合成関数の2次偏導関数

■問題

$3$\displaystyle{z=f $ x,y $ \text{\hspace{1}},\text{\hspace{1}}\text{\hspace{1}}x=r\cos {\theta}\text{\hspace{1}},\text{\hspace{1}}\text{\hspace{1}}y=r\sin {\theta}\text{\hspace{1}}\text{\hspace{1}}}$ のとき

$3$\displaystyle{\text{\hspace{1}}\text{\hspace{1}}\frac{\hspace{2} {{\partial}}^{2}z \hspace{2}}{\hspace{2} {\partial}{x}^{2} \hspace{2}}+\frac{\hspace{2} {{\partial}}^{2}z \hspace{2}}{\hspace{2} {\partial}{y}^{2} \hspace{2}}=\frac{\hspace{2} {{\partial}}^{2}z \hspace{2}}{\hspace{2} {\partial}{r}^{2} \hspace{2}}+\frac{\hspace{2}1\hspace{2}}{\hspace{2}r\hspace{2}}\frac{\hspace{2} {\partial}z \hspace{2}}{\hspace{2} {\partial}r \hspace{2}}+\frac{\hspace{2}1\hspace{2}}{\hspace{2} {r}^{2} \hspace{2}}\frac{\hspace{2} {{\partial}}^{2}z \hspace{2}}{\hspace{2} {\partial}{{\theta}}^{2} \hspace{2}}\text{\hspace{1}}\text{\hspace{1}}}$

となることを示せ．

■ヒント

右辺を変形して左辺を導く．

$3$\displaystyle{x}$ ， $3$\displaystyle{y}$ を $3$\displaystyle{r}$ と $3$\displaystyle{{\theta}}$ でそれぞれ2回偏微分する．求めた式を合成関数の1次偏導関数の公式と合成関数の2次偏導関数の公式に代入する．

■解説

$3$\displaystyle{x}$ を $3$\displaystyle{r}$ で偏微分すると

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

となる．これを更に $3$\displaystyle{r}$ で偏微分すると

$3$\displaystyle{\frac{\hspace{2} {{\partial}}^{2}x \hspace{2}}{\hspace{2} {\partial}{r}^{2} \hspace{2}}=\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}r \hspace{2}} $ \frac{\hspace{2} {\partial}x \hspace{2}}{\hspace{2} {\partial}r \hspace{2}} $ =\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}r \hspace{2}} $ \cos {\theta} $ =0}$

となる．

同様の手順で $3$\displaystyle{y}$ を $3$\displaystyle{r}$ で偏微分を繰り返すと

$3$\displaystyle{\frac{\hspace{2} {\partial}y \hspace{2}}{\hspace{2} {\partial}r \hspace{2}}=\sin {\theta}}$

$3$\displaystyle{\frac{\hspace{2} {{\partial}}^{2}y \hspace{2}}{\hspace{2} {\partial}{r}^{2} \hspace{2}}=\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}r \hspace{2}} $ \frac{\hspace{2} {\partial}y \hspace{2}}{\hspace{2} {\partial}r \hspace{2}} $ =\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}r \hspace{2}} $ \sin {\theta} $ =0}$

となる．

$3$\displaystyle{{\theta}}$ についても同様に偏微分すると

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

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

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

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

以上の結果と合成関数の1次偏導関数の公式と合成関数の2次偏導関数の公式より

$3$\displaystyle{\frac{\hspace{2} {{\partial}}^{2}z \hspace{2}}{\hspace{2} {\partial}{r}^{2} \hspace{2}}=\hspace{1}{f}_{ xx }{ $ \frac{\hspace{2} {\partial}x \hspace{2}}{\hspace{2} {\partial}r \hspace{2}} $ }^{2}+2\hspace{1}{f}_{ xy }\frac{\hspace{2} {\partial}x \hspace{2}}{\hspace{2} {\partial}r \hspace{2}}\frac{\hspace{2} {\partial}y \hspace{2}}{\hspace{2} {\partial}r \hspace{2}}+\hspace{1}{f}_{ yy }{ $ \frac{\hspace{2} {\partial}y \hspace{2}}{\hspace{2} {\partial}r \hspace{2}} $ }^{2}+\hspace{1}{f}_{x}\frac{\hspace{2} {{\partial}}^{2}x \hspace{2}}{\hspace{2} {\partial}{r}^{2} \hspace{2}}+\hspace{1}{f}_{y}\frac{\hspace{2} {{\partial}}^{2}y \hspace{2}}{\hspace{2} {\partial}{r}^{2} \hspace{2}}}$

