偏微分の順序交換

${\displaystyle z=f\left(x,y\right)}$  について， ${\displaystyle \frac{{\partial }^{2}z}{\partial y\partial x}}$ ， ${\displaystyle \frac{{\partial }^{2}z}{\partial x\partial y}}$ が共に連続ならば

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

表現をかえると

${\displaystyle f_{xy}\left(x,y\right)=f_{yx}\left(x,y\right)}$  

が成り立つ．

■証明

偏導関数の定義より

${\displaystyle f_x\left(x,y\right)=\frac{\partial f}{\partial x}}$ ${\displaystyle =\underset{h\to 0}{\lim }\frac{f\left(x+h,y\right)-f\left(x,y\right)}{h}}$  

${\displaystyle f_y\left(x,y\right)=\frac{\partial f}{\partial y}}$ ${\displaystyle =\underset{k\to 0}{\lim }\frac{f\left(x,y+k\right)-f\left(x,y\right)}{k}}$  

また

同様にして

ここで

${\displaystyle \Delta =f\left(x+h,y+k\right)-f\left(x,y+k\right)}$ ${\displaystyle -f\left(x+h,y\right)}$ ${\displaystyle +f\left(x,y\right)}$  ･･････(3)

とおき，

${\displaystyle g_1\left(x\right)=f\left(x,y+k\right)-f\left(x,y\right)}$  

とおくと，(3)は

${\displaystyle \Delta =g_1\left(x+h\right)-g_1\left(x\right)}$  

平均値の定理より

${\displaystyle \Delta ={g^{\prime }}_1\left(x+{\theta }_{1}h\right)h}$     ( ${\displaystyle 0<{\theta }_1<1}$ )

ここで ${\displaystyle {g^{\prime }}_1\left(x\right)}$ は

${\displaystyle {g^{\prime }}_1\left(x\right)=\frac{d}{dx}{g}_1\left(x\right)}$  

${\displaystyle =\left\{\frac{d}{dx}{g}_1\left(x+{\theta }_{1}h\right)\right\}h}$  

${\displaystyle =\frac{\partial }{\partial x}\left\{f\left(x,y+k\right)-f\left(x,y\right)\right\}}$  

          ${\displaystyle =f_x\left(x,y+k\right)-f_x\left(x,y\right)}$  

よって

${\displaystyle g_2\left(y\right)=f_x\left(x+{\theta }_{1}h,y\right)}$  とおくと(4)は，

${\displaystyle \Delta =\left\{g_2\left(y+k\right)-g_2\left(y\right)\right\}h}$  

平均値の定理より

${\displaystyle g_2\left(y+k\right)-g_2\left(y\right)={g^{\prime }}_2\left(y+{\theta }_{2}k\right)k}$  

となるので

${\displaystyle \Delta =\left\{{g^{\prime }}_2\left(y+{\theta }_{2}k\right)k\right\}h}$     ( ${\displaystyle 0<{\theta }_2<1}$ )

${\displaystyle =\left\{{g^{\prime }}_2\left(y+{\theta }_{2}k\right)\right\}hk}$  

ここで ${\displaystyle {g^{\prime }}_2\left(y\right)}$ は

${\displaystyle {g^{\prime }}_2\left(y\right)=\frac{d}{dy}{g}_2\left(y\right)}$  

${\displaystyle =\frac{\partial }{\partial y}{f}_x\left(x+{\theta }_{1}h,y\right)}$  

${\displaystyle =f_{xy}\left(x+{\theta }_{1}h,y\right)}$  

よって

${\displaystyle \Delta =f_{xy}\left(x+{\theta }_{1}h,y+{\theta }_{2}k\right)hk}$  ･･････(5)

次に， ${\displaystyle h_1\left(y\right)=f\left(x+h,y\right)-f\left(x,y\right)}$  とおくと(3)は

${\displaystyle \Delta =h_1\left(y+k\right)-h_1\left(y\right)}$  

平均値の定理より

${\displaystyle \Delta ={h^{\prime }}_1\left(y+{\varphi }_{1}k\right)k}$     ( ${\displaystyle 0<{\varphi }_1<1}$ )

ここで ${\displaystyle {h^{\prime }}_1\left(y\right)}$ は

${\displaystyle {h^{\prime }}_1\left(y\right)=\frac{d}{dy}{h}_1\left(y\right)}$  

${\displaystyle =\frac{\partial }{\partial y}\left\{f\left(x+h,y\right)-f\left(x,y\right)\right\}}$  

${\displaystyle =f_y\left(x+h,y\right)-f_y\left(x,y\right)}$  

よって

${\displaystyle h_2\left(x\right)=f_y\left(x,y+{\varphi }_{1}k\right)}$  とおくと(6)は

${\displaystyle \Delta =\left\{h_2\left(x+h\right)-h_2\left(x\right)\right\}k}$  

平均値の定理より

${\displaystyle h_2\left(x+h\right)-h_2\left(x\right)={h^{\prime }}_2\left(x+{\varphi }_{2}h\right)h}$  

となるので

${\displaystyle \Delta =\left\{{h^{\prime }}_2\left(x+{\varphi }_{2}h\right)h\right\}k}$     ( ${\displaystyle 0<{\varphi }_2<1}$ )

    ${\displaystyle =\left\{{h^{\prime }}_2\left(x+{\varphi }_{2}h\right)\right\}hk}$  

ここで ${\displaystyle {h^{\prime }}_2\left(x\right)}$ は

${\displaystyle {h^{\prime }}_2\left(x\right)=\frac{d}{dx}{h}_2\left(x\right)}$  

${\displaystyle =\frac{\partial }{\partial x}{f}_y\left(x,y+{\varphi }_{1}k\right)}$  

${\displaystyle =f_{yx}\left(x,y+{\varphi }_{1}k\right)}$  

よって

${\displaystyle \Delta =f_{yx}\left(x+{\varphi }_{2}h,y+{\varphi }_{1}k\right)hk}$  ･･････(7)

(1)，(5)より

${\displaystyle \underset{\begin{array}{l}h\to 0\\ k\to 0\end{array}}{\lim }\frac{\Delta }{hk}=\underset{\begin{array}{l}h\to 0\\ k\to 0\end{array}}{\lim }{f}_{xy}\left(x+{\theta }_{1}h,y+{\theta }_{2}k\right)}$ ${\displaystyle =f_{xy}\left(x,y\right)}$  ･･････(8)

(2)，(7)より

${\displaystyle \underset{\begin{array}{l}h\to 0\\ k\to 0\end{array}}{\lim }\frac{\Delta }{hk}=\underset{\begin{array}{l}h\to 0\\ k\to 0\end{array}}{\lim }{f}_{yx}\left(x+{\varphi }_{2}h,y+{\varphi }_{1}k\right)}$ ${\displaystyle =f_{yx}\left(x,y\right)}$  ･･････(9)

(8)，(9)より 

${\displaystyle f_{xy}\left(x,y\right)=f_{yx}\left(x,y\right)}$  

 

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最終更新日： 2025年4月22日