partial^2 }{\partial r^2} -\frac{\cos{\theta}\sin{\theta}\cos^2{\psi}}{r^2}\frac{\partial }{\partial\theta} +\frac{\cos{\theta}\sin{\theta}\cos^2{\psi}}{r}\frac{\partial^2 }{\partial r \partial \theta} +\frac{\cancel{\sin{\theta}}\cos{\psi}\sin{\psi}}{r^2\cancel{\sin{\theta}}}\frac{\partial }{\partial \psi} -\frac{\cancel{\sin{\theta}}\cos{\psi}\sin{\psi}}{r\cancel{\sin{\theta}}}\frac{\partial^2 }{\partial r \partial \psi} \\&=& \sin^2{\theta}\cos^2{\psi}\frac{\partial^2 }{\partial r^2} -\frac{\cos{\theta}\sin{\theta}\cos^2{\psi}}{r^2}\frac{\partial }{\partial\theta} +\frac{\cos{\theta}\sin{\theta}\cos^2{\psi}}{r}\frac{\partial^2 }{\partial r \partial \theta} +\frac{\cos{\psi}\sin{\psi}}{r^2}\frac{\partial }{\partial \psi} -\frac{\cos{\psi}\sin{\psi}}{r}\frac{\partial^2 }{\partial r \partial \psi} \end{eqnarray}$$

$$\begin{eqnarray} \\B_x&=&\frac{\partial }{\partial \theta}\left(\sin{\theta}\cos{\psi}\frac{\partial }{\partial r}\right) +\frac{\partial }{\partial \theta}\left(\frac{\cos{\theta}\cos{\psi}}{r}\frac{\partial }{\partial \theta}\right) -\frac{\partial }{\partial \theta}\left(\frac{\sin{\psi}}{r\sin{\theta}}\frac{\partial }{\partial \psi}\right) \\&=& \frac{\partial }{\partial \theta}\left(\sin{\theta}\cos{\psi}\right)\cdot\frac{\partial }{\partial r} +\sin{\theta}\cos{\psi}\cdot\frac{\partial }{\partial \theta}\left(\frac{\partial }{\partial r}\right) \\&&+\frac{\partial }{\partial \theta}\left(\frac{\cos{\theta}\cos{\psi}}{r}\right) \cdot\frac{\partial}{\partial \theta} +\frac{\cos{\theta}\cos{\psi}}{r}\cdot\frac{\partial }{\partial \theta}\left(\frac{\partial }{\partial \theta}\right) \\&&-\frac{\partial }{\partial \theta}\left(\frac{\sin{\psi}}{r\sin{\theta}}\right) \cdot\frac{\partial }{\partial \psi} -\frac{\sin{\psi}}{r\sin{\theta}}\cdot\frac{\partial }{\partial \theta}\left(\frac{\partial }{\partial \psi}\right) \\&=& \left(\cos{\theta}\cos{\psi}\right)\frac{\partial }{\partial r} +\sin{\theta}\cos{\psi}\frac{\partial^2 }{\partial \theta \partial r} -\frac{\sin{\theta}\cos{\psi}}{r}\frac{\partial}{\partial \theta} +\frac{\cos{\theta}\cos{\psi}}{r}\frac{\partial^2 }{\partial \theta^2} +\frac{\cos{\theta}\sin{\psi}}{r\sin^2{\theta}}\frac{\partial }{\partial \psi} -\frac{\sin{\psi}}{r\sin{\theta}}\frac{\partial^2 }{\partial \theta \partial \psi} \end{eqnarray}$$ $$\begin{eqnarray} \frac{\cos{\theta}\cos{\psi}}{r}\cdot B_x &=&\frac{\cos{\theta}\cos{\psi}}{r}\left\{ \left(\cos{\theta}\cos{\psi}\right)\frac{\partial }{\partial r} +\sin{\theta}\cos{\psi}\frac{\partial^2 }{\partial \theta \partial r} -\frac{\sin{\theta}\cos{\psi}}{r}\frac{\partial}{\partial \theta} +\frac{\cos{\theta}\cos{\psi}}{r}\frac{\partial^2 }{\partial \theta^2} +\frac{\cos{\theta}\sin{\psi}}{r\sin^2{\theta}}\frac{\partial }{\partial \psi} -\frac{\sin{\psi}}{r\sin{\theta}}\frac{\partial^2 }{\partial \theta \partial \psi} \right\} \\&=&\frac{\cos^2{\theta}\cos^2{\psi}}{r}\frac{\partial }{\partial r} +\frac{\cos{\theta}\sin{\theta}\cos^2{\psi}}{r}\frac{\partial^2 }{\partial \theta \partial r} -\frac{\cos{\theta}\sin{\theta}\cos^2{\psi}}{r^2}\frac{\partial}{\partial \theta} +\frac{\cos^2{\theta}\cos^2{\psi}}{r^2}\frac{\partial^2 }{\partial \theta^2} +\frac{\cos^2{\theta}\cos{\psi}\sin{\psi}}{r^2\sin^2{\theta}}\frac{\partial }{\partial \psi} -\frac{\cos{\theta}\cos{\psi}\sin{\psi}}{r^2\sin{\theta}}\frac{\partial^2 }{\partial \theta \partial \psi} \end{eqnarray}$$

$$\begin{eqnarray} \\C_x&=& \frac{\partial }{\partial \psi}\left(\sin{\theta}\cos{\psi}\frac{\partial }{\partial r}\right) +\frac{\partial }{\partial \psi}\left(\frac{\cos{\theta}\cos{\psi}}{r}\frac{\partial }{\partial \theta}\right) -\frac{\partial }{\partial \psi}\left(\frac{\sin{\psi}}{r\sin{\theta}}\frac{\partial }{\partial \psi}\right) \\&=& \frac{\partial }{\partial \psi}\left(\sin{\theta}\cos{\psi}\right)\cdot\frac{\partial }{\partial r} +\sin{\theta}\cos{\psi}\cdot\frac{\partial }{\partial \psi}\left(\frac{\partial }{\partial r}\right) \\&&+\frac{\partial }{\partial \psi}\left(\frac{\cos{\theta}\cos{\psi}}{r}\right)\cdot\frac{\partial }{\partial \theta} +\frac{\cos{\theta}\cos{\psi}}{r}\cdot\frac{\partial }{\partial \psi}\left(\frac{\partial }{\partial \theta}\right) \\&&-\frac{\partial }{\partial \psi}\left(\frac{\sin{\psi}}{r\sin{\theta}}\right)\cdot\frac{\partial }{\partial \psi} -\frac{\sin{\psi}}{r\sin{\theta}}\cdot\frac{\partial }{\partial \psi}\left(\frac{\partial }{\partial \psi}\right) \\&=& \left(-\sin{\theta}\sin{\psi}\right)\frac{\partial }{\partial r} +\sin{\theta}\cos{\psi}\frac{\partial^2 }{\partial \psi\partial r} \\&&+\left(-\frac{\cos{\theta}\sin{\psi}}{r}\right)\frac{\partial }{\partial\theta} +\frac{\cos{\theta}\cos{\psi}}{r}\frac{\partial^2 }{\partial \psi \partial \theta} \\&&-\left(\frac{\cos{\psi}}{r\sin{\theta}}\right)\frac{\partial }{\partial\psi} -\frac{\sin{\psi}}{r\sin{\theta}}\cdot\frac{\partial^2 }{\partial \psi^2} \\&=& -\sin{\theta}\sin{\psi}\frac{\partial }{\partial r} +\sin{\theta}\cos{\psi}\frac{\partial^2 }{\partial \psi\partial r} -\frac{\cos{\theta}\sin{\psi}}{r}\frac{\partial }{\partial\theta} +\frac{\cos{\theta}\cos{\psi}}{r}\frac{\partial^2 }{\partial \psi \partial \theta} -\frac{\cos{\psi}}{r\sin{\theta}}\frac{\partial }{\partial\psi} -\frac{\sin{\psi}}{r\sin{\theta}}\frac{\partial^2 }{\partial \psi^2} \end{eqnarray}$$ $$\begin{eqnarray} -\frac{\sin{\psi}}{r\sin{\theta}}\cdot C_x &=&-\frac{\sin{\psi}}{r\sin{\theta}}\left\{ -\sin{\theta}\sin{\psi}\frac{\partial }{\partial r} +\sin{\theta}\cos{\psi}\frac{\partial^2 }{\partial \psi\partial r} -\frac{\cos{\theta}\sin{\psi}}{r}\frac{\partial }{\partial\theta} +\frac{\cos{\theta}\cos{\psi}}{r}\frac{\partial^2 }{\partial \psi \partial \theta} -\frac{\cos{\psi}}{r\sin{\theta}}\frac{\partial }{\partial\psi} -\frac{\sin{\psi}}{r\sin{\theta}}\frac{\partial^2 }{\partial \psi^2} \right\} \\&=&\frac{\cancel{\sin{\theta}}\sin^2{\psi}}{r\cancel{\sin{\theta}}}\frac{\partial }{\partial r} -\frac{\cancel{\sin{\theta}}\cos{\psi}\sin{\psi}}{r\cancel{\sin{\theta}}}\frac{\partial^2 }{\partial \psi\partial r} +\frac{\cos{\theta}\sin^2{\psi}}{r^2\sin{\theta}}\frac{\partial }{\partial\theta} -\frac{\cos{\theta}\cos{\psi}\sin{\psi}}{r^2\sin{\theta}}\frac{\partial^2 }{\partial \psi \partial \theta} +\frac{\cos{\psi}\sin{\psi}}{r^2\sin^2{\theta}}\frac{\partial }{\partial\psi} +\frac{\sin^2{\psi}}{r^2\sin^2{\theta}}\frac{\partial^2 }{\partial \psi^2} \\&=&\frac{\sin^2{\psi}}{r}\frac{\partial }{\partial r} -\frac{\cos{\psi}\sin{\psi}}{r}\frac{\partial^2 }{\partial \psi\partial r} +\frac{\cos{\theta}\sin^2{\psi}}{r^2\sin{\theta}}\frac{\partial }{\partial\theta} -\frac{\cos{\theta}\cos{\psi}\sin{\psi}}{r^2\sin{\theta}}\frac{\partial^2 }{\partial \psi \partial \theta} +\frac{\cos{\psi}\sin{\psi}}{r^2\sin^2{\theta}}\frac{\partial }{\partial\psi} +\frac{\sin^2{\psi}}{r^2\sin^2{\theta}}\frac{\partial^2 }{\partial \psi^2} \end{eqnarray}$$

