合成関数の2次偏導関数の導出

$3$\displaystyle{z=f $ x,y $ }$ で $3$\displaystyle{x={\varphi} $ u,v $ ,y={\psi} $ u,v $ }$ ならば

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

もしくは，

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

■導出

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

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

$3$\displaystyle{ =\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}u \hspace{2}} $ \frac{\hspace{2} {\partial}z \hspace{2}}{\hspace{2} {\partial}x \hspace{2}} $ \cdot \frac{\hspace{2} {\partial}x \hspace{2}}{\hspace{2} {\partial}v \hspace{2}}+\frac{\hspace{2} {\partial}z \hspace{2}}{\hspace{2} {\partial}x \hspace{2}}\cdot \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}u \hspace{2}} $ \frac{\hspace{2} {\partial}x \hspace{2}}{\hspace{2} {\partial}v \hspace{2}} $ }$ $3$\displaystyle{ +\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}u \hspace{2}} $ \frac{\hspace{2} {\partial}z \hspace{2}}{\hspace{2} {\partial}y \hspace{2}} $ \cdot \frac{\hspace{2} {\partial}y \hspace{2}}{\hspace{2} {\partial}v \hspace{2}} }$ $3$\displaystyle{ +\frac{\hspace{2} {\partial}z \hspace{2}}{\hspace{2} {\partial}y \hspace{2}}\cdot \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}u \hspace{2}} $ \frac{\hspace{2} {\partial}y \hspace{2}}{\hspace{2} {\partial}v \hspace{2}} $ }$

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

$3$\displaystyle{z}$ が合成関数であるので， $3$\displaystyle{\text{\hspace{1}}\text{\hspace{1}}\frac{\hspace{2} {\partial}z \hspace{2}}{\hspace{2} {\partial}x \hspace{2}}\text{\hspace{1}},\text{\hspace{1}}\text{\hspace{1}}\frac{\hspace{2} {\partial}z \hspace{2}}{\hspace{2} {\partial}y \hspace{2}}\text{\hspace{1}}\text{\hspace{1}}}$ も共に合成関数となる．これらを $3$\displaystyle{\text{\hspace{1}}\text{\hspace{1}}u\text{\hspace{1}}\text{\hspace{1}}}$ で偏微分すると，合成関数の偏微分より，

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

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

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

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

これらを代入して，

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

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

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

偏微分の順序交換が成り立つ，すなわち $3$\displaystyle{ \hspace{1}{f}_{ xy }=\hspace{1}{f}_{ yx } }$ より

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

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最終更新日： 2023年1月21日