合成関数の2次偏導関数

■問題

$3$\displaystyle{z}$ に関する方程式

$3$\displaystyle{\frac{\hspace{2} {{\partial}}^{2}z \hspace{2}}{\hspace{2} {\partial}{t}^{2} \hspace{2}}={c}^{2} $ \frac{\hspace{2} {{\partial}}^{2}z \hspace{2}}{\hspace{2} {\partial}{r}^{2} \hspace{2}}+\frac{\hspace{2}2\hspace{2}}{\hspace{2}r\hspace{2}}\frac{\hspace{2} {\partial}z \hspace{2}}{\hspace{2} {\partial}r \hspace{2}} $ }$

において， $3$\displaystyle{z=\frac{\hspace{2}1\hspace{2}}{\hspace{2}r\hspace{2}}u}$ とおき， $3$\displaystyle{u}$ に関する方程式に変換すると

$3$\displaystyle{\frac{\hspace{2} {{\partial}}^{2}u \hspace{2}}{\hspace{2} {\partial}{t}^{2} \hspace{2}}={c}^{2}\frac{\hspace{2} {{\partial}}^{2}u \hspace{2}}{\hspace{2} {\partial}{r}^{2} \hspace{2}}}$

となることを示せ．

■ヒント

$3$\displaystyle{z}$ は $3$\displaystyle{r}$ と $3$\displaystyle{t}$ を変数とする2変数関数となる．である． $3$\displaystyle{z=\frac{\hspace{2}1\hspace{2}}{\hspace{2}r\hspace{2}}u}$ より $3$\displaystyle{u=rz}$ 変形する． $3$\displaystyle{u}$ も $3$\displaystyle{r}$ と $3$\displaystyle{t}$ を変数とする2変数関数となる． $3$\displaystyle{u}$ を $3$\displaystyle{t}$ と $3$\displaystyle{r}$ で偏微分する．このとき，合成関数の偏導関数の公式を用いる．

■解説

$3$\displaystyle{u=rz}$ において $3$\displaystyle{u}$ を $3$\displaystyle{t}$ で偏微分する．

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

さらに $3$\displaystyle{u}$ を， $3$\displaystyle{t}$ で偏微分する．

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

となる．よって

$3$\displaystyle{ \frac{\hspace{2} {{\partial}}^{2}z \hspace{2}}{\hspace{2} {\partial}{t}^{2} \hspace{2}}=\frac{\hspace{2}1\hspace{2}}{\hspace{2}r\hspace{2}}\frac{\hspace{2} {{\partial}}^{2}u \hspace{2}}{\hspace{2} {\partial}{t}^{2} \hspace{2}} }$ 　･･････(1)

となる．

次に， $3$\displaystyle{u=rz}$ において $3$\displaystyle{u}$ を $3$\displaystyle{r}$ で偏微分する．

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

さらに $3$\displaystyle{u}$ を， $3$\displaystyle{r}$ で偏微分する．

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

よって

$3$\displaystyle{\frac{\hspace{2} {{\partial}}^{2}z \hspace{2}}{\hspace{2} {\partial}{r}^{2} \hspace{2}}+\frac{\hspace{2}2\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}}\frac{\hspace{2} {{\partial}}^{2}u \hspace{2}}{\hspace{2} {\partial}{r}^{2} \hspace{2}}}$ 　･･････(2)

となる．したがって

に(1)，(2)を代入しすると

$3$\displaystyle{\frac{\hspace{2}1\hspace{2}}{\hspace{2}r\hspace{2}}\frac{\hspace{2} {{\partial}}^{2}u \hspace{2}}{\hspace{2} {\partial}{t}^{2} \hspace{2}}={c}^{2} $ \frac{\hspace{2}1\hspace{2}}{\hspace{2}r\hspace{2}}\frac{\hspace{2} {{\partial}}^{2}u \hspace{2}}{\hspace{2} {\partial}{r}^{2} \hspace{2}} $ }$

となる．

ホーム>>カテゴリー分類>>微分>>偏微分>>問題演習>>合成関数の2次偏導関数

学生スタッフ作成

最終更新日： 2023年9月1日