$$\begin{eqnarray} \frac{\partial^2 }{\partial x^2}&=& \sin{\theta}\cos{\psi}\cdot A_x +\frac{\cos{\theta}\cos{\psi}}{r}\cdot B_x -\frac{\sin{\psi}}{r\sin{\theta}}\cdot C_x \\&=& \sin^2{\theta}\cos^2{\psi}\frac{\partial^2 }{\partial r^2} -\frac{\cos{\theta}\sin{\theta}\cos^2{\psi}}{r^2}\frac{\partial }{\partial\theta} +\frac{\cos{\theta}\sin{\theta}\cos^2{\psi}}{r}\frac{\partial^2 }{\partial r \partial \theta} +\frac{\cos{\psi}\sin{\psi}}{r^2}\frac{\partial }{\partial \psi} -\frac{\cos{\psi}\sin{\psi}}{r}\frac{\partial^2 }{\partial r\partial \psi} \\&&+\frac{\cos^2{\theta}\cos^2{\psi}}{r}\frac{\partial }{\partial r} +\frac{\cos{\theta}\sin{\theta}\cos^2{\psi}}{r}\frac{\partial^2 }{\partial \theta \partial r} -\frac{\cos{\theta}\sin{\theta}\cos^2{\psi}}{r^2}\frac{\partial}{\partial \theta} +\frac{\cos^2{\theta}\cos^2{\psi}}{r^2}\frac{\partial^2 }{\partial \theta^2} +\frac{\cos^2{\theta}\cos{\psi}\sin{\psi}}{r^2\sin^2{\theta}}\frac{\partial }{\partial \psi} -\frac{\cos{\theta}\cos{\psi}\sin{\psi}}{r^2\sin{\theta}}\frac{\partial^2 }{\partial \theta \partial \psi} \\&&+\frac{\sin^2{\psi}}{r}\frac{\partial }{\partial r} -\frac{\cos{\psi}\sin{\psi}}{r}\frac{\partial^2 }{\partial \psi\partial r} +\frac{\cos{\theta}\sin^2{\psi}}{r^2\sin{\theta}}\frac{\partial }{\partial\theta} -\frac{\cos{\theta}\cos{\psi}\sin{\psi}}{r^2\sin{\theta}}\frac{\partial^2 }{\partial \psi \partial \theta} +\frac{\cos{\psi}\sin{\psi}}{r^2\sin^2{\theta}}\frac{\partial }{\partial\psi} +\frac{\sin^2{\psi}}{r^2\sin^2{\theta}}\frac{\partial^2 }{\partial \psi^2} \\&=& \sin^2{\theta}\cos^2{\psi}\frac{\partial^2 }{\partial r^2} +\frac{\cos^2{\theta}\cos^2{\psi}}{r^2}\frac{\partial^2 }{\partial \theta^2} +\frac{\sin^2{\psi}}{r^2\sin^2{\theta}}\frac{\partial^2 }{\partial \psi^2} \\&&+\frac{\cos{\theta}\sin{\theta}\cos^2{\psi}}{r}\frac{\partial^2 }{\partial r \partial \theta} +\frac{\cos{\theta}\sin{\theta}\cos^2{\psi}}{r}\frac{\partial^2 }{\partial \theta \partial r} \\&&-\frac{\cos{\psi}\sin{\psi}}{r}\frac{\partial^2 }{\partial r\partial \psi} -\frac{\cos{\psi}\sin{\psi}}{r}\frac{\partial^2 }{\partial\psi\partial r} \\&&-\frac{\cos{\theta}\cos{\psi}\sin{\psi}}{r^2\sin{\theta}}\frac{\partial^2 }{\partial \theta \partial \psi} -\frac{\cos{\theta}\cos{\psi}\sin{\psi}}{r^2\sin{\theta}}\frac{\partial^2 }{\partial \psi \partial \theta} \\&& +\frac{\cos^2{\theta}\cos^2{\psi}}{r}\frac{\partial }{\partial r} +\frac{\sin^2{\psi}}{r}\frac{\partial }{\partial r} \\&& -\frac{\cos{\theta}\sin{\theta}\cos^2{\psi}}{r^2}\frac{\partial }{\partial\theta} -\frac{\cos{\theta}\sin{\theta}\cos^2{\psi}}{r^2}\frac{\partial}{\partial \theta} +\frac{\cos{\theta}\sin^2{\psi}}{r^2\sin{\theta}}\frac{\partial }{\partial\theta} \\&&+\frac{\cos{\psi}\sin{\psi}}{r^2}\frac{\partial }{\partial \psi} +\frac{\cos^2{\theta}\cos{\psi}\sin{\psi}}{r^2\sin^2{\theta}}\frac{\partial }{\partial \psi} +\frac{\cos{\psi}\sin{\psi}}{r^2\sin^2{\theta}}\frac{\partial }{\partial\psi} \\&=& \sin^2{\theta}\cos^2{\psi}\frac{\partial^2 }{\partial r^2} +\frac{\cos^2{\theta}\cos^2{\psi}}{r^2}\frac{\partial^2 }{\partial \theta^2} +\frac{\sin^2{\psi}}{r^2\sin^2{\theta}}\frac{\partial^2 }{\partial \psi^2} \\&&+2\frac{\cos{\theta}\sin{\theta}\cos^2{\psi}}{r} \frac{\partial^2}{\partial r \partial \theta} \;\cdots\;\frac{\partial^2 }{\partial r \partial \theta}=\frac{\partial^2 }{\partial \theta \partial r}を仮定 \\&&-2\frac{\cos{\psi}\sin{\psi}}{r}\frac{\partial^2 }{\partial r\partial \psi} \;\cdots\;\frac{\partial^2 }{\partial r \partial \psi}=\frac{\partial^2 }{\partial \psi \partial r}を仮定 \\&&-2\frac{\cos{\theta}\cos{\psi}\sin{\psi}}{r^2\sin{\theta}}\frac{\partial^2 }{\partial\theta\partial\psi} \;\cdots\;\frac{\partial^2 }{\partial\theta\partial\psi}=\frac{\partial^2 }{\partial\psi\partial\theta}を仮定 \\&&+\frac{\cos^2{\theta}\cos^2{\psi}+\sin^2{\psi}}{r}\frac{\partial }{\partial r} \\&& +\left( -2\frac{\cos{\theta}\sin{\theta}\cos^2{\psi}}{r^2} +\frac{\cos{\theta}\sin^2{\psi}}{r^2\sin{\theta}} \right)\frac{\partial }{\partial\theta} \\&&+\frac{\cos{\psi}\sin{\psi}}{r^2\sin^2{\theta}}\left( \sin^2{\theta} +\cos^2{\theta} +1 \right)\frac{\partial }{\partial\psi} \;\cdots\;\sin^2{\theta}+\cos^2{\theta}=1 \\&=& \sin^2{\theta}\cos^2{\psi}\frac{\partial^2 }{\partial r^2} +\frac{\cos^2{\theta}\cos^2{\psi}}{r^2}\frac{\partial^2 }{\partial \theta^2} +\frac{\sin^2{\psi}}{r^2\sin^2{\theta}}\frac{\partial^2 }{\partial \psi^2} \\&&+2\frac{\cos{\theta}\sin{\theta}\cos^2{\psi}}{r} \frac{\partial^2}{\partial r \partial \theta} -2\frac{\cos{\psi}\sin{\psi}}{r}\frac{\partial^2 }{\partial r\partial \psi} -2\frac{\cos{\theta}\cos{\psi}\sin{\psi}}{r^2\sin{\theta}}\frac{\partial^2 }{\partial\theta\partial\psi} \\&&+\frac{\cos^2{\theta}\cos^2{\psi}+\sin^2{\psi}}{r}\frac{\partial }{\partial r} +\left( -2\frac{\cos{\theta}\sin{\theta}\cos^2{\psi}}{r^2} +\frac{\cos{\theta}\sin^2{\psi}}{r^2\sin{\theta}} \right)\frac{\partial }{\partial\theta} +2\frac{\cos{\psi}\sin{\psi}}{r^2\sin^2{\theta}}\frac{\partial}{\partial\psi} \end{eqnarray}$$

\(\frac{\partial^2 }{\partial y^2}\)を求める

$$\begin{eqnarray} \frac{\partial^2 }{\partial y^2}&=& \frac{\partial }{\partial y}\frac{\partial }{\partial y} \\&=& \left( \sin{\theta}\sin{\psi}\frac{\partial }{\partial r} +\frac{\cos{\theta}\sin{\psi}}{r}\frac{\partial }{\partial \theta} +\frac{\cos{\psi}}{r\sin{\theta}}\frac{\partial }{\partial \psi} \right) \left( \sin{\theta}\sin{\psi}\frac{\partial }{\partial r} +\frac{\cos{\theta}\sin{\psi}}{r}\frac{\partial }{\partial \theta} +\frac{\cos{\psi}}{r\sin{\theta}}\frac{\partial }{\partial \psi} \right) \\&=& \sin{\theta}\sin{\psi}\frac{\partial }{\partial r} \color{red}{\left( \sin{\theta}\sin{\psi}\frac{\partial }{\partial r} +\frac{\cos{\theta}\sin{\psi}}{r}\frac{\partial }{\partial \theta} +\frac{\cos{\psi}}{r\sin{\theta}}\frac{\partial }{\partial \psi} \right)} \\&&+\frac{\cos{\theta}\sin{\psi}}{r} \color{green}{\frac{\partial }{\partial \theta}\left( \sin{\theta}\sin{\psi}\frac{\partial }{\partial r} +\frac{\cos{\theta}\sin{\psi}}{r}\frac{\partial }{\partial \theta} +\frac{\cos{\psi}}{r\sin{\theta}}\frac{\partial }{\partial \psi} \right)} \\&&+\frac{\cos{\psi}}{r\sin{\theta}} \color{blue}{\frac{\partial }{\partial \psi}\left( \sin{\theta}\sin{\psi}\frac{\partial }{\partial r} +\frac{\cos{\theta}\sin{\psi}}{r}\frac{\partial }{\partial \theta} +\frac{\cos{\psi}}{r\sin{\theta}}\frac{\partial }{\partial \psi} \right)} \\&=& \sin{\theta}\sin{\psi}\frac{\partial }{\partial r}\color{red}{A_y}\color{black}{} +\frac{\cos{\theta}\sin{\psi}}{r}\color{green}{B_y}\color{black}{} +\frac{\cos{\psi}}{r\sin{\theta}}\color{blue}{C_y}\color{black}{} \end{eqnarray}$$

