$z$ による関数 $ψ$ の2階偏微分

関数 $ψ$ を $z$ で偏微分する．

$\frac{\partial ψ}{\partial z} = \frac{\partial ψ}{\partial r} \frac{\partial r}{\partial z} + \frac{\partial ψ}{\partial θ} \frac{\partial θ}{\partial z}$

$z$ に関しても $x$ と同様に計算して

$\frac{\partial^{2} ψ}{\partial z^{2}} = \frac{\partial}{\partial z} (\frac{\partial ψ}{\partial r} \frac{\partial r}{\partial z} + \frac{\partial ψ}{\partial θ} \frac{\partial θ}{\partial z})$

$= \{\frac{\partial}{\partial z} (\frac{\partial ψ}{\partial r})\} \frac{\partial r}{\partial z} + \frac{\partial ψ}{\partial r} \{\frac{\partial}{\partial z} (\frac{\partial r}{\partial z})\}$ $+ \{\frac{\partial}{\partial z} (\frac{\partial ψ}{\partial θ}) \frac{\partial θ}{\partial z}\} + \frac{\partial ψ}{\partial θ} \{\frac{\partial}{\partial z} (\frac{\partial θ}{\partial z})\}$

$= \{\frac{\partial}{\partial r} (\frac{\partial ψ}{\partial r}) \frac{\partial r}{\partial z} + \frac{\partial}{\partial θ} (\frac{\partial ψ}{\partial r}) \frac{\partial θ}{\partial z}\} \frac{\partial r}{\partial z} + \frac{\partial ψ}{\partial r} \frac{\partial^{2} r}{\partial z^{2}}$ $+ \{\frac{\partial}{\partial r} (\frac{\partial ψ}{\partial θ}) \frac{\partial r}{\partial z} + \frac{\partial}{\partial θ} (\frac{\partial ψ}{\partial θ}) \frac{\partial θ}{\partial z}\} \frac{\partial θ}{\partial z} + \frac{\partial ψ}{\partial θ} \frac{\partial^{2} θ}{\partial z^{2}}$

$= \frac{\partial^{2} ψ}{\partial r^{2}} {(\frac{\partial r}{\partial z})}^{2} + \frac{\partial^{2} ψ}{\partial θ \partial r} \frac{\partial θ}{\partial z} \frac{\partial r}{\partial z} + \frac{\partial ψ}{\partial r} \frac{\partial^{2} r}{\partial z^{2}} + \frac{\partial^{2} ψ}{\partial r \partial θ} \frac{\partial r}{\partial z} \frac{\partial θ}{\partial z} + \frac{\partial^{2} ψ}{\partial^{2} θ} {(\frac{\partial θ}{\partial z})}^{2} + \frac{\partial ψ}{\partial θ} \frac{\partial^{2} θ}{\partial z^{2}}$

$= \frac{\partial^{2} ψ}{\partial r^{2}} {(\frac{\partial r}{\partial z})}^{2} + \frac{\partial^{2} ψ}{\partial^{2} θ} {(\frac{\partial θ}{\partial z})}^{2} + 2 \frac{\partial^{2} ψ}{\partial r \partial θ} \frac{\partial r}{\partial z} \frac{\partial θ}{\partial z} + \frac{\partial ψ}{\partial r} \frac{\partial^{2} r}{\partial z^{2}} + \frac{\partial ψ}{\partial θ} \frac{\partial^{2} θ}{\partial z^{2}}$ 　･･････(1)

が得られる．

$x^{2} + y^{2} + z^{2} = r^{2}$ (このページの(4)式)の両辺を $z$ で偏微分して

$2 z$ $= 2 r \frac{\partial r}{\partial z}$

$\frac{z}{r}$ $= \frac{\partial r}{\partial z}$

$\frac{\partial r}{\partial z}$ $= \frac{r \cos θ}{r}$

$\frac{\partial r}{\partial z}$ $= \cos θ$ 　　･･････(2)

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

$- \frac{{(x^{2} + y^{2})}^{- \frac{1}{2}}}{z^{2}}$ $= \frac{1}{\cos^{2} θ} \frac{\partial θ}{\partial z}$

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

が得られる

$\frac{\partial^{2} r}{\partial z^{2}}$ $= \frac{\partial}{\partial z} (\frac{\partial r}{\partial z})$

(2)を代入して

$= \frac{\partial}{\partial z} \cos θ$

$= \frac{\partial}{\partial r} \cos θ \frac{\partial r}{\partial z} + \frac{\partial}{\partial θ} \cos θ \frac{\partial θ}{\partial z} + \frac{\partial}{\partial φ} \cos θ \frac{\partial φ}{\partial z}$

$= - \sin θ \frac{\partial θ}{\partial z}$

(3)を代入して

$= - \sin θ (- \frac{\sin θ}{r})$

$= \frac{\sin^{2} θ}{r}$ 　　･･････(4)

が得られる．

$\frac{\partial^{2} θ}{\partial z^{2}}$ $= \frac{\partial}{\partial z} (\frac{\partial θ}{\partial z})$

(3)を代入して

$= \frac{\partial}{\partial z} (- \frac{\sin θ}{r})$

$= \frac{\partial}{\partial r} (- \frac{\sin θ}{r}) \frac{\partial r}{\partial z} + \frac{\partial}{\partial θ} (- \frac{\sin θ}{r}) \frac{\partial θ}{\partial z}$

$= \frac{\sin θ}{r^{2}} \frac{\partial r}{\partial z} - \frac{\cos θ}{r} \frac{\partial θ}{\partial z}$

(2)，(3)を代入して

$= \frac{\sin θ}{r^{2}} \cos θ - \frac{\cos θ}{r} (- \frac{\sin θ}{r})$

$= \frac{\sin θ \cos θ}{r^{2}} + \frac{\sin θ \cos θ}{r^{2}}$

$= \frac{2sin θ \cos θ}{r^{2}}$ 　　･･････(5)

が得られる．

ここで，(2)～(5)を用いて(1)を式変形すると

$\frac{\partial^{2} ψ}{\partial z^{2}}$ $= \frac{\partial^{2} ψ}{\partial r^{2}} {(\cos θ)}^{2} + \frac{\partial^{2} ψ}{\partial^{2} θ} {(- \frac{\sin θ}{r})}^{2} + 2 \frac{\partial^{2} ψ}{\partial r \partial θ} (\cos θ) (- \frac{\sin θ}{r})$

$+ \frac{\partial ψ}{\partial r} (\frac{\sin^{2} θ}{r}) + \frac{\partial ψ}{\partial θ} (\frac{2 \sin θ \cos θ}{r^{2}})$

$= \cos^{2} θ \frac{\partial^{2} ψ}{\partial r^{2}} + \frac{\sin^{2} θ}{r^{2}} \frac{\partial^{2} ψ}{\partial^{2} θ} - \frac{2 \cos θ \sin θ}{r} \frac{\partial^{2} ψ}{\partial r \partial θ}$ $+ \frac{\sin^{2} θ}{r} \frac{\partial ψ}{\partial r} + \frac{2 \sin θ \cos θ}{r^{2}} \frac{\partial ψ}{\partial θ}$

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)>> $z$ による関数 $ψ$ の2階偏微分

最終更新日：2026年4月2日

z による関数 ψ の2階偏微分

$z$ による関数 $ψ$ の2階偏微分