$y$ による関数 $ψ$ の2階偏微分

関数 $ψ$ を $y$ で偏微分する．

$\frac{\partial ψ}{\partial y}$ $= \frac{\partial ψ}{\partial r} \frac{\partial r}{\partial y} + \frac{\partial ψ}{\partial θ} \frac{\partial θ}{\partial y} + \frac{\partial ψ}{\partial φ} \frac{\partial φ}{\partial y}$

$y$ に関しても $x$ と同様に計算して

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

$+ 2 \frac{\partial^{2} ψ}{\partial r \partial θ} \frac{\partial r}{\partial y} \frac{\partial θ}{\partial y} + 2 \frac{\partial^{2} ψ}{\partial θ \partial φ} \frac{\partial θ}{\partial y} \frac{φ}{\partial y} + 2 \frac{\partial^{2} ψ}{\partial φ \partial r} \frac{\partial φ}{\partial y} \frac{\partial r}{\partial y}$

$+ \frac{\partial ψ}{\partial r} \frac{\partial^{2} r}{\partial y^{2}} + \frac{\partial ψ}{\partial θ} \frac{\partial^{2} θ}{\partial y^{2}} + \frac{\partial ψ}{\partial φ} \frac{\partial^{2} φ}{\partial y^{2}}$ ･･････(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

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

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

$\frac{\partial r}{\partial y}$ $= \frac{r \sin θ \sin φ}{r}$

$\frac{\partial r}{\partial y}$ $= \sin θ \sin φ$ ･･････(2)

が得られる． $\frac{y}{x} = \tan φ$ (このページの(3)式)の両辺を $y$ で偏微分して

$\frac{1}{x}$ $= \frac{1}{\cos^{2} φ} \frac{\partial φ}{\partial y}$

$(\tan^{2} φ + 1) \frac{\partial φ}{\partial y}$ $= \frac{1}{x}$

$(\frac{y^{2}}{x^{2}} + 1) \frac{\partial φ}{\partial y}$ $= \frac{1}{x}$

$(\frac{y^{2} + x^{2}}{x^{2}}) \frac{\partial φ}{\partial y}$ $= \frac{1}{x}$

$(\frac{r^{2} - z^{2}}{x^{2}}) \frac{\partial φ}{\partial y}$ $= \frac{1}{x}$

$\frac{\partial φ}{\partial y}$ $= \frac{x}{r^{2} - z^{2}}$ $= \frac{r \sin θ \cos φ}{r^{2} - r^{2} \cos^{2} θ}$ $= \frac{r \sin θ \cos φ}{r^{2} (1 - \cos^{2} θ)}$ $= \frac{r \sin θ \cos φ}{r^{2} \sin^{2} θ}$ $= \frac{\cos φ}{r \sin θ}$ ･･････(3)

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

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

$\frac{\partial θ}{\partial y}$ $= \frac{y \cos^{2} θ}{z \sqrt{x^{2} + y^{2}}}$

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

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

$= \frac{\cos θ \sin φ}{r}$ ･･････(4)

が得られる．

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

(2)を代入して

$= \frac{\partial}{\partial y} (\sin θ \sin φ)$

$= \frac{\partial}{\partial r} (\sin θ \sin φ) \frac{\partial r}{\partial y} + \frac{\partial}{\partial θ} (\sin θ \sin φ) \frac{\partial θ}{\partial y} + \frac{\partial}{\partial φ} (\sin θ \sin φ) \frac{\partial φ}{\partial y}$

$= \cos θ \sin φ \frac{\partial θ}{\partial y} + \sin θ \cos φ \frac{\partial φ}{\partial y}$

$= \cos θ \sin φ (\frac{\cos θ \sin φ}{r}) + \sin θ \cos φ (\frac{\cos φ}{r \sin θ})$

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

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

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

(4)を代入して

$= \frac{\partial}{\partial y} (\frac{\cos θ \cos φ}{r})$

$= \frac{\partial}{\partial r} (\frac{\cos θ \sin φ}{r}) \frac{\partial r}{\partial y} + \frac{\partial}{\partial θ} (\frac{\cos θ \sin φ}{r}) \frac{\partial θ}{\partial y} + \frac{\partial}{\partial φ} (\frac{\cos θ \sin φ}{r}) \frac{\partial φ}{\partial y}$

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

(2)､(4)､(3)を代入して

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

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

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

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

$= \frac{\cos θ (- 2 \sin^{2} θ \sin^{2} φ + \cos^{2} φ)}{r^{2} \sin θ}$ ･･････(6)

が得られる．

$\frac{\partial^{2} φ}{\partial y^{2}}$ $= \frac{\partial}{\partial y} (\frac{\partial φ}{\partial y})$

(3)を代入して

$= \frac{\partial}{\partial y} (\frac{\cos φ}{r \sin θ})$

$= \frac{\partial}{\partial r} (\frac{\cos φ}{r \sin θ}) \frac{\partial r}{\partial y} + \frac{\partial}{\partial θ} (\frac{\cos φ}{r \sin θ}) \frac{\partial θ}{\partial y} + \frac{\partial}{\partial φ} (\frac{\cos φ}{r \sin θ}) \frac{\partial φ}{\partial y}$

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

(2)，(4)，(3)を代入して

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

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

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

$= - \frac{2 \sin φ \cos φ}{r^{2} \sin^{2} θ}$ ･･････(7)

が得られる．ここで，(2)～(7)を(1)に代入し式を整理すると

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

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

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

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

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

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)>> $y$ による関数 $ψ$ の2階偏微分

最終更新日：2026年4月2日

y による関数 ψ の2階偏微分

$y$ による関数 $ψ$ の2階偏微分