$$\begin{eqnarray} A_y&=&\frac{\partial }{\partial r} \left( \sin{\theta}\sin{\psi}\frac{\partial }{\partial r} +\frac{\cos{\theta}\sin{\psi}}{r}\frac{\partial }{\partial \theta} +\frac{\cos{\psi}}{r\sin{\theta}}\frac{\partial }{\partial \psi} \right) \\&=&\frac{\partial }{\partial r}\left(\sin{\theta}\sin{\psi}\right)\cdot\frac{\partial }{\partial r} +\sin{\theta}\sin{\psi}\cdot\frac{\partial }{\partial r}\left(\frac{\partial }{\partial r}\right) \\&&+\frac{\partial }{\partial r}\left(\frac{\cos{\theta}\sin{\psi}}{r}\right) \cdot\frac{\partial }{\partial\theta} +\frac{\cos{\theta}\sin{\psi}}{r} \cdot \frac{\partial }{\partial r}\left( \frac{\partial }{\partial\theta} \right) \\&&+\frac{\partial }{\partial r}\left(\frac{\cos{\psi}}{r\sin{\theta}}\right)\cdot\frac{\partial }{\partial \psi} +\frac{\cos{\psi}}{r\sin{\theta}}\cdot\frac{\partial }{\partial r}\left(\frac{\partial }{\partial \psi}\right) \\&=&\left(0\right)\frac{\partial }{\partial r} +\sin{\theta}\sin{\psi}\frac{\partial^2 }{\partial r^2} \\&&+\left(-\frac{\cos{\theta}\sin{\psi}}{r^2}\right)\frac{\partial }{\partial \theta} +\frac{\cos{\theta}\sin{\psi}}{r}\frac{\partial^2 }{\partial r\partial\theta} \\&&+\left(-\frac{\cos{\psi}}{r^2\sin{\theta}}\right)\frac{\partial }{\partial\psi} +\frac{\cos{\psi}}{r\sin{\theta}}\frac{\partial^2 }{\partial r\partial\psi} \\&=&\sin{\theta}\sin{\psi}\frac{\partial^2 }{\partial r^2} -\frac{\cos{\theta}\sin{\psi}}{r^2}\frac{\partial }{\partial \theta} +\frac{\cos{\theta}\sin{\psi}}{r}\frac{\partial^2 }{\partial r\partial\theta} -\frac{\cos{\psi}}{r^2\sin{\theta}}\frac{\partial }{\partial\psi} +\frac{\cos{\psi}}{r\sin{\theta}}\frac{\partial^2 }{\partial r\partial\psi} \end{eqnarray}$$ $$\begin{eqnarray} \sin{\theta}\sin{\psi}\cdot A_y &=& \sin{\theta}\sin{\psi}\left( \sin{\theta}\sin{\psi}\frac{\partial^2 }{\partial r^2} -\frac{\cos{\theta}\sin{\psi}}{r^2}\frac{\partial }{\partial \theta} +\frac{\cos{\theta}\sin{\psi}}{r}\frac{\partial^2 }{\partial r\partial\theta} -\frac{\cos{\psi}}{r^2\sin{\theta}}\frac{\partial }{\partial\psi} +\frac{\cos{\psi}}{r\sin{\theta}}\frac{\partial^2 }{\partial r\partial\psi} \right) \\&=&\sin^2{\theta}\sin^2{\psi}\frac{\partial^2 }{\partial r^2} -\frac{\cos{\theta}\sin{\theta}\sin^2{\psi}}{r^2}\frac{\partial }{\partial \theta} +\frac{\cos{\theta}\sin{\theta}\sin^2{\psi}}{r}\frac{\partial^2 }{\partial r\partial\theta} -\frac{\cancel{\sin{\theta}}\cos{\psi}\sin{\psi}}{r^2\cancel{\sin{\theta}}}\frac{\partial }{\partial\psi} +\frac{\cancel{\sin{\theta}}\cos{\psi}\sin{\psi}}{r\cancel{\sin{\theta}}} \frac{\partial^2 }{\partial r\partial\psi} \\&=&\sin^2{\theta}\sin^2{\psi}\frac{\partial^2 }{\partial r^2} -\frac{\cos{\theta}\sin{\theta}\sin^2{\psi}}{r^2}\frac{\partial }{\partial \theta} +\frac{\cos{\theta}\sin{\theta}\sin^2{\psi}}{r}\frac{\partial^2 }{\partial r\partial\theta} -\frac{\cos{\psi}\sin{\psi}}{r^2}\frac{\partial }{\partial\psi} +\frac{\cos{\psi}\sin{\psi}}{r}\frac{\partial^2 }{\partial r\partial\psi} \end{eqnarray}$$

$$\begin{eqnarray} B_y&=&\frac{\partial }{\partial \theta}\left( \sin{\theta}\sin{\psi}\frac{\partial }{\partial r} +\frac{\cos{\theta}\sin{\psi}}{r}\frac{\partial }{\partial \theta} +\frac{\cos{\psi}}{r\sin{\theta}}\frac{\partial }{\partial \psi} \right) \\&=&\frac{\partial }{\partial \theta}\left(\sin{\theta}\sin{\psi}\right)\cdot\frac{\partial }{\partial r} +\sin{\theta}\sin{\psi}\cdot\frac{\partial }{\partial \theta}\left(\frac{\partial }{\partial r}\right) \\&&+\frac{\partial }{\partial \theta}\left(\frac{\cos{\theta}\sin{\psi}}{r}\right)\cdot\frac{\partial }{\partial \theta} +\frac{\cos{\theta}\sin{\psi}}{r} \cdot \frac{\partial }{\partial \theta}\left( \frac{\partial }{\partial\theta} \right) \\&&+\frac{\partial }{\partial \theta}\left(\frac{\cos{\psi}}{r\sin{\theta}}\right)\cdot\frac{\partial }{\partial \psi} +\frac{\cos{\psi}}{r\sin{\theta}}\cdot\frac{\partial }{\partial \theta}\left(\frac{\partial }{\partial \psi}\right) \\&=&\cos{\theta}\sin{\psi}\frac{\partial }{\partial r} +\sin{\theta}\sin{\psi}\frac{\partial^2 }{\partial \theta\partial r} -\frac{\sin{\theta}\sin{\psi}}{r}\frac{\partial}{\partial \theta} +\frac{\cos{\theta}\sin{\psi}}{r}\frac{\partial^2 }{\partial \theta^2} -\frac{\cos{\theta}\cos{\psi}}{r\sin^2{\theta}}\frac{\partial}{\partial\psi} +\frac{\cos{\psi}}{r\sin{\theta}}\frac{\partial^2 }{\partial \theta \partial\psi} \end{eqnarray}$$ $$\begin{eqnarray} \frac{\cos{\theta}\sin{\psi}}{r}\cdot B_y &=&\frac{\cos{\theta}\sin{\psi}}{r}\left( \cos{\theta}\sin{\psi}\frac{\partial }{\partial r} +\sin{\theta}\sin{\psi}\frac{\partial^2 }{\partial \theta\partial r} -\frac{\sin{\theta}\sin{\psi}}{r}\frac{\partial}{\partial \theta} +\frac{\cos{\theta}\sin{\psi}}{r}\frac{\partial^2 }{\partial \theta^2} -\frac{\cos{\theta}\cos{\psi}}{r\sin^2{\theta}}\frac{\partial}{\partial\psi} +\frac{\cos{\psi}}{r\sin{\theta}}\frac{\partial^2 }{\partial \theta \partial\psi} \right) \\&=& \frac{\cos^2{\theta}\sin^2{\psi}}{r}\frac{\partial }{\partial r} +\frac{\cos{\theta}\sin{\theta}\sin^2{\psi}}{r}\frac{\partial^2 }{\partial \theta\partial r} -\frac{\cos{\theta}\sin{\theta}\sin^2{\psi}}{r^2}\frac{\partial}{\partial \theta} +\frac{\cos^2{\theta}\sin^2{\psi}}{r^2}\frac{\partial^2 }{\partial \theta^2} -\frac{\cos^2{\theta}\cos{\psi}\sin{\psi}}{r^2\sin^2{\theta}}\frac{\partial}{\partial\psi} +\frac{\cos{\theta}\cos{\psi}\sin{\psi}}{r^2\sin{\theta}}\frac{\partial^2 }{\partial\theta\partial\psi} \end{eqnarray}$$

