# 合成関数の2次偏導関数

## ■問題

$z=f\left(x,y\right)\text{\hspace{0.17em}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}x=r\mathrm{cos}\theta \text{\hspace{0.17em}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}y=r\mathrm{sin}\theta \text{\hspace{0.17em}}\text{\hspace{0.17em}}$ のとき

$\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{{\partial }^{2}z}{\partial {x}^{2}}+\frac{{\partial }^{2}z}{\partial {y}^{2}}=\frac{{\partial }^{2}z}{\partial {r}^{2}}+\frac{1}{r}\frac{\partial z}{\partial r}+\frac{1}{{r}^{2}}\frac{{\partial }^{2}z}{\partial {\theta }^{2}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}$

となることを示せ．

## ■ヒント

$x$$y$$r$$\theta$でそれぞれ2回偏微分する． 求めた式を 合成関数の1次偏導関数の公式合成関数の2次偏導関数の公式に 代入する．

## ■解説

$x$$r$偏微分すると

$\frac{\partial x}{\partial r}=cos\theta$

となる．これを更に $r$偏微分すると

$\frac{{\partial }^{2}x}{\partial {r}^{2}}=\frac{\partial }{\partial r}\left(\frac{\partial x}{\partial r}\right)=\frac{\partial }{\partial r}\left(cos\theta \right)=0$

となる．

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

$\frac{{\partial }^{2}y}{\partial {r}^{2}}=\frac{\partial }{\partial r}\left(\frac{\partial y}{\partial r}\right)=\frac{\partial }{\partial r}\left(sin\theta \right)=0$

となる．

$\theta$ についても同様に偏微分すると

$\frac{\partial x}{\partial \theta }=-r\mathrm{sin}\theta$

$\frac{{\partial }^{2}x}{\partial {\theta }^{2}}=\frac{\partial }{\partial \theta }\left(\frac{\partial x}{\partial \theta }\right)$ $=\frac{\partial }{\partial \theta }\left(-r\mathrm{sin}\theta \right)=-r\mathrm{cos}\theta$

$\frac{\partial y}{\partial \theta }=r\mathrm{cos}\theta$

$\frac{{\partial }^{2}y}{\partial {\theta }^{2}}=\frac{\partial }{\partial \theta }\left(\frac{\partial y}{\partial \theta }\right)$ $=\frac{\partial }{\partial \theta }\left(-r\mathrm{cos}\theta \right)=-r\mathrm{sin}\theta$

$\frac{{\partial }^{2}z}{\partial {r}^{2}}={f}_{xx}{\left(\frac{\partial x}{\partial r}\right)}^{2}+2{f}_{xy}\frac{\partial x}{\partial r}\frac{\partial y}{\partial r}+{f}_{yy}{\left(\frac{\partial y}{\partial r}\right)}^{2}+{f}_{x}\frac{{\partial }^{2}x}{\partial {r}^{2}}+{f}_{y}\frac{{\partial }^{2}y}{\partial {r}^{2}}$

$={f}_{xx}{\left(\mathrm{cos}\theta \right)}^{2}+2{f}_{xy}\mathrm{cos}\theta \mathrm{sin}\theta +{f}_{yy}{\left(\mathrm{sin}\theta \right)}^{2}+{f}_{x}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}0+{f}_{y}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}0$

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

$\frac{1}{r}\frac{\partial z}{\partial r}=\frac{1}{r}\left\{{f}_{x}\frac{\partial x}{\partial r}+{f}_{y}\frac{\partial y}{\partial r}\right\}$

$=\frac{1}{r}\left\{{f}_{x}\left(r\mathrm{cos}\theta \right)+{f}_{y}\left(r\mathrm{sin}\theta \right)\right\}$

$=\frac{1}{r}\left\{r{f}_{x}\mathrm{cos}\theta +r{f}_{y}\mathrm{sin}\theta \right\}$

$={f}_{x}\mathrm{cos}\theta +{f}_{y}\mathrm{sin}\theta$　･･････(2)

$\frac{1}{{r}^{2}}\frac{{\partial }^{2}z}{\partial {\theta }^{2}}=\frac{1}{{r}^{2}}\left\{{f}_{xx}{\left(\frac{\partial x}{\partial \theta }\right)}^{2}+2{f}_{xy}\frac{\partial x}{\partial \theta }\frac{\partial y}{\partial \theta }+{f}_{yy}{\left(\frac{\partial y}{\partial \theta }\right)}^{2}+{f}_{x}\frac{{\partial }^{2}x}{\partial {\theta }^{2}}+{f}_{y}\frac{{\partial }^{2}y}{\partial {\theta }^{2}}\right\}$

$=\frac{1}{{r}^{2}}\left\{{f}_{xx}{\left(-r\mathrm{sin}\theta \right)}^{2}+2{f}_{xy}\left(-r\mathrm{sin}\theta \right)·r\mathrm{cos}\theta +{f}_{yy}{\left(r\mathrm{cos}\theta \right)}^{2}+{f}_{x}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\left(-{r}^{2}\mathrm{cos}\theta \right)+{f}_{y}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\left(-{r}^{2}\mathrm{sin}\theta \right)\right\}$

$=\frac{1}{{r}^{2}}\left({r}^{2}{f}_{xx}{\mathrm{sin}}^{2}\theta -2{r}^{2}{f}_{xy}\mathrm{sin}\theta \mathrm{cos}\theta +{r}^{2}{f}_{yy}{\mathrm{cos}}^{2}\theta -{r}^{2}{f}_{x}\mathrm{cos}\theta -{r}^{2}{f}_{y}sin\theta \right)$

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

したがって，(1)，(2)，(3)より

$\frac{{\partial }^{2}z}{\partial {r}^{2}}+\frac{1}{r}\frac{\partial z}{\partial r}+\frac{1}{{r}^{2}}\frac{{\partial }^{2}z}{\partial {\theta }^{2}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}$

$=\left({f}_{xx}{\mathrm{cos}}^{2}\theta +2{f}_{xy}\mathrm{sin}\theta \mathrm{cos}\theta +{f}_{yy}{\mathrm{sin}}^{2}\theta \right)+\left({f}_{x}\mathrm{cos}\theta +{f}_{y}\mathrm{sin}\theta \right)$

$\text{\hspace{0.17em}}+\left({f}_{xx}{\mathrm{sin}}^{2}\theta -2{f}_{xy}\mathrm{sin}\theta \mathrm{cos}\theta +{f}_{yy}{\mathrm{cos}}^{2}\theta -{f}_{x}\mathrm{cos}\theta -{f}_{y}sin\theta \right)$

$={f}_{xx}\left({\mathrm{cos}}^{2}\theta +{\mathrm{sin}}^{2}\theta \right)+{f}_{yy}\left({\mathrm{cos}}^{2}\theta +{\mathrm{sin}}^{2}\theta \right)$

$={f}_{xx}+{f}_{yy}$

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

となり

$\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{{\partial }^{2}z}{\partial {x}^{2}}+\frac{{\partial }^{2}z}{\partial {y}^{2}}=\frac{{\partial }^{2}z}{\partial {r}^{2}}+\frac{1}{r}\frac{\partial z}{\partial r}+\frac{1}{{r}^{2}}\frac{{\partial }^{2}z}{\partial {\theta }^{2}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}$

が成り立つ．

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