$3$\displaystyle{ z }$ による関数 $3$\displaystyle{{\psi}}$ の2階偏微分

関数 $3$\displaystyle{{\psi}}$ を $3$\displaystyle{ z }$ で偏微分する．

$3$\displaystyle{ \frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}z \hspace{2}} }$ $3$\displaystyle{ =\frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}r \hspace{2}}\frac{\hspace{2} {\partial}r \hspace{2}}{\hspace{2} {\partial}z \hspace{2}}+ \frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}{\theta} \hspace{2}}\frac{\hspace{2} {\partial}{\theta} \hspace{2}}{\hspace{2} {\partial}z \hspace{2}}+ \frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}{\phi} \hspace{2}}\frac{\hspace{2} {\partial}{\phi} \hspace{2}}{\hspace{2} {\partial}z \hspace{2}} }$

$3$\displaystyle{ z }$ に関しても $3$\displaystyle{x}$ と同様に計算して

$3$\displaystyle{ \frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}{z}^{2} \hspace{2}} }$ $3$\displaystyle{ =\frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}{r}^{2} \hspace{2}}{\left({\frac{\hspace{2} {\partial}r \hspace{2}}{\hspace{2} {\partial}z \hspace{2}}}\right)}^{2} }$ $3$\displaystyle{ + \frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}{\theta}{\partial}r \hspace{2}}\frac{\hspace{2} {\partial}{\theta} \hspace{2}}{\hspace{2} {\partial}z \hspace{2}}\frac{\hspace{2} {\partial}r \hspace{2}}{\hspace{2} {\partial}z \hspace{2}} }$ $3$\displaystyle{ + \frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}r \hspace{2}}\frac{\hspace{2} {{\partial}}^{2}r \hspace{2}}{\hspace{2} {\partial}{z}^{2} \hspace{2}} }$ $3$\displaystyle{ + \frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}r{\partial}{\theta} \hspace{2}}\frac{\hspace{2} {\partial}r \hspace{2}}{\hspace{2} {\partial}z \hspace{2}}\frac{\hspace{2} {\partial}{\theta} \hspace{2}}{\hspace{2} {\partial}z \hspace{2}} }$ $3$\displaystyle{ + \frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}{{\theta}}^{2} \hspace{2}}{\left({\frac{\hspace{2} {\partial}{\theta} \hspace{2}}{\hspace{2} {\partial}z \hspace{2}}}\right)}^{2}+ \frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}{\theta} \hspace{2}}\frac{\hspace{2} {{\partial}}^{2}{\theta} \hspace{2}}{\hspace{2} {\partial}{z}^{2} \hspace{2}} }$ $3$\displaystyle{ + \frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}r{\partial}{\phi} \hspace{2}}\frac{\hspace{2} {\partial}r \hspace{2}}{\hspace{2} {\partial}z \hspace{2}}\frac{\hspace{2} {\partial}{\phi} \hspace{2}}{\hspace{2} {\partial}z \hspace{2}} }$ $3$\displaystyle{ + \frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}{{\phi}}^{2} \hspace{2}}{\left({\frac{\hspace{2} {\partial}{\phi} \hspace{2}}{\hspace{2} {\partial}z \hspace{2}}}\right)}^{2} }$ $3$\displaystyle{ + \frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}{\phi} \hspace{2}}\frac{\hspace{2} {{\partial}}^{2}{\phi} \hspace{2}}{\hspace{2} {\partial}{z}^{2} \hspace{2}} }$ 　･･････(1)

が得られる．(4)の両辺を $3$\displaystyle{ z }$ で偏微分して

$3$\displaystyle{ 2z }$ $3$\displaystyle{ =2r\frac{\hspace{2} {\partial}r \hspace{2}}{\hspace{2} {\partial}z \hspace{2}} }$

