演習問題

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

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

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

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

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

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

となることを示せ．

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

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

となることを示せ．

${\displaystyle z=f\left(x,y\right),x=r\cos \theta ,y=r\sin \theta }$ のとき

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

となることを示せ．