$$\begin{eqnarray} C_y&=&\frac{\partial }{\partial \psi}\left( \sin{\theta}\sin{\psi}\frac{\partial }{\partial r} +\frac{\cos{\theta}\sin{\psi}}{r}\frac{\partial }{\partial \theta} +\frac{\cos{\psi}}{r\sin{\theta}}\frac{\partial }{\partial \psi} \right) \\&=&\frac{\partial }{\partial \psi}\left(\sin{\theta}\sin{\psi}\right)\cdot\frac{\partial }{\partial r} +\sin{\theta}\sin{\psi}\cdot\frac{\partial }{\partial \psi}\left(\frac{\partial }{\partial r}\right) \\&&+\frac{\partial }{\partial \psi}\left(\frac{\cos{\theta}\sin{\psi}}{r}\right)\cdot\frac{\partial }{\partial \theta} +\frac{\cos{\theta}\sin{\psi}}{r} \cdot \frac{\partial }{\partial \psi}\left( \frac{\partial }{\partial\theta} \right) \\&&+\frac{\partial }{\partial \psi}\left(\frac{\cos{\psi}}{r\sin{\theta}}\right)\cdot\frac{\partial }{\partial \psi} +\frac{\cos{\psi}}{r\sin{\theta}}\cdot\frac{\partial }{\partial \psi}\left(\frac{\partial }{\partial \psi}\right) \\&=&\left(\sin{\theta}\cos{\psi}\right)\frac{\partial }{\partial r} +\sin{\theta}\sin{\psi}\frac{\partial^2 }{\partial \psi\partial r} \\&&+\left(\frac{\cos{\theta}\cos{\psi}}{r}\right)\frac{\partial}{\partial \theta} +\frac{\cos{\theta}\sin{\psi}}{r} \frac{\partial^2 }{\partial\psi\partial\theta} \\&&+\left(-\frac{\sin{\psi}}{r\sin{\theta}}\right)\frac{\partial}{\partial\psi} +\frac{\cos{\psi}}{r\sin{\theta}}\frac{\partial^2 }{\partial \psi^2} \\&=&\sin{\theta}\cos{\psi}\frac{\partial }{\partial r} +\sin{\theta}\sin{\psi}\frac{\partial^2 }{\partial \psi\partial r} +\frac{\cos{\theta}\cos{\psi}}{r}\frac{\partial}{\partial \theta} +\frac{\cos{\theta}\sin{\psi}}{r} \frac{\partial^2 }{\partial\psi\partial\theta} -\frac{\sin{\psi}}{r\sin{\theta}}\frac{\partial}{\partial\psi} +\frac{\cos{\psi}}{r\sin{\theta}}\frac{\partial^2 }{\partial \psi^2} \end{eqnarray}$$ $$\begin{eqnarray} \frac{\cos{\psi}}{r\sin{\theta}}\cdot C_y &=&\frac{\cos{\psi}}{r\sin{\theta}} \left( \sin{\theta}\cos{\psi}\frac{\partial }{\partial r} +\sin{\theta}\sin{\psi}\frac{\partial^2 }{\partial \psi\partial r} +\frac{\cos{\theta}\cos{\psi}}{r}\frac{\partial}{\partial \theta} +\frac{\cos{\theta}\sin{\psi}}{r} \frac{\partial^2 }{\partial\psi\partial\theta} -\frac{\sin{\psi}}{r\sin{\theta}}\frac{\partial}{\partial\psi} +\frac{\cos{\psi}}{r\sin{\theta}}\frac{\partial^2 }{\partial \psi^2} \right) \\&=& \frac{\cancel{\sin{\theta}}\cos^2{\psi}}{r\cancel{\sin{\theta}}}\frac{\partial }{\partial r} +\frac{\cancel{\sin{\theta}}\cos{\psi}\sin{\psi}}{r\cancel{\sin{\theta}}}\frac{\partial^2 }{\partial \psi\partial r} +\frac{\cos{\theta}\cos^2{\psi}}{r^2\sin{\theta}}\frac{\partial}{\partial \theta} +\frac{\cos{\theta}\cos{\psi}\sin{\psi}}{r^2\sin{\theta}} \frac{\partial^2}{\partial\psi\partial\theta} -\frac{\cos{\psi}\sin{\psi}}{r^2\sin^2{\theta}}\frac{\partial}{\partial\psi} +\frac{\cos^2{\psi}}{r^2\sin^2{\theta}}\frac{\partial^2 }{\partial \psi^2} \\&=& \frac{\cos^2{\psi}}{r}\frac{\partial }{\partial r} +\frac{\cos{\psi}\sin{\psi}}{r}\frac{\partial^2 }{\partial\psi\partial r} +\frac{\cos{\theta}\cos^2{\psi}}{r^2\sin{\theta}}\frac{\partial}{\partial \theta} +\frac{\cos{\theta}\cos{\psi}\sin{\psi}}{r^2\sin{\theta}} \frac{\partial^2}{\partial\psi\partial\theta} -\frac{\cos{\psi}\sin{\psi}}{r^2\sin^2{\theta}}\frac{\partial}{\partial\psi} +\frac{\cos^2{\psi}}{r^2\sin^2{\theta}}\frac{\partial^2 }{\partial \psi^2} \end{eqnarray}$$

$$\begin{eqnarray} \frac{\partial^2 }{\partial y^2} &=& \sin{\theta}\sin{\psi}\frac{\partial }{\partial r}A_y +\frac{\cos{\theta}\sin{\psi}}{r}B_y +\frac{\cos{\psi}}{r\sin{\theta}}C_y \\&=& \sin^2{\theta}\sin^2{\psi}\frac{\partial^2 }{\partial r^2} -\frac{\cos{\theta}\sin{\theta}\sin^2{\psi}}{r^2}\frac{\partial }{\partial \theta} +\frac{\cos{\theta}\sin{\theta}\sin^2{\psi}}{r}\frac{\partial^2 }{\partial r\partial\theta} -\frac{\cos{\psi}\sin{\psi}}{r^2}\frac{\partial }{\partial\psi} +\frac{\cos{\psi}\sin{\psi}}{r}\frac{\partial^2 }{\partial r\partial\psi} \\&&+\frac{\cos^2{\theta}\sin^2{\psi}}{r}\frac{\partial }{\partial r} +\frac{\cos{\theta}\sin{\theta}\sin^2{\psi}}{r}\frac{\partial^2 }{\partial \theta\partial r} -\frac{\cos{\theta}\sin{\theta}\sin^2{\psi}}{r^2}\frac{\partial}{\partial \theta} +\frac{\cos^2{\theta}\sin^2{\psi}}{r^2}\frac{\partial^2 }{\partial \theta^2} -\frac{\cos^2{\theta}\cos{\psi}\sin{\psi}}{r^2\sin^2{\theta}}\frac{\partial}{\partial\psi} +\frac{\cos{\theta}\cos{\psi}\sin{\psi}}{r^2\sin{\theta}}\frac{\partial^2 }{\partial\theta\partial\psi} \\&&+\frac{\cos^2{\psi}}{r}\frac{\partial }{\partial r} +\frac{\cos{\psi}\sin{\psi}}{r}\frac{\partial^2 }{\partial\psi\partial r} +\frac{\cos{\theta}\cos^2{\psi}}{r^2\sin{\theta}}\frac{\partial}{\partial \theta} +\frac{\cos{\theta}\cos{\psi}\sin{\psi}}{r^2\sin{\theta}} \frac{\partial^2}{\partial\psi\partial\theta} -\frac{\cos{\psi}\sin{\psi}}{r^2\sin^2{\theta}}\frac{\partial}{\partial\psi} +\frac{\cos^2{\psi}}{r^2\sin^2{\theta}}\frac{\partial^2 }{\partial \psi^2} \\&=& \sin^2{\theta}\sin^2{\psi}\frac{\partial^2 }{\partial r^2} +\frac{\cos^2{\theta}\sin^2{\psi}}{r^2}\frac{\partial^2 }{\partial \theta^2} +\frac{\cos^2{\psi}}{r^2\sin^2{\theta}}\frac{\partial^2 }{\partial \psi^2} \\&&+\frac{\cos{\theta}\sin{\theta}\sin^2{\psi}}{r}\frac{\partial^2 }{\partial r\partial\theta} +\frac{\cos{\theta}\sin{\theta}\sin^2{\psi}}{r}\frac{\partial^2 }{\partial \theta\partial r} \\&&+\frac{\cos{\psi}\sin{\psi}}{r}\frac{\partial^2 }{\partial r\partial\psi} +\frac{\cos{\psi}\sin{\psi}}{r}\frac{\partial^2 }{\partial\psi\partial r} \\&&+\frac{\cos{\theta}\cos{\psi}\sin{\psi}}{r^2\sin{\theta}}\frac{\partial^2 }{\partial\theta\partial\psi} +\frac{\cos{\theta}\cos{\psi}\sin{\psi}}{r^2\sin{\theta}} \frac{\partial^2}{\partial\psi\partial\theta} \\&& +\frac{\cos^2{\theta}\sin^2{\psi}}{r}\frac{\partial }{\partial r} +\frac{\cos^2{\psi}}{r}\frac{\partial }{\partial r} \\&&-\frac{\cos{\theta}\sin{\theta}\sin^2{\psi}}{r^2}\frac{\partial}{\partial \theta} -\frac{\cos{\theta}\sin{\theta}\sin^2{\psi}}{r^2}\frac{\partial}{\partial \theta} +\frac{\cos{\theta}\cos^2{\psi}}{r^2\sin{\theta}}\frac{\partial}{\partial \theta} \\&&-\frac{\cos{\psi}\sin{\psi}}{r^2}\frac{\partial }{\partial\psi} -\frac{\cos^2{\theta}\cos{\psi}\sin{\psi}}{r^2\sin^2{\theta}}\frac{\partial}{\partial\psi} -\frac{\cos{\psi}\sin{\psi}}{r^2\sin^2{\theta}}\frac{\partial}{\partial\psi} \\&=& \sin^2{\theta}\sin^2{\psi}\frac{\partial^2 }{\partial r^2} +\frac{\cos^2{\theta}\sin^2{\psi}}{r^2}\frac{\partial^2 }{\partial \theta^2} +\frac{\cos^2{\psi}}{r^2\sin^2{\theta}}\frac{\partial^2 }{\partial \psi^2} \\&&+2\frac{\cos{\theta}\sin{\theta}\sin^2{\psi}}{r}\frac{\partial^2 }{\partial r\partial\theta} \;\cdots\;\frac{\partial^2 }{\partial r \partial \theta}=\frac{\partial^2 }{\partial \theta \partial r}を仮定 \\&&+2\frac{\cos{\psi}\sin{\psi}}{r}\frac{\partial^2 }{\partial r\partial\psi} \;\cdots\;\frac{\partial^2 }{\partial r \partial \psi}=\frac{\partial^2 }{\partial \psi \partial r}を仮定 \\&&+2\frac{\cos{\theta}\cos{\psi}\sin{\psi}}{r^2\sin{\theta}}\frac{\partial^2 }{\partial\theta\partial\psi} \;\cdots\;\frac{\partial^2 }{\partial \theta \partial \psi}=\frac{\partial^2 }{\partial \psi \partial \theta}を仮定 \\&& +\frac{\cos^2{\theta}\sin^2{\psi}+\cos^2{\psi}}{r}\frac{\partial }{\partial r} \\&&+\left( -2\frac{\cos{\theta}\sin{\theta}\sin^2{\psi}}{r^2} +\frac{\cos{\theta}\cos^2{\psi}}{r^2\sin{\theta}} \right)\frac{\partial}{\partial \theta} \\&&-\frac{\cos{\psi}\sin{\psi}}{r^2\sin^2{\theta}}\left( \sin^2{\theta} +\cos^2{\theta} +1 \right)\frac{\partial}{\partial\psi} \;\cdots\;\sin^2{\theta}+\cos^2{\theta}=1 \\&=& \sin^2{\theta}\sin^2{\psi}\frac{\partial^2 }{\partial r^2} +\frac{\cos^2{\theta}\sin^2{\psi}}{r^2}\frac{\partial^2 }{\partial \theta^2} +\frac{\cos^2{\psi}}{r^2\sin^2{\theta}}\frac{\partial^2 }{\partial \psi^2} \\&&+2\frac{\cos{\theta}\sin{\theta}\sin^2{\psi}}{r}\frac{\partial^2 }{\partial r\partial\theta} +2\frac{\cos{\psi}\sin{\psi}}{r}\frac{\partial^2 }{\partial r\partial\psi} +2\frac{\cos{\theta}\cos{\psi}\sin{\psi}}{r^2\sin{\theta}}\frac{\partial^2 }{\partial\theta\partial\psi} \\&&+\frac{\cos^2{\theta}\sin^2{\psi}+\cos^2{\psi}}{r}\frac{\partial }{\partial r} +\left( -2\frac{\cos{\theta}\sin{\theta}\sin^2{\psi}}{r^2} +\frac{\cos{\theta}\cos^2{\psi}}{r^2\sin{\theta}} \right)\frac{\partial}{\partial \theta} -2\frac{\cos{\psi}\sin{\psi}}{r^2\sin^2{\theta}}\frac{\partial}{\partial\psi} \end{eqnarray}$$

