偏微分の順序交換

$3$\displaystyle{z=f $ x,y $ }$ について， $3$\displaystyle{\frac{\hspace{2} {{\partial}}^{2}z \hspace{2}}{\hspace{2} {\partial}y{\partial}x \hspace{2}}}$ ， $3$\displaystyle{\frac{\hspace{2} {{\partial}}^{2}z \hspace{2}}{\hspace{2} {\partial}x{\partial}y \hspace{2}}}$ が共に連続ならば

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

表現をかえると

$3$\displaystyle{\hspace{1}{f}_{ xy } $ x,y $ =\hspace{1}{f}_{ yx } $ x,y $ }$

が成り立つ．

■証明

偏導関数の定義より

$3$\displaystyle{\hspace{1}{f}_{x} $ x,y $ =\frac{\hspace{2} {\partial}f \hspace{2}}{\hspace{2} {\partial}x \hspace{2}}}$ $3$\displaystyle{={ \lim }\limits_{ h\rightarrow 0 }\frac{\hspace{2} f $ x+h,y $ - f $ x,y $ \hspace{2}}{\hspace{2}h\hspace{2}}}$

$3$\displaystyle{\hspace{1}{f}_{y} $ x,y $ =\frac{\hspace{2} {\partial}f \hspace{2}}{\hspace{2} {\partial}y \hspace{2}}}$ $3$\displaystyle{={ \lim }\limits_{ k\rightarrow 0 }\frac{\hspace{2} f $ x,y+k $ - f $ x,y $ \hspace{2}}{\hspace{2}k\hspace{2}}}$

また

$3$\displaystyle{\hspace{1}{f}_{ xy }=\frac{\hspace{2} {\partial}\hspace{1}{f}_{x} \hspace{2}}{\hspace{2} {\partial}y \hspace{2}}={ \lim }\limits_{ k\rightarrow 0 }\frac{\hspace{2} \hspace{1}{f}_{x} $ x,y+k $ - \hspace{1}{f}_{x} $ x,y $ \hspace{2}}{\hspace{2}k\hspace{2}}}$

$3$\displaystyle{={ \lim }\limits_{ k\rightarrow 0 }\frac{\hspace{2} { \lim }\limits_{ h\rightarrow 0 }\frac{\hspace{2} f $ x+h,y+k $ - f $ x,y+k $ \hspace{2}}{\hspace{2}h\hspace{2}}- { \lim }\limits_{ h\rightarrow 0 }\frac{\hspace{2} f $ x+h,y $ - f $ x,y $ \hspace{2}}{\hspace{2}h\hspace{2}} \hspace{2}}{\hspace{2}k\hspace{2}}}$

$3$\displaystyle{={ \lim }\limits_{ k\rightarrow 0 } \[ \frac{\hspace{2}1\hspace{2}}{\hspace{2}k\hspace{2}} \{ { \lim }\limits_{ h\rightarrow 0 }\frac{\hspace{2} f $ x+h,y+k $ - f $ x,y+k $ - f $ x+h,y $ +f $ x,y $ \hspace{2}}{\hspace{2}h\hspace{2}} \} \] }$

$3$\displaystyle{={ \lim }\limits_{ k\rightarrow 0 } \[ { \lim }\limits_{ h\rightarrow 0 }\frac{\hspace{2}1\hspace{2}}{\hspace{2} hk \hspace{2}} \{ f $ x+h,y+k $ - f $ x,y+k $ - f $ x+h,y $ +f $ x,y $ \} \] }$ ･･････(1)

同様にして

$3$\displaystyle{\hspace{1}{f}_{ yx }=\frac{\hspace{2} {\partial}\hspace{1}{f}_{y} \hspace{2}}{\hspace{2} {\partial}x \hspace{2}}={ \lim }\limits_{ h\rightarrow 0 }\frac{\hspace{2} \hspace{1}{f}_{y} $ x+h,y $ - \hspace{1}{f}_{y} $ x,y $ \hspace{2}}{\hspace{2}h\hspace{2}}}$

