# 演習問題

$z=f\left(x,y\right)\text{\hspace{0.17em}},\text{\hspace{0.17em}}x=a+ht\text{\hspace{0.17em}},\text{\hspace{0.17em}}y=b+kt\text{\hspace{0.17em}}$ のとき， $\frac{dz}{dt}\text{\hspace{0.17em}}$ を求めよ．

$z={x}^{2}+{y}^{2}\text{\hspace{0.17em}},\text{\hspace{0.17em}}x=t-sint\text{\hspace{0.17em}},\text{\hspace{0.17em}}y=1-cost\text{\hspace{0.17em}}$ のとき， $\frac{dz}{dt}$ を求めよ．

$z=xy\text{\hspace{0.17em}},\text{\hspace{0.17em}}x=2{t}^{2}+1\text{\hspace{0.17em}},\text{\hspace{0.17em}}y={t}^{2}+3t+1\text{\hspace{0.17em}}$ のとき， $\frac{dz}{dt}\text{\hspace{0.17em}}$ を求めよ．

$z=xtany\text{\hspace{0.17em}},\text{\hspace{0.17em}}x={sin}^{-1}2t\text{\hspace{0.17em}},\text{\hspace{0.17em}}y={cos}^{-1}2t\text{\hspace{0.17em}}$ のとき， $\frac{dz}{dt}\text{\hspace{0.17em}}$ を求めよ．

$z=f\left(x,y\right)$ , $x=rcos\theta$ , $y=rsin\theta \text{\hspace{0.17em}}$平面の極座標変換） ならば

${\left(\frac{dz}{dx}\right)}^{2}+{\left(\frac{dz}{dy}\right)}^{2}$ $={\left(\frac{dz}{dr}\right)}^{2}+\frac{1}{{r}^{2}}{\left(\frac{dz}{d\theta }\right)}^{2}$

となることを示せ．

$z=f\left(x,y\right)$, $x=u\mathrm{cos}\alpha -v\mathrm{sin}\alpha$, $y=u\mathrm{sin}\alpha +v\mathrm{cos}\alpha$ならば

${\left(\frac{\partial z}{\partial x}\right)}^{2}+{\left(\frac{\partial z}{\partial y}\right)}^{2}$ $={\left(\frac{\partial z}{\partial u}\right)}^{2}+{\left(\frac{\partial z}{\partial v}\right)}^{2}$

となることを示せ．

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