\(\frac{\partial^2 }{\partial z^2}\)を求める

$$\begin{eqnarray} \frac{\partial^2 }{\partial z^2}&=& \frac{\partial }{\partial z}\frac{\partial }{\partial z} \\&=&\left( \cos{\theta}\frac{\partial }{\partial r} -\frac{\sin{\theta}}{r}\frac{\partial }{\partial \theta} \right) \left( \cos{\theta}\frac{\partial }{\partial r} -\frac{\sin{\theta}}{r}\frac{\partial }{\partial \theta} \right) \\&=& \cos{\theta}\color{red}{\frac{\partial }{\partial r} \left( \cos{\theta}\frac{\partial }{\partial r} -\frac{\sin{\theta}}{r}\frac{\partial }{\partial \theta} \right)}\color{black}{} -\frac{\sin{\theta}}{r}\color{green}{\frac{\partial }{\partial \theta} \left( \cos{\theta}\frac{\partial }{\partial r} -\frac{\sin{\theta}}{r}\frac{\partial }{\partial \theta} \right)} \\&=&\cos{\theta}\color{red}{A_z}\color{black}{}-\frac{\sin{\theta}}{r}\color{green}{B_z}\color{black}{} \end{eqnarray}$$

$$\begin{eqnarray} A_z&=&\frac{\partial }{\partial r} \left( \cos{\theta}\frac{\partial }{\partial r} -\frac{\sin{\theta}}{r}\frac{\partial }{\partial \theta} \right) \\&=& \frac{\partial }{\partial r} \left(\cos{\theta}\right)\cdot\frac{\partial }{\partial r} +\cos{\theta}\cdot\frac{\partial }{\partial r}\left(\frac{\partial }{\partial r}\right) \\&&+\frac{\partial }{\partial r}\left(-\frac{\sin{\theta}}{r}\right)\cdot\frac{\partial }{\partial \theta} -\frac{\sin{\theta}}{r}\cdot\frac{\partial }{\partial r}\left(\frac{\partial }{\partial \theta}\right) \\&=& \left(0\right)\frac{\partial }{\partial r} +\cos{\theta}\frac{\partial^2 }{\partial r^2} \\&&+\left(\frac{\sin{\theta}}{r^2}\right)\frac{\partial }{\partial \theta} -\frac{\sin{\theta}}{r}\frac{\partial^2 }{\partial r\partial \theta} \\&=& \cos{\theta}\frac{\partial^2 }{\partial r^2} +\frac{\sin{\theta}}{r^2}\frac{\partial }{\partial \theta} -\frac{\sin{\theta}}{r}\frac{\partial^2 }{\partial r\partial \theta} \end{eqnarray}$$ $$\begin{eqnarray} \cos{\theta}\cdot A_z&=&\cos{\theta}\left( \cos{\theta}\frac{\partial^2 }{\partial r^2} +\frac{\sin{\theta}}{r^2}\frac{\partial }{\partial \theta} -\frac{\sin{\theta}}{r}\frac{\partial^2 }{\partial r\partial \theta} \right) \\&=&\cos^2{\theta}\frac{\partial^2 }{\partial r^2} +\frac{\cos{\theta}\sin{\theta}}{r^2}\frac{\partial }{\partial \theta} -\frac{\cos{\theta}\sin{\theta}}{r}\frac{\partial^2 }{\partial r\partial \theta} \end{eqnarray}$$

$$\begin{eqnarray} B_z&=&\frac{\partial }{\partial \theta} \left( \cos{\theta}\frac{\partial }{\partial r} -\frac{\sin{\theta}}{r}\frac{\partial }{\partial \theta} \right) \\&=& \frac{\partial }{\partial \theta} \left(\cos{\theta}\right)\cdot\frac{\partial }{\partial r} +\cos{\theta}\cdot\frac{\partial }{\partial \theta}\left(\frac{\partial }{\partial r}\right) \\&&+\frac{\partial }{\partial \theta} \left(-\frac{\sin{\theta}}{r}\right)\cdot\frac{\partial }{\partial\theta} -\frac{\sin{\theta}}{r}\cdot\frac{\partial }{\partial \theta}\left(\frac{\partial }{\partial \theta}\right) \\&=& \left(-\sin{\theta}\right)\frac{\partial }{\partial r} +\cos{\theta}\frac{\partial^2 }{\partial \theta\partial r} \\&&+\left(-\frac{\cos{\theta}}{r}\right)\frac{\partial }{\partial\theta} -\frac{\sin{\theta}}{r}\frac{\partial^2 }{\partial \theta^2} \\&=& -\sin{\theta}\frac{\partial }{\partial r} +\cos{\theta}\frac{\partial^2 }{\partial \theta\partial r} -\frac{\cos{\theta}}{r}\frac{\partial }{\partial\theta} -\frac{\sin{\theta}}{r}\frac{\partial^2 }{\partial \theta^2} \end{eqnarray}$$ $$\begin{eqnarray} -\frac{\sin{\theta}}{r}B_z &=&-\frac{\sin{\theta}}{r}\left( -\sin{\theta}\frac{\partial }{\partial r} +\cos{\theta}\frac{\partial^2 }{\partial \theta\partial r} -\frac{\cos{\theta}}{r}\frac{\partial }{\partial\theta} -\frac{\sin{\theta}}{r}\frac{\partial^2 }{\partial \theta^2} \right) \\&=& \frac{\sin^2{\theta}}{r} \frac{\partial }{\partial r} -\frac{\cos{\theta}\sin{\theta}}{r} \frac{\partial^2 }{\partial \theta\partial r} +\frac{\cos{\theta}\sin{\theta}}{r^2} \frac{\partial }{\partial\theta} +\frac{\sin^2{\theta}}{r^2} \frac{\partial^2 }{\partial \theta^2} \end{eqnarray}$$

$$\begin{eqnarray} \frac{\partial^2 }{\partial z^2} &=& \cos{\theta}\cdot A_z-\frac{\sin{\theta}}{r}\cdot B_z \\&=& \cos^2{\theta}\frac{\partial^2 }{\partial r^2} +\frac{\cos{\theta}\sin{\theta}}{r^2}\frac{\partial }{\partial \theta} -\frac{\cos{\theta}\sin{\theta}}{r}\frac{\partial^2 }{\partial r\partial \theta} \\&&+\frac{\sin^2{\theta}}{r} \frac{\partial }{\partial r} -\frac{\cos{\theta}\sin{\theta}}{r} \frac{\partial^2 }{\partial \theta\partial r} +\frac{\cos{\theta}\sin{\theta}}{r^2} \frac{\partial }{\partial\theta} +\frac{\sin^2{\theta}}{r^2} \frac{\partial^2 }{\partial \theta^2} \\&=& \cos^2{\theta}\frac{\partial^2 }{\partial r^2} +\frac{\sin^2{\theta}}{r^2} \frac{\partial^2 }{\partial \theta^2} \\&&-\frac{\cos{\theta}\sin{\theta}}{r}\frac{\partial^2 }{\partial r\partial \theta} -\frac{\cos{\theta}\sin{\theta}}{r} \frac{\partial^2 }{\partial \theta\partial r} \\&&+\frac{\sin^2{\theta}}{r} \frac{\partial }{\partial r} \\&&+\frac{\cos{\theta}\sin{\theta}}{r^2} \frac{\partial }{\partial\theta} +\frac{\cos{\theta}\sin{\theta}}{r^2} \frac{\partial }{\partial\theta} \\&=& \cos^2{\theta}\frac{\partial^2 }{\partial r^2} +\frac{\sin^2{\theta}}{r^2}\frac{\partial^2 }{\partial \theta^2} \\&&-2\frac{\cos{\theta}\sin{\theta}}{r}\frac{\partial^2 }{\partial r\partial \theta} \;\cdots\;\frac{\partial^2 }{\partial r\partial \theta}=\frac{\partial^2 }{\partial \theta\partial r}を仮定 \\&&+\frac{\sin^2{\theta}}{r} \frac{\partial }{\partial r} \\&&+2\frac{\cos{\theta}\sin{\theta}}{r^2} \frac{\partial }{\partial\theta} \\&=& \cos^2{\theta}\frac{\partial^2 }{\partial r^2} +\frac{\sin^2{\theta}}{r^2}\frac{\partial^2 }{\partial \theta^2} -2\frac{\cos{\theta}\sin{\theta}}{r}\frac{\partial^2 }{\partial r\partial \theta} +\frac{\sin^2{\theta}}{r} \frac{\partial }{\partial r} +2\frac{\cos{\theta}\sin{\theta}}{r^2} \frac{\partial }{\partial\theta} \end{eqnarray}$$