$3$\displaystyle{ \frac{\hspace{2}z\hspace{2}}{\hspace{2}r\hspace{2}} }$ $3$\displaystyle{ =\frac{\hspace{2} {\partial}r \hspace{2}}{\hspace{2} {\partial}z \hspace{2}} }$

$3$\displaystyle{ \frac{\hspace{2} {\partial}r \hspace{2}}{\hspace{2} {\partial}z \hspace{2}} }$ $3$\displaystyle{ =\frac{\hspace{2} r\cos {\theta} \hspace{2}}{\hspace{2}r\hspace{2}} }$

$3$\displaystyle{ \frac{\hspace{2} {\partial}r \hspace{2}}{\hspace{2} {\partial}z \hspace{2}} }$ $3$\displaystyle{ =\cos {\theta} }$ 　　･･････(2)

$3$\displaystyle{\frac{\hspace{2}y\hspace{2}}{\hspace{2}x\hspace{2}}=\tan {\phi}}$ (このページの(3)式) の両辺を $3$\displaystyle{ z }$ で偏微分して

$3$\displaystyle{ 0= \frac{\hspace{2}1\hspace{2}}{\hspace{2} {\cos }^{2}{\phi} \hspace{2}}\frac{\hspace{2} {\partial}{\phi} \hspace{2}}{\hspace{2} {\partial}z \hspace{2}} }$ $3$\displaystyle{}$

$3$\displaystyle{ \frac{\hspace{2} {\partial}{\phi} \hspace{2}}{\hspace{2} {\partial}z \hspace{2}} }$ $3$\displaystyle{ =0 }$ 　　･･････(3)

$3$\displaystyle{\frac{\hspace{2} \sqrt{ {x}^{2}+{y}^{2} } \hspace{2}}{\hspace{2}z\hspace{2}}=\tan {\theta}}$ (このページの(2)式) の両辺を $3$\displaystyle{ z }$ で偏微分して

$3$\displaystyle{ - \frac{\hspace{2}{ \left({ {x}^{2}+ {y}^{2} }\right) }^{ - \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} }\hspace{2}}{\hspace{2}{z}^{2}\hspace{2}} }$ $3$\displaystyle{ =\frac{\hspace{2}1\hspace{2}}{\hspace{2} {\cos }^{2}{\theta} \hspace{2}}\frac{\hspace{2} {\partial}{\theta} \hspace{2}}{\hspace{2} {\partial}z \hspace{2}} }$

$3$\displaystyle{ \frac{\hspace{2} {\partial}{\theta} \hspace{2}}{\hspace{2} {\partial}z \hspace{2}} }$ $3$\displaystyle{ =- \frac{\hspace{2} {\cos }^{2}{\theta}\sqrt{ {x}^{2}+ {y}^{2} } \hspace{2}}{\hspace{2}{z}^{2}\hspace{2}} }$ $3$\displaystyle{\displaystyle{=- \frac{\hspace{2} {\cos }^{2}{\theta}r\sin {\theta} \hspace{2}}{\hspace{2} {r}^{2}{\cos }^{2}{\theta} \hspace{2}}}}$ $3$\displaystyle{\displaystyle{=- \frac{\hspace{2} \sin {\theta} \hspace{2}}{\hspace{2}r\hspace{2}}}}$ 　　･･････(4)

が得られる

$3$\displaystyle{ \frac{\hspace{2} {{\partial}}^{2}r \hspace{2}}{\hspace{2} {\partial}{z}^{2} \hspace{2}} }$ $3$\displaystyle{=\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}z \hspace{2}}\left({ \frac{\hspace{2} {\partial}r \hspace{2}}{\hspace{2} {\partial}z \hspace{2}} }\right)}$

(2)を代入して

$3$\displaystyle{ =\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}z \hspace{2}}\cos {\theta} }$