$3$\displaystyle{={ \lim }\limits_{ h\rightarrow 0 }\frac{\hspace{2} { \lim }\limits_{ k\rightarrow 0 }\frac{\hspace{2} f $ x+h,y+k $ - f $ x+h,y $ \hspace{2}}{\hspace{2}k\hspace{2}}- { \lim }\limits_{ k\rightarrow 0 }\frac{\hspace{2} f $ x,y+k $ - f $ x,y $ \hspace{2}}{\hspace{2}k\hspace{2}} \hspace{2}}{\hspace{2}h\hspace{2}}}$

$3$\displaystyle{={ \lim }\limits_{ h\rightarrow 0 } \[ \frac{\hspace{2}1\hspace{2}}{\hspace{2}h\hspace{2}} \{ { \lim }\limits_{ k\rightarrow 0 }\frac{\hspace{2} f $ x+h,y+k $ - f $ x+h,y $ \hspace{2}}{\hspace{2}k\hspace{2}}- { \lim }\limits_{ k\rightarrow 0 }\frac{\hspace{2} f $ x,y+k $ - f $ x,y $ \hspace{2}}{\hspace{2}k\hspace{2}} \} \] }$

$3$\displaystyle{={ \lim }\limits_{ h\rightarrow 0 } \[ { \lim }\limits_{ k\rightarrow 0 }\frac{\hspace{2}1\hspace{2}}{\hspace{2} hk \hspace{2}} \{ f $ x+h,y+k $ - f $ x,y+k $ - f $ x+h,y $ +f $ x,y $ \} \] }$ ･･････(2)

ここで

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

とおき，

$3$\displaystyle{\hspace{1}{g}_{1} $x$ =f $ x,y+k $ - f $ x,y $ }$

とおくと，(3)は

$3$\displaystyle{{\Delta}=\hspace{1}{g}_{1} $ x+h $ - \hspace{1}{g}_{1} $x$ }$

平均値の定理より

$3$\displaystyle{{\Delta}=\hspace{1}{{g}^{\prime }}_{1} $ x+\hspace{1}{{\theta}}_{1}h $ h}$ ( $3$\displaystyle{0< \hspace{1}{{\theta}}_{1}< 1}$ )

ここで $3$\displaystyle{\hspace{1}{{g}^{\prime }}_{1} $x$ }$ は

$3$\displaystyle{\hspace{1}{{g}^{\prime }}_{1} $x$ =\frac{\hspace{2}d\hspace{2}}{\hspace{2} dx \hspace{2}}\hspace{1}{g}_{1} $x$ }$

$3$\displaystyle{= \{ \frac{\hspace{2}d\hspace{2}}{\hspace{2} dx \hspace{2}}\hspace{1}{g}_{1} $ x+\hspace{1}{{\theta}}_{1}h $ \} h}$

$3$\displaystyle{=\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}} \{ f $ x,y+k $ - f $ x,y $ \} }$

$3$\displaystyle{=\hspace{1}{f}_{x} $ x,y+k $ - \hspace{1}{f}_{x} $ x,y $ }$

よって

$3$\displaystyle{{\Delta}= \{ \hspace{1}{f}_{x} $ x+\hspace{1}{{\theta}}_{1}h,y+k $ - \hspace{1}{f}_{x} $ x+\hspace{1}{{\theta}}_{1}h,y $ \} h}$ ･･････(4)

$3$\displaystyle{\hspace{1}{g}_{2} $y$ =\hspace{1}{f}_{x} $ x+\hspace{1}{{\theta}}_{1}h,y $ }$ とおくと(4)は，

$3$\displaystyle{{\Delta}= \{ \hspace{1}{g}_{2} $ y+k $ - \hspace{1}{g}_{2} $y$ \} h}$

平均値の定理より

$3$\displaystyle{\hspace{1}{g}_{2} $ y+k $ - \hspace{1}{g}_{2} $y$ =\hspace{1}{{g}^{\prime }}_{2} $ y+\hspace{1}{{\theta}}_{2}k $ k}$

となるので

$3$\displaystyle{{\Delta}= \{ \hspace{1}{{g}^{\prime }}_{2} $ y+\hspace{1}{{\theta}}_{2}k $ k \} h}$ ( $3$\displaystyle{0< \hspace{1}{{\theta}}_{2}< 1}$ )