\(\Delta\left(=\laplacian\right)\)を求める

$$\begin{eqnarray} \Delta&=& \frac{\partial^2 }{\partial x^2} +\frac{\partial^2 }{\partial y^2} +\frac{\partial^2 }{\partial z^2} \\&=& \sin^2{\theta}\cos^2{\psi}\frac{\partial^2 }{\partial r^2} +\frac{\cos^2{\theta}\cos^2{\psi}}{r^2}\frac{\partial^2 }{\partial \theta^2} +\frac{\sin^2{\psi}}{r^2\sin^2{\theta}}\frac{\partial^2 }{\partial \psi^2} +2\frac{\cos{\theta}\sin{\theta}\cos^2{\psi}}{r}\frac{\partial^2}{\partial r \partial \theta} -2\frac{\cos{\psi}\sin{\psi}}{r}\frac{\partial^2 }{\partial r\partial \psi} -2\frac{\cos{\theta}\cos{\psi}\sin{\psi}}{r^2\sin{\theta}}\frac{\partial^2 }{\partial\theta\partial\psi} +\frac{\cos^2{\theta}\cos^2{\psi}+\sin^2{\psi}}{r}\frac{\partial }{\partial r} +\left( -2\frac{\cos{\theta}\sin{\theta}\cos^2{\psi}}{r^2} +\frac{\cos{\theta}\sin^2{\psi}}{r^2\sin{\theta}} \right)\frac{\partial }{\partial\theta} +2\frac{\cos{\psi}\sin{\psi}}{r^2\sin^2{\theta}}\frac{\partial}{\partial\psi} \\&&+\sin^2{\theta}\sin^2{\psi}\frac{\partial^2 }{\partial r^2} +\frac{\cos^2{\theta}\sin^2{\psi}}{r^2}\frac{\partial^2 }{\partial \theta^2} +\frac{\cos^2{\psi}}{r^2\sin^2{\theta}}\frac{\partial^2 }{\partial \psi^2} +2\frac{\cos{\theta}\sin{\theta}\sin^2{\psi}}{r}\frac{\partial^2 }{\partial r\partial\theta} +2\frac{\cos{\psi}\sin{\psi}}{r}\frac{\partial^2 }{\partial r\partial\psi} +2\frac{\cos{\theta}\cos{\psi}\sin{\psi}}{r^2\sin{\theta}}\frac{\partial^2 }{\partial\theta\partial\psi} +\frac{\cos^2{\theta}\sin^2{\psi}+\cos^2{\psi}}{r}\frac{\partial }{\partial r} +\left( -2\frac{\cos{\theta}\sin{\theta}\sin^2{\psi}}{r^2} +\frac{\cos{\theta}\cos^2{\psi}}{r^2\sin{\theta}} \right)\frac{\partial}{\partial \theta} -2\frac{\cos{\psi}\sin{\psi}}{r^2\sin^2{\theta}}\frac{\partial}{\partial\psi} \\&&+\cos^2{\theta}\frac{\partial^2 }{\partial r^2} +\frac{\sin^2{\theta}}{r^2}\frac{\partial^2 }{\partial \theta^2} -2\frac{\cos{\theta}\sin{\theta}}{r}\frac{\partial^2 }{\partial r\partial \theta} +\frac{\sin^2{\theta}}{r} \frac{\partial }{\partial r} +2\frac{\cos{\theta}\sin{\theta}}{r^2} \frac{\partial }{\partial\theta} \\&=& \sin^2{\theta}\cos^2{\psi}\frac{\partial^2 }{\partial r^2} +\sin^2{\theta}\sin^2{\psi}\frac{\partial^2 }{\partial r^2} +\cos^2{\theta}\frac{\partial^2 }{\partial r^2} \\&& +\frac{\cos^2{\theta}\cos^2{\psi}}{r^2}\frac{\partial^2 }{\partial \theta^2} +\frac{\cos^2{\theta}\sin^2{\psi}}{r^2}\frac{\partial^2 }{\partial \theta^2} +\frac{\sin^2{\theta}}{r^2}\frac{\partial^2 }{\partial \theta^2} \\&& +\frac{\sin^2{\psi}}{r^2\sin^2{\theta}}\frac{\partial^2 }{\partial \psi^2} +\frac{\cos^2{\psi}}{r^2\sin^2{\theta}}\frac{\partial^2 }{\partial \psi^2} \\&& +2\frac{\cos{\theta}\sin{\theta}\cos^2{\psi}}{r}\frac{\partial^2}{\partial r\partial\theta} +2\frac{\cos{\theta}\sin{\theta}\sin^2{\psi}}{r}\frac{\partial^2}{\partial r\partial\theta} -2\frac{\cos{\theta}\sin{\theta}}{r}\frac{\partial^2}{\partial r\partial\theta} \\&& \cancel{-2\frac{\cos{\psi}\sin{\psi}}{r}\frac{\partial^2 }{\partial r\partial\psi}} \cancel{+2\frac{\cos{\psi}\sin{\psi}}{r}\frac{\partial^2 }{\partial r\partial\psi}} \\&& \cancel{ -2\frac{\cos{\theta}\cos{\psi}\sin{\psi}}{r^2\sin{\theta}}\frac{\partial^2 }{\partial\theta\partial\psi} } \cancel{ +2\frac{\cos{\theta}\cos{\psi}\sin{\psi}}{r^2\sin{\theta}}\frac{\partial^2 }{\partial\theta\partial\psi} } \\&& +\frac{\cos^2{\theta}\cos^2{\psi}+\sin^2{\psi}}{r}\frac{\partial }{\partial r} +\frac{\cos^2{\theta}\sin^2{\psi}+\cos^2{\psi}}{r}\frac{\partial }{\partial r} +\frac{\sin^2{\theta}}{r} \frac{\partial }{\partial r} \\&& +\left( -2\frac{\cos{\theta}\sin{\theta}\cos^2{\psi}}{r^2} +\frac{\cos{\theta}\sin^2{\psi}}{r^2\sin{\theta}} \right)\frac{\partial }{\partial\theta} +\left( -2\frac{\cos{\theta}\sin{\theta}\sin^2{\psi}}{r^2} +\frac{\cos{\theta}\cos^2{\psi}}{r^2\sin{\theta}} \right)\frac{\partial}{\partial \theta} +2\frac{\cos{\theta}\sin{\theta}}{r^2} \frac{\partial }{\partial\theta} \\&& \cancel{+2\frac{\cos{\psi}\sin{\psi}}{r^2\sin^2{\theta}}\frac{\partial}{\partial\psi}} \cancel{-2\frac{\cos{\psi}\sin{\psi}}{r^2\sin^2{\theta}}\frac{\partial}{\partial\psi}} \\&=& \sin^2{\theta}\left(\cos^2{\psi}+\sin^2{\psi}\right)\frac{\partial^2 }{\partial r^2} +\cos^2{\theta}\frac{\partial^2 }{\partial r^2} \;\cdots\;\cos^2{\psi}+\sin^2{\psi}=1 \\&& +\frac{\cos^2{\theta}}{r^2}\left( \cos^2{\psi} +\sin^2{\psi} \right)\frac{\partial^2 }{\partial \theta^2} +\frac{\sin^2{\theta}}{r^2}\frac{\partial^2 }{\partial \theta^2} \;\cdots\;\cos^2{\psi}+\sin^2{\psi}=1 \\&& +\frac{1}{r^2\sin^2{\theta}}\left(\sin^2{\psi}+\cos^2{\psi}\right)\frac{\partial^2 }{\partial \psi^2} \;\cdots\;\cos^2{\psi}+\sin^2{\psi}=1 \\&& +2\frac{\cos{\theta}\sin{\theta}}{r}\left( \cos^2{\psi}+\sin^2{\psi} \right)\frac{\partial^2}{\partial r\partial\theta} -2\frac{\cos{\theta}\sin{\theta}}{r}\frac{\partial^2}{\partial r\partial\theta} \;\cdots\;\cos^2{\psi}+\sin^2{\psi}=1 \\&& +\frac{ \cos^2{\theta}\left(\cos^2{\psi}+\sin^2{\psi}\right)+\left(\cos^2{\psi}+\sin^2{\psi}\right) +\sin^2{\theta} }{r}\frac{\partial }{\partial r} \;\cdots\;\cos^2{\psi}+\sin^2{\psi}=1 \\&& +\left\{ -2\frac{\cos{\theta}\sin{\theta}}{r^2}\left(\cos^2{\psi}+\sin^2{\psi}\right) +2\frac{\cos{\theta}\sin{\theta}}{r^2} +\frac{\cos{\theta}}{r^2\sin{\theta}}\left(\cos^2{\psi}+\sin^2{\psi}\right) \right\}\frac{\partial }{\partial\theta} \;\cdots\;\cos^2{\psi}+\sin^2{\psi}=1 \\&=& \left(\cos^2{\theta}+\sin^2{\theta}\right)\frac{\partial^2 }{\partial r^2} \\&& +\frac{1}{r^2}\left( \cos^2{\theta} +\sin^2{\theta} \right)\frac{\partial^2 }{\partial \theta^2} \;\cdots\;\cos^2{\theta}+\sin^2{\theta}=1 \\&& +\frac{1}{r^2\sin^2{\theta}}\frac{\partial^2 }{\partial \psi^2} \\&& \cancel{+2\frac{\cos{\theta}\sin{\theta}}{r}\frac{\partial^2}{\partial r\partial\theta}} \cancel{-2\frac{\cos{\theta}\sin{\theta}}{r}\frac{\partial^2}{\partial r\partial\theta}} \\&& +\frac{\cos^2{\theta}+\sin^2{\theta}+1}{r}\frac{\partial }{\partial r} \;\cdots\;\cos^2{\theta}+\sin^2{\theta}=1 \\&& +\left\{ \cancel{-2\frac{\cos{\theta}\sin{\theta}}{r^2}} \cancel{+2\frac{\cos{\theta}\sin{\theta}}{r^2}} +\frac{\cos{\theta}}{r^2\sin{\theta}} \right\}\frac{\partial }{\partial\theta} \\&=& \frac{\partial^2 }{\partial r^2} +\frac{1}{r^2}\frac{\partial^2 }{\partial \theta^2} +\frac{1}{r^2\sin^2{\theta}}\frac{\partial^2 }{\partial \psi^2} +\frac{2}{r}\frac{\partial }{\partial r} +\frac{\cos{\theta}}{r^2\sin{\theta}}\frac{\partial }{\partial\theta} \end{eqnarray}$$

$$\begin{eqnarray} \frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial }{\partial r}\right) &=& \frac{1}{r^2}\left\{ \frac{\partial }{\partial r}\left(r^2\right)\cdot\frac{\partial }{\partial r} +r^2\cdot\frac{\partial }{\partial r}\left(\frac{\partial }{\partial r}\right) \right\} \\&=& \frac{1}{r^2}\left\{ 2r\frac{\partial }{\partial r} +r^2\frac{\partial^2}{\partial r^2} \right\} \\&=& \frac{2}{r}\frac{\partial }{\partial r} +\frac{\cancel{r^2}}{\cancel{r^2}}\frac{\partial^2}{\partial r^2} \\&=& \frac{\partial^2}{\partial r^2} +\frac{2}{r}\frac{\partial }{\partial r} \end{eqnarray}$$

