合成関数の2次偏導関数の導出

${\displaystyle z=f\left(x,y\right)}$ で ${\displaystyle x=\phi \left(u,v\right),y=\psi \left(u,v\right)}$ ならば

${\displaystyle \frac{{\partial }^{2}z}{\partial u\partial v}=\frac{{\partial }^{2}z}{\partial v\partial u}=f_{xx}\frac{\partial x}{\partial u}\frac{\partial x}{\partial v}}$${\displaystyle +f_{xy}(\frac{\partial x}{\partial u}\frac{\partial y}{\partial v}}$${\displaystyle +\frac{\partial x}{\partial v}\frac{\partial y}{\partial u})}$${\displaystyle +f_{yy}\frac{\partial y}{\partial u}\frac{\partial y}{\partial v}}$${\displaystyle +f_x\frac{{\partial }^{2}x}{\partial u\partial v}+f_y\frac{{\partial }^{2}y}{\partial u\partial v}}$

もしくは，

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

■導出

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

${\displaystyle =\frac{\partial }{\partial u}\left(f_x\frac{\partial x}{\partial v}+f_y\frac{\partial y}{\partial v}\right)}$

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

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

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

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

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

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

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

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

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

これらを代入して，

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

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

${\displaystyle =f_{xx}\frac{\partial x}{\partial u}\frac{\partial x}{\partial v}+f_{xy}\frac{\partial y}{\partial u}\frac{\partial x}{\partial v}}$${\displaystyle +f_x\frac{{\partial }^{2}x}{\partial u\partial v}}$${\displaystyle +f_{yx}\frac{\partial x}{\partial u}\frac{\partial y}{\partial v}+f_{yy}\frac{\partial y}{\partial u}\frac{\partial y}{\partial v}}$${\displaystyle +f_y\frac{{\partial }^{2}y}{\partial u\partial v}}$

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

${\displaystyle =f_{xx}\frac{\partial x}{\partial u}\frac{\partial x}{\partial v}+f_{xy}\left(\frac{\partial x}{\partial u}\frac{\partial y}{\partial v}+\frac{\partial x}{\partial v}\frac{\partial y}{\partial u}\right)}$${\displaystyle +f_{yy}\frac{\partial y}{\partial u}\frac{\partial y}{\partial v}}$${\displaystyle +f_x\frac{{\partial }^{2}x}{\partial u\partial v}+f_y\frac{{\partial }^{2}y}{\partial u\partial v}}$

　

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最終更新日： 2023年1月21日