$3$\displaystyle{=\hspace{1}{f}_{ xx }{ $ \cos {\theta} $ }^{2}+2\hspace{1}{f}_{ xy }\cos {\theta}\sin {\theta}+\hspace{1}{f}_{ yy }{ $ \sin {\theta} $ }^{2}+\hspace{1}{f}_{x}\text{\hspace{1}}\cdot \text{\hspace{1}}0+\hspace{1}{f}_{y}\text{\hspace{1}}\cdot \text{\hspace{1}}0}$

$3$\displaystyle{=\hspace{1}{f}_{ xx }{\cos }^{2}{\theta}+2\hspace{1}{f}_{ xy }\sin {\theta}\cos {\theta}+\hspace{1}{f}_{ yy }{\sin }^{2}{\theta}}$ 　･･････(1)

$3$\displaystyle{\frac{\hspace{2}1\hspace{2}}{\hspace{2}r\hspace{2}}\frac{\hspace{2} {\partial}z \hspace{2}}{\hspace{2} {\partial}r \hspace{2}}=\frac{\hspace{2}1\hspace{2}}{\hspace{2}r\hspace{2}} \{ \hspace{1}{f}_{x}\frac{\hspace{2} {\partial}x \hspace{2}}{\hspace{2} {\partial}r \hspace{2}}+\hspace{1}{f}_{y}\frac{\hspace{2} {\partial}y \hspace{2}}{\hspace{2} {\partial}r \hspace{2}} \} }$

$3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}r\hspace{2}} \{ \hspace{1}{f}_{x} $ r\cos {\theta} $ +\hspace{1}{f}_{y} $ r\sin {\theta} $ \} }$

$3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}r\hspace{2}} \{ r\hspace{1}{f}_{x}\cos {\theta}+r\hspace{1}{f}_{y}\sin {\theta} \} }$

$3$\displaystyle{=\hspace{1}{f}_{x}\cos {\theta}+\hspace{1}{f}_{y}\sin {\theta}}$ 　･･････(2)

$3$\displaystyle{\frac{\hspace{2}1\hspace{2}}{\hspace{2} {r}^{2} \hspace{2}}\frac{\hspace{2} {{\partial}}^{2}z \hspace{2}}{\hspace{2} {\partial}{{\theta}}^{2} \hspace{2}}=\frac{\hspace{2}1\hspace{2}}{\hspace{2} {r}^{2} \hspace{2}} \{ \hspace{1}{f}_{ xx }{ $ \frac{\hspace{2} {\partial}x \hspace{2}}{\hspace{2} {\partial}{\theta} \hspace{2}} $ }^{2}+2\hspace{1}{f}_{ xy }\frac{\hspace{2} {\partial}x \hspace{2}}{\hspace{2} {\partial}{\theta} \hspace{2}}\frac{\hspace{2} {\partial}y \hspace{2}}{\hspace{2} {\partial}{\theta} \hspace{2}}+\hspace{1}{f}_{ yy }{ $ \frac{\hspace{2} {\partial}y \hspace{2}}{\hspace{2} {\partial}{\theta} \hspace{2}} $ }^{2}+\hspace{1}{f}_{x}\frac{\hspace{2} {{\partial}}^{2}x \hspace{2}}{\hspace{2} {\partial}{{\theta}}^{2} \hspace{2}}+\hspace{1}{f}_{y}\frac{\hspace{2} {{\partial}}^{2}y \hspace{2}}{\hspace{2} {\partial}{{\theta}}^{2} \hspace{2}} \} }$