$$\begin{eqnarray} \frac{1}{r^2\sin{\theta}}\frac{\partial}{\partial \theta} \left( \sin{\theta}\frac{\partial }{\partial\theta} \right) &=& \frac{1}{r^2\sin{\theta}} \left\{ \frac{\partial }{\partial\theta}\left(\sin{\theta}\right)\cdot\frac{\partial }{\partial\theta} +\sin{\theta}\cdot\frac{\partial }{\partial\theta}\left(\frac{\partial }{\partial\theta}\right) \right\} \\&=& \frac{1}{r^2\sin{\theta}} \left\{ \cos{\theta}\frac{\partial }{\partial\theta} +\sin{\theta}\frac{\partial^2 }{\partial\theta^2} \right\} \\&=& \frac{\cos{\theta}}{r^2\sin{\theta}}\frac{\partial }{\partial\theta} +\frac{\cancel{\sin{\theta}}}{r^2\cancel{\sin{\theta}}}\frac{\partial^2 }{\partial\theta^2} \\&=& \frac{1}{r^2}\frac{\partial^2 }{\partial\theta^2} +\frac{\cos{\theta}}{r^2\sin{\theta}}\frac{\partial }{\partial\theta} \end{eqnarray}$$

$$\begin{eqnarray} \Delta&=& \frac{\partial^2 }{\partial r^2} +\frac{1}{r^2}\frac{\partial^2 }{\partial \theta^2} +\frac{1}{r^2\sin^2{\theta}}\frac{\partial^2 }{\partial \psi^2} +\frac{2}{r}\frac{\partial }{\partial r} +\frac{\cos{\theta}}{r^2\sin{\theta}}\frac{\partial }{\partial\theta} \\&=& \left\{ \frac{\partial^2 }{\partial r^2} +\frac{2}{r}\frac{\partial }{\partial r} \right\} +\left\{ \frac{1}{r^2}\frac{\partial^2 }{\partial \theta^2} +\frac{\cos{\theta}}{r^2\sin{\theta}}\frac{\partial }{\partial\theta} \right\} +\frac{1}{r^2\sin^2{\theta}}\frac{\partial^2 }{\partial \psi^2} \\&=& \frac{1}{r^2} \frac{\partial }{\partial r} \left( r^2\frac{\partial }{\partial r} \right) +\frac{1}{r^2\sin{\theta}}\frac{\partial}{\partial \theta} \left( \sin{\theta}\frac{\partial }{\partial\theta} \right) +\frac{1}{r^2\sin^2{\theta}}\frac{\partial^2 }{\partial \psi^2} \end{eqnarray}$$

sin(mx)sin(nx), cos(mx)cos(nx), sin(mx)cos(nx) の0から2πまでの積分

準備

準備の準備

$$\begin{eqnarray} \cos{\left(\alpha+\beta\right)}&=&\href{https://shikitenkai.blogspot.com/2019/06/blog-post.html}{ \cos{\left(\alpha\right)}\cos{\left(\beta\right)}-\sin{\left(\alpha\right)}\sin{\left(\beta\right)} } \\\cos{\left(\alpha-\beta\right)}&=&\cos{\left(\alpha\right)}\cos{\left(\beta\right)}+\sin{\left(\alpha\right)}\sin{\left(\beta\right)} \\\sin{\left(\alpha+\beta\right)}&=&\href{https://shikitenkai.blogspot.com/2019/06/blog-post.html}{ \sin{\left(\alpha\right)}\cos{\left(\beta\right)}+\cos{\left(\alpha\right)}\sin{\left(\beta\right)} } \\\sin{\left(\alpha-\beta\right)}&=&\sin{\left(\alpha\right)}\cos{\left(\beta\right)}-\cos{\left(\alpha\right)}\sin{\left(\beta\right)} \end{eqnarray}$$

準備1

$$\begin{eqnarray} \\\cos{\left(\alpha-\beta\right)}-\cos{\left(\alpha+\beta\right)}&=&\cos{\left(\alpha\right)}\cos{\left(\beta\right)}+\sin{\left(\alpha\right)}\sin{\left(\beta\right)} -\left\{\cos{\left(\alpha\right)}\cos{\left(\beta\right)}-\sin{\left(\alpha\right)}\sin{\left(\beta\right)}\right\} \;\cdots\;準備の準備 \\&=&\cancel{\cos{\left(\alpha\right)}\cos{\left(\beta\right)}}+\sin{\left(\alpha\right)}\sin{\left(\beta\right)} \cancel{-\cos{\left(\alpha\right)}\cos{\left(\beta\right)}}+\sin{\left(\alpha\right)}\sin{\left(\beta\right)} \\&=&2\sin{\left(\alpha\right)}\sin{\left(\beta\right)} \\\sin{\left(\alpha\right)}\sin{\left(\beta\right)}&=&\frac{\cos{\left(\alpha-\beta\right)}-\cos{\left(\alpha+\beta\right)}}{2} \end{eqnarray}$$

準備2

$$\begin{eqnarray} \\\cos{\left(\alpha-\beta\right)}+\cos{\left(\alpha+\beta\right)}&=&\cos{\left(\alpha\right)}\cos{\left(\beta\right)}+\sin{\left(\alpha\right)}\sin{\left(\beta\right)} +\left\{\cos{\left(\alpha\right)}\cos{\left(\beta\right)}-\sin{\left(\alpha\right)}\sin{\left(\beta\right)}\right\} \;\cdots\;準備の準備 \\&=&\cos{\left(\alpha\right)}\cos{\left(\beta\right)}+\cancel{\sin{\left(\alpha\right)}\sin{\left(\beta\right)}} +\cos{\left(\alpha\right)}\cos{\left(\beta\right)}-\cancel{\sin{\left(\alpha\right)}\sin{\left(\beta\right)}} \\&=&2\cos{\left(\alpha\right)}\cos{\left(\beta\right)} \\\cos{\left(\alpha\right)}\cos{\left(\beta\right)}&=&\frac{\cos{\left(\alpha-\beta\right)}+\cos{\left(\alpha+\beta\right)}}{2} \end{eqnarray}$$

準備3

$$\begin{eqnarray} \cos{\left(2x\right)}&=&\cos{\left(x+x\right)} \\&=&\cos(x)\cos(x)-\sin(x)\sin(x)\;\cdots\;準備の準備 \\&=&\cos^2{\left(x\right)}-\sin^2(x) \\&=&(1-\sin^2(x))-\sin^2(x) \\&=&1-2\sin^2(x) \\\cos{\left(2x\right)}-1&=&-2\sin^2(x) \\1-\cos{\left(2x\right)}&=&2\sin^2(x) \\\sin^2(x)&=&\frac{1-\cos(2x)}{2}=\frac{1}{2}-\frac{\cos(2x)}{2} \end{eqnarray}$$

準備4

$$\begin{eqnarray} \cos{\left(2x\right)}&=&\cos{\left(x+x\right)} \\&=&\cos(x)\cos(x)-\sin(x)\sin(x)\;\cdots\;準備の準備 \\&=&\cos^2{\left(x\right)}-\sin^2(x) \\&=&\cos^2{\left(x\right)}-\left(1-\cos^2{\left(x\right)}\right) \\&=&2\cos^2(x)-1 \\\cos{\left(2x\right)}+1&=&2\cos^2(x) \\\cos^2(x)&=&\frac{1+\cos(2x)}{2}=\frac{1}{2}+\frac{\cos(2x)}{2} \end{eqnarray}$$

準備5

$$\begin{eqnarray} \\\sin{\left(\alpha-\beta\right)}+\sin{\left(\alpha+\beta\right)}&=& \sin{\left(\alpha\right)}\cos{\left(\beta\right)}-\cos{\left(\alpha\right)}\sin{\left(\beta\right)} +\left\{\sin{\left(\alpha\right)}\cos{\left(\beta\right)}+\cos{\left(\alpha\right)}\sin{\left(\beta\right)}\right\} \;\cdots\;準備の準備 \\&=&\sin{\left(\alpha\right)}\cos{\left(\beta\right)}\cancel{-\cos{\left(\alpha\right)}\sin{\left(\beta\right)}} +\sin{\left(\alpha\right)}\cos{\left(\beta\right)}\cancel{+\cos{\left(\alpha\right)}\sin{\left(\beta\right)}} \\&=&2\sin{\left(\alpha\right)}\cos{\left(\beta\right)} \\\sin{\left(\alpha\right)}\cos{\left(\beta\right)}&=&\frac{\sin{\left(\alpha-\beta\right)}+\sin{\left(\alpha+\beta\right)}}{2} \end{eqnarray}$$

準備6

$$\begin{eqnarray} \sin{\left(2x\right)}&=&\sin{\left(x+x\right)} \\&=&\sin{\left(x\right)}\cos{\left(x\right)}+\cos{\left(x\right)}\sin{\left(x\right)} \;\cdots\;準備の準備 \\&=&2\sin{\left(x\right)}\cos{\left(x\right)} \\\sin{\left(x\right)}\cos{\left(x\right)}&=&\frac{\sin{\left(2x\right)}}{2} \end{eqnarray}$$

\(\sin{\left(m\;x\right)}\sin{\left(n\;x\right)}\)

\(m,n\in\mathbb{N},m\ne n\)

$$\begin{eqnarray} &&\int_{0}^{2\pi}\sin{\left(m\;x\right)}\sin{\left(n\;x\right)}\mathrm{d}x\;\cdots\;m,n\in\mathbb{N},m\ne n \\&=&\int_{0}^{2\pi}\frac{1}{2}\left[ \cos{\left(mx-nx\right)} -\cos{\left(mx+nx\right)} \right]\mathrm{d}x\;\cdots\;準備1 \\&=&\int_{0}^{2\pi}\frac{1}{2}\left[ \cos{\left(\left(m-n\right)\;x\right)} -\cos{\left(\left(m+n\right)\;x\right)} \right]\mathrm{d}x \\&=&\frac{1}{2}\left[ \int_{0}^{2\pi}\cos{\left(\left(m-n\right)\;x\right)}\mathrm{d}x -\int_{0}^{2\pi}\cos{\left(\left(m+n\right)\;x\right)}\mathrm{d}x \right] \\&=&\frac{1}{2}\left[ \left[\frac{1}{m-n}\sin{\left(\left(m-n\right)\;x\right)}\right]_{0}^{2\pi} -\left[\frac{1}{m+n}\sin{\left(\left(m+n\right)\;x\right)}\right]_{0}^{2\pi} \right] \\&=&\frac{1}{2}\left[ \frac{1}{m-n} \left[\sin{\left(\left(m-n\right)\;2\pi\right)} - \sin{\left(\left(m-n\right)\;0\right)}\right] -\frac{1}{m+n} \left[\sin{\left(\left(m+n\right)\;2\pi\right)} - \sin{\left(\left(m+n\right)\;0\right)}\right] \right] \\&=&\frac{1}{2}\left[ \frac{1}{m-n}\left[0-0\right] -\frac{1}{m+n}\left[0-0\right] \right] \\&=&\frac{1}{2}\left[ \frac{0}{m-n} -\frac{0}{m+n} \right] \\&=&\frac{1}{2}\;0 \\&=&0 \end{eqnarray}$$