$3$\displaystyle{ =\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}r \hspace{2}}\cos {\theta}\frac{\hspace{2} {\partial}r \hspace{2}}{\hspace{2} {\partial}z \hspace{2}}+ \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}{\theta} \hspace{2}}\cos {\theta}\frac{\hspace{2} {\partial}{\theta} \hspace{2}}{\hspace{2} {\partial}z \hspace{2}}+ \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}{\phi} \hspace{2}}\cos {\theta}\frac{\hspace{2} {\partial}{\phi} \hspace{2}}{\hspace{2} {\partial}z \hspace{2}} }$

$3$\displaystyle{ =- \sin {\theta}\frac{\hspace{2} {\partial}{\theta} \hspace{2}}{\hspace{2} {\partial}z \hspace{2}} }$

(4)を代入して

$3$\displaystyle{ =- \sin {\theta}\left({ - \frac{\hspace{2} \sin {\theta} \hspace{2}}{\hspace{2}r\hspace{2}} }\right) }$

$3$\displaystyle{\displaystyle{=\frac{\hspace{2} {\sin }^{2}{\theta} \hspace{2}}{\hspace{2}r\hspace{2}}}}$ 　　･･････(5)

が得られる．

$3$\displaystyle{ \frac{\hspace{2} {{\partial}}^{2}{\theta} \hspace{2}}{\hspace{2} {\partial}{z}^{2} \hspace{2}} }$

(4)を代入して

$3$\displaystyle{=\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}z \hspace{2}}\left({ \frac{\hspace{2} {\partial}{\theta} \hspace{2}}{\hspace{2} {\partial}z \hspace{2}} }\right)}$ $3$\displaystyle{ =\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}z \hspace{2}}\left({ - \frac{\hspace{2} \sin {\theta} \hspace{2}}{\hspace{2}r\hspace{2}} }\right) }$

$3$\displaystyle{ =\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}r \hspace{2}}\left({ - \frac{\hspace{2} \sin {\theta} \hspace{2}}{\hspace{2}r\hspace{2}} }\right)\frac{\hspace{2} {\partial}r \hspace{2}}{\hspace{2} {\partial}z \hspace{2}}+ \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}{\theta} \hspace{2}}\left({ - \frac{\hspace{2} \sin {\theta} \hspace{2}}{\hspace{2}r\hspace{2}} }\right)\frac{\hspace{2} {\partial}{\theta} \hspace{2}}{\hspace{2} {\partial}z \hspace{2}}+ \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}{\phi} \hspace{2}}\left({ - \frac{\hspace{2} \sin {\theta} \hspace{2}}{\hspace{2}r\hspace{2}} }\right)\frac{\hspace{2} {\partial}{\phi} \hspace{2}}{\hspace{2} {\partial}z \hspace{2}} }$

$3$\displaystyle{ =\frac{\hspace{2} \sin {\theta} \hspace{2}}{\hspace{2}{r}^{2}\hspace{2}}\frac{\hspace{2} {\partial}r \hspace{2}}{\hspace{2} {\partial}z \hspace{2}}- \frac{\hspace{2} \cos {\theta} \hspace{2}}{\hspace{2}r\hspace{2}}\frac{\hspace{2} {\partial}{\theta} \hspace{2}}{\hspace{2} {\partial}z \hspace{2}} }$

(2)，(4)を代入して

$3$\displaystyle{ =\frac{\hspace{2} \sin {\theta} \hspace{2}}{\hspace{2}{r}^{2}\hspace{2}}\cos {\theta}- \frac{\hspace{2} \cos {\theta} \hspace{2}}{\hspace{2}r\hspace{2}}\left({ - \frac{\hspace{2} \sin {\theta} \hspace{2}}{\hspace{2}r\hspace{2}} }\right) }$

$3$\displaystyle{ =\frac{\hspace{2} \sin {\theta}\cos {\theta} \hspace{2}}{\hspace{2}{r}^{2}\hspace{2}}+ \frac{\hspace{2} \sin {\theta}\cos {\theta} \hspace{2}}{\hspace{2}{r}^{2}\hspace{2}} }$