$3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2} {r}^{2} \hspace{2}} \{ \hspace{1}{f}_{ xx }{ $ - r\sin {\theta} $ }^{2}+2\hspace{1}{f}_{ xy } $ - r\sin {\theta} $ \cdot r\cos {\theta}+\hspace{1}{f}_{ yy }{ $ r\cos {\theta} $ }^{2}+\hspace{1}{f}_{x}\text{\hspace{1}}\cdot \text{\hspace{1}} $ - {r}^{2}\cos {\theta} $ +\hspace{1}{f}_{y}\text{\hspace{1}}\cdot \text{\hspace{1}} $ - {r}^{2}\sin {\theta} $ \} }$

$3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2} {r}^{2} \hspace{2}} $ {r}^{2}\hspace{1}{f}_{ xx }{ \sin }^{2}{\theta}- 2{r}^{2}\hspace{1}{f}_{ xy }\sin {\theta}\cos {\theta}+{r}^{2}\hspace{1}{f}_{ yy }{ \cos }^{2}{\theta}- {r}^{2}\hspace{1}{f}_{x}\cos {\theta}- {r}^{2}\hspace{1}{f}_{y}sin{\theta} $ }$

$3$\displaystyle{=\hspace{1}{f}_{ xx }{\sin }^{2}{\theta}- 2\hspace{1}{f}_{ xy }\sin {\theta}\cos {\theta}+\hspace{1}{f}_{ yy }{\cos }^{2}{\theta}- \hspace{1}{f}_{x}\cos {\theta}- \hspace{1}{f}_{y}sin{\theta}}$ 　･･････(3)

したがって，(1)，(2)，(3)より

$3$\displaystyle{\frac{\hspace{2} {{\partial}}^{2}z \hspace{2}}{\hspace{2} {\partial}{r}^{2} \hspace{2}}+\frac{\hspace{2}1\hspace{2}}{\hspace{2}r\hspace{2}}\frac{\hspace{2} {\partial}z \hspace{2}}{\hspace{2} {\partial}r \hspace{2}}+\frac{\hspace{2}1\hspace{2}}{\hspace{2} {r}^{2} \hspace{2}}\frac{\hspace{2} {{\partial}}^{2}z \hspace{2}}{\hspace{2} {\partial}{{\theta}}^{2} \hspace{2}}\text{\hspace{1}}\text{\hspace{1}}}$

$3$\displaystyle{= $ \hspace{1}{f}_{ xx }{ \cos }^{2}{\theta}+2\hspace{1}{f}_{ xy }\sin {\theta}\cos {\theta}+\hspace{1}{f}_{ yy }{ \sin }^{2}{\theta} $ + $ \hspace{1}{f}_{x}\cos {\theta}+\hspace{1}{f}_{y}\sin {\theta} $ }$

$3$\displaystyle{\text{\hspace{1}}+ $ \hspace{1}{f}_{ xx }{ \sin }^{2}{\theta}- 2\hspace{1}{f}_{ xy }\sin {\theta}\cos {\theta}+\hspace{1}{f}_{ yy }{ \cos }^{2}{\theta}- \hspace{1}{f}_{x}\cos {\theta}- \hspace{1}{f}_{y}sin{\theta} $ }$

$3$\displaystyle{=\hspace{1}{f}_{ xx } $ { \cos }^{2}{\theta}+{ \sin }^{2}{\theta} $ +\hspace{1}{f}_{ yy } $ { \cos }^{2}{\theta}+{ \sin }^{2}{\theta} $ }$

$3$\displaystyle{=\hspace{1}{f}_{ xx }+\hspace{1}{f}_{ yy }}$

$3$\displaystyle{=\frac{\hspace{2} {{\partial}}^{2}z \hspace{2}}{\hspace{2} {\partial}{x}^{2} \hspace{2}}+\frac{\hspace{2} {{\partial}}^{2}z \hspace{2}}{\hspace{2} {\partial}{y}^{2} \hspace{2}}}$

となり

が成り立つ．

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最終更新日： 2023年9月1日