$3$\displaystyle{= \{ \hspace{1}{{g}^{\prime }}_{2} $ y+\hspace{1}{{\theta}}_{2}k $ \} hk}$

ここで $3$\displaystyle{\hspace{1}{{g}^{\prime }}_{2} $y$ }$ は

$3$\displaystyle{\hspace{1}{{g}^{\prime }}_{2} $y$ =\frac{\hspace{2}d\hspace{2}}{\hspace{2} dy \hspace{2}}\hspace{1}{g}_{2} $y$ }$

$3$\displaystyle{=\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}y \hspace{2}}\hspace{1}{f}_{x} $ x+\hspace{1}{{\theta}}_{1}h,y $ }$

$3$\displaystyle{=\hspace{1}{f}_{ xy } $ x+\hspace{1}{{\theta}}_{1}h,y $ }$

よって

$3$\displaystyle{{\Delta}=\hspace{1}{f}_{ xy } $ x+\hspace{1}{{\theta}}_{1}h,y+\hspace{1}{{\theta}}_{2}k $ hk}$ ･･････(5)

次に， $3$\displaystyle{\hspace{1}{h}_{1} $y$ =f $ x+h,y $ - f $ x,y $ }$ とおくと(3)は

$3$\displaystyle{{\Delta}=\hspace{1}{h}_{1} $ y+k $ - \hspace{1}{h}_{1} $y$ }$

平均値の定理より

$3$\displaystyle{{\Delta}=\hspace{1}{{h}^{\prime }}_{1} $ y+\hspace{1}{{\phi}}_{1}k $ k}$ ( $3$\displaystyle{0< \hspace{1}{{\phi}}_{1}< 1}$ )

ここで $3$\displaystyle{\hspace{1}{{h}^{\prime }}_{1} $y$ }$ は

$3$\displaystyle{\hspace{1}{{h}^{\prime }}_{1} $y$ =\frac{\hspace{2}d\hspace{2}}{\hspace{2} dy \hspace{2}}\hspace{1}{h}_{1} $y$ }$

$3$\displaystyle{=\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}y \hspace{2}} \{ f $ x+h,y $ - f $ x,y $ \} }$

$3$\displaystyle{=\hspace{1}{f}_{y} $ x+h,y $ - \hspace{1}{f}_{y} $ x,y $ }$

よって

$3$\displaystyle{{\Delta}= \{ \hspace{1}{f}_{y} $ x+h,y+\hspace{1}{{\phi}}_{1}k $ - \hspace{1}{f}_{y} $ x,y+\hspace{1}{{\phi}}_{1}k $ \} k}$ ･･････(6)

$3$\displaystyle{\hspace{1}{h}_{2} $x$ =\hspace{1}{f}_{y} $ x,y+\hspace{1}{{\phi}}_{1}k $ }$ とおくと(6)は

$3$\displaystyle{{\Delta}= \{ \hspace{1}{h}_{2} $ x+h $ - \hspace{1}{h}_{2} $x$ \} k}$

平均値の定理より

$3$\displaystyle{\hspace{1}{h}_{2} $ x+h $ - \hspace{1}{h}_{2} $x$ =\hspace{1}{{h}^{\prime }}_{2} $ x+\hspace{1}{{\phi}}_{2}h $ h}$

となるので

$3$\displaystyle{{\Delta}= \{ \hspace{1}{{h}^{\prime }}_{2} $ x+\hspace{1}{{\phi}}_{2}h $ h \} k}$ ( $3$\displaystyle{0< \hspace{1}{{\phi}}_{2}< 1}$ )

$3$\displaystyle{= \{ \hspace{1}{{h}^{\prime }}_{2} $ x+\hspace{1}{{\phi}}_{2}h $ \} hk}$

ここで $3$\displaystyle{\hspace{1}{{h}^{\prime }}_{2} $x$ }$ は

$3$\displaystyle{\hspace{1}{{h}^{\prime }}_{2} $x$ =\frac{\hspace{2}d\hspace{2}}{\hspace{2} dx \hspace{2}}\hspace{1}{h}_