$3$\displaystyle{ =\frac{\hspace{2} 2\sin {\theta}\cos {\theta} \hspace{2}}{\hspace{2}{r}^{2}\hspace{2}} }$ 　　･･････(6)

が得られる．(3)より

$3$\displaystyle{ \frac{\hspace{2} {{\partial}}^{2}{\phi} \hspace{2}}{\hspace{2} {\partial}{z}^{2} \hspace{2}} }$ $3$\displaystyle{ =0 }$ 　　･･････(26)

が得られる．ここで，(21)～(26)を用いて(20)を式変形すると

$3$\displaystyle{ \frac{\hspace{2} {{\partial}}^{2}ψ \hspace{2}}{\hspace{2} {\partial}{z}^{2} \hspace{2}} }$
TeXに変換設定していない数学記号や，特殊文字が含まれています。今後直していきます。
$3$\displaystyle{ =\frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}{r}^{2} \hspace{2}}{\left({ \cos {\theta} }\right)}^{2}+ \frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}{\theta}{\partial}r \hspace{2}}\displaystyle{\left({ - \frac{\hspace{2} \sin {\theta} \hspace{2}}{\hspace{2}r\hspace{2}} }\right)\cos {\theta}+ \frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}r \hspace{2}}}\frac{\hspace{2} {\sin }^{2}{\theta} \hspace{2}}{\hspace{2}r\hspace{2}} }$

$3$\displaystyle{ + \frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}r{\partial}{\theta} \hspace{2}}\cos {\theta}\left({ - \frac{\hspace{2} \sin {\theta} \hspace{2}}{\hspace{2}r\hspace{2}} }\right)+ \frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}{{\theta}}^{2} \hspace{2}}{\left({ - \frac{\hspace{2} \sin {\theta} \hspace{2}}{\hspace{2}r\hspace{2}} }\right)}^{2}+ \frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}{\theta} \hspace{2}}\frac{\hspace{2} 2\sin {\theta}\cos {\theta} \hspace{2}}{\hspace{2}{r}^{2}\hspace{2}} }$

$3$\displaystyle{ + 0+ 0+ 0 }$

$3$\displaystyle{ =\frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}{r}^{2} \hspace{2}}{\cos }^{2}{\theta}- \frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}{\theta}{\partial}r \hspace{2}}\displaystyle{\frac{\hspace{2} \sin {\theta}\cos {\theta} \hspace{2}}{\hspace{2}r\hspace{2}}+ \frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}r \hspace{2}}}\frac{\hspace{2} {\sin }^{2}{\theta} \hspace{2}}{\hspace{2}r\hspace{2}} }$

$3$\displaystyle{ - \frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}r{\partial}{\theta} \hspace{2}}\frac{\hspace{2} \sin {\theta}\cos {\theta} \hspace{2}}{\hspace{2}r\hspace{2}}+\frac{\hspace{2} {{\partial}}^{2}{\psi} \hspace{2}}{\hspace{2} {\partial}{{\theta}}^{2} \hspace{2}}\frac{\hspace{2} {\sin }^{2}{\theta} \hspace{2}}{\hspace{2}{r}^{2}\hspace{2}}+ \frac{\hspace{2} {\partial}{\psi} \hspace{2}}{\hspace{2} {\partial}{\theta} \hspace{2}}\frac{\hspace{2} 2\sin {\theta}\cos {\theta} \hspace{2}}{\hspace{2}{r}^{2}\hspace{2}} }$

が得られる．

ホーム>>カテゴリー別分類>>微分>>偏微分>>極座標表示におけるラプラシアン(3次元)>> $3$\displaystyle{ z }$ による関数 $3$\displaystyle{{\psi}}$ の2階偏微分

最終更新日：2023年1月16日

による関数 の2階偏微分

$3$\displaystyle{ z }$ による関数 $3$\displaystyle{{\psi}}$ の2階偏微分