\(m,n\in\mathbb{N},m= n\)

$$\begin{eqnarray} &&\int_{0}^{2\pi}\sin{\left(m\;x\right)}\sin{\left(n\;x\right)}\mathrm{d}x\;\cdots\;m,n\in\mathbb{N},m=n \\&=&\int_{0}^{2\pi}\sin{\left(m\;x\right)}\sin{\left(m\;x\right)}\mathrm{d}x \\&=&\int_{0}^{2\pi}\sin^2{\left(m\;x\right)}\mathrm{d}x \\&=&\frac{1}{m}\int_{0}^{2m\pi}\sin^2{\left(u\right)}\mathrm{d}u\;\cdots\;u=mx,\frac{\mathrm{d}u}{\mathrm{d}x}=m,\mathrm{d}x=\frac{1}{m}\mathrm{d}u \\&=&\frac{1}{m}\int_{0}^{2m\pi}\left[\frac{1}{2}-\frac{1}{2}\cos{\left(2u\right)}\right]\mathrm{d}u\;\cdots\;準備3 \\&=&\frac{1}{2m}\left[\int_{0}^{2m\pi}\mathrm{d}u-\int_{0}^{2m\pi}\cos{\left(2u\right)}\mathrm{d}u\right] \\&=&\frac{1}{2m}\left[\left[u\right]_{0}^{2m\pi}-\left[\frac{1}{2}\sin{\left(2u\right)}\right]_0^{2m\pi}\right] \\&=&\frac{1}{2m}\left[\left[2m\pi-0\right]-\left[\frac{1}{2}\sin{\left(2\cdot2m\pi\right)}-\frac{1}{2}\sin{\left(2\cdot0\right)}\right]\right] \\&=&\frac{1}{2m}\left[2m\pi-\left[0-0\right]\right] \\&=&\frac{1}{\cancel{2m}}\cancel{2m}\pi \\&=&\pi \end{eqnarray}$$

\(\cos{\left(m\;x\right)}\cos{\left(n\;x\right)}\)

\(m,n\in\mathbb{N},m\ne n\)

$$\begin{eqnarray} &&\int_{0}^{2\pi}\cos{\left(m\;x\right)}\cos{\left(n\;x\right)}\mathrm{d}x\;\cdots\;m,n\in\mathbb{N},m\ne n \\&=&\int_{0}^{2\pi}\frac{1}{2}\left[ \cos{\left(mx-nx\right)} +\cos{\left(mx+nx\right)} \right]\mathrm{d}x\;\cdots\;準備2 \\&=&\int_{0}^{2\pi}\frac{1}{2}\left[ \cos{\left(\left(m-n\right)\;x\right)} +\cos{\left(\left(m+n\right)\;x\right)} \right]\mathrm{d}x \\&=&\frac{1}{2}\left[ \int_{0}^{2\pi}\cos{\left(\left(m-n\right)\;x\right)}\mathrm{d}x +\int_{0}^{2\pi}\cos{\left(\left(m+n\right)\;x\right)}\mathrm{d}x \right] \\&=&\frac{1}{2}\left[ \left[\frac{1}{m-n}\sin{\left(\left(m-n\right)\;x\right)}\right]_{0}^{2\pi} +\left[\frac{1}{m+n}\sin{\left(\left(m+n\right)\;x\right)}\right]_{0}^{2\pi} \right] \\&=&\frac{1}{2}\left[ \frac{1}{m-n} \left[\sin{\left(\left(m-n\right)\;2\pi\right)} - \sin{\left(\left(m-n\right)\;0\right)}\right] +\frac{1}{m+n} \left[\sin{\left(\left(m+n\right)\;2\pi\right)} - \sin{\left(\left(m+n\right)\;0\right)}\right] \right] \\&=&\frac{1}{2}\left[ \frac{1}{m-n}\left[0-0\right] +\frac{1}{m+n}\left[0-0\right] \right] \\&=&\frac{1}{2}\left[ \frac{0}{m-n} +\frac{0}{m+n} \right] \\&=&\frac{1}{2}\;0 \\&=&0 \end{eqnarray}$$

\(m,n\in\mathbb{N},m= n\)

$$\begin{eqnarray} &&\int_{0}^{2\pi}\cos{\left(m\;x\right)}\cos{\left(n\;x\right)}\mathrm{d}x\;\cdots\;m,n\in\mathbb{N},m=n \\&=&\int_{0}^{2\pi}\cos{\left(m\;x\right)}\cos{\left(m\;x\right)}\mathrm{d}x \\&=&\int_{0}^{2\pi}\cos^2{\left(m\;x\right)}\mathrm{d}x \\&=&\frac{1}{m}\int_{0}^{2m\pi}\cos^2{\left(u\right)}\mathrm{d}u\;\cdots\;u=mx,\frac{\mathrm{d}u}{\mathrm{d}x}=m,\mathrm{d}x=\frac{1}{m}\mathrm{d}u \\&=&\frac{1}{m}\int_{0}^{2m\pi}\left[\frac{1}{2}+\frac{1}{2}\cos{\left(2u\right)}\right]\mathrm{d}u\;\cdots\;準備4 \\&=&\frac{1}{2m}\left[\int_{0}^{2m\pi}\mathrm{d}u+\int_{0}^{2m\pi}\cos{\left(2u\right)}\mathrm{d}u\right] \\&=&\frac{1}{2m}\left[\left[u\right]_{0}^{2m\pi}+\left[\frac{1}{2}\sin{\left(2u\right)}\right]_0^{2m\pi}\right] \\&=&\frac{1}{2m}\left[\left[2m\pi-0\right]+\left[\frac{1}{2}\sin{\left(2\cdot2m\pi\right)}-\frac{1}{2}\sin{\left(2\cdot0\right)}\right]\right] \\&=&\frac{1}{2m}\left[2m\pi+\left[0-0\right]\right] \\&=&\frac{1}{\cancel{2m}}\cancel{2m}\pi \\&=&\pi \end{eqnarray}$$

\(\sin{\left(m\;x\right)}\cos{\left(n\;x\right)}\)

\(m,n\in\mathbb{N},m\ne n\)

$$\begin{eqnarray} &&\int_{0}^{2\pi}\sin{\left(m\;x\right)}\cos{\left(n\;x\right)}\mathrm{d}x\;\cdots\;m,n\in\mathbb{N},m\ne n \\&=&\int_{0}^{2\pi}\frac{1}{2}\left[ \sin{\left(mx-nx\right)} +\sin{\left(mx+nx\right)} \right]\mathrm{d}x\;\cdots\;準備5 \\&=&\int_{0}^{2\pi}\frac{1}{2}\left[ \sin{\left((m-n)x\right)} +\sin{\left((m+n)x\right)} \right]\mathrm{d}x \\&=&\frac{1}{2}\left[ \int_{0}^{2\pi}\sin{\left((m-n)x\right)}\mathrm{d}x +\int_{0}^{2\pi}\sin{\left((m+n)x\right)}\mathrm{d}x \right] \\&=&\frac{1}{2}\left[ \left[ \frac{-1}{m-n} \cos{\left((m-n)x\right)} \right]_{0}^{2\pi} +\left[ \frac{-1}{m+n} \cos{\left((m+n)x\right)} \right]_{0}^{2\pi} \right] \\&=&\frac{1}{2}\left[ \frac{-1}{m-n}\left[ \cos{\left((m-n)2\pi\right)}-\cos{\left((m-n)0\right)} \right] +\frac{-1}{m+n}\left[ \cos{\left((m-n)2\pi\right)}-\cos{\left((m-n)0\right)} \right] \right] \\&=&\frac{1}{2}\left[ \frac{-1}{m-n}\left[1-1\right] +\frac{-1}{m+n}\left[1-1\right] \right] \\&=&\frac{1}{2}\left[ \frac{-1}{m-n}0 +\frac{-1}{m+n}0 \right] \\&=&\frac{1}{2}\left[ 0 +0 \right] \\&=&\frac{1}{2}0 \\&=&0 \end{eqnarray}$$

\(m,n\in\mathbb{N},m= n\)

$$\begin{eqnarray} &&\int_{0}^{2\pi}\sin{\left(m\;x\right)}\cos{\left(n\;x\right)}\mathrm{d}x\;\cdots\;m,n\in\mathbb{N},m= n \\&=&\int_{0}^{2\pi}\sin{\left(m\;x\right)}\cos{\left(m\;x\right)}\mathrm{d}x \\&=&\int_{0}^{2\pi}\frac{1}{2}\sin{\left(2m\;x\right)}\mathrm{d}x\;\cdots\;準備6 \\&=&\frac{1}{2}\int_{0}^{4m\pi}\frac{1}{2m}\sin{\left(u\right)}\mathrm{d}u\;\cdots\;u=2mx,\;\frac{\mathrm{d}u}{\mathrm{d}x}=2m,\;\mathrm{d}x=\frac{1}{2m}\mathrm{d}u \\&=&\frac{1}{4m}\left[-\cos{\left(u\right)}\right]_{0}^{4m\pi} \\&=&\frac{1}{4m}\left[-\cos{\left(4m\pi\right)}-\left(-\cos{\left(0\right)}\right)\right] \\&=&\frac{1}{4m}\left[-1-\left(-1\right)\right] \\&=&\frac{1}{4m}\left[-1+1\right] \\&=&\frac{1}{4m}0 \\&=&0 \end{eqnarray}$$

まとめ

$$\begin{eqnarray} m,n\in\mathbb{N} \\\int_{0}^{2\pi}\sin{\left(m\;x\right)}\sin{\left(n\;x\right)}\mathrm{d}x&=& \begin{cases} \pi\;\cdots\;m=n \\0\;\cdots\;m\ne n \end{cases} \\\int_{0}^{2\pi}\cos{\left(m\;x\right)}\cos{\left(n\;x\right)}\mathrm{d}x&=& \begin{cases} \pi\;\cdots\;m=n \\0\;\cdots\;m\ne n \end{cases} \\\int_{0}^{2\pi}\sin{\left(m\;x\right)}\cos{\left(n\;x\right)}\mathrm{d}x&=&0 \end{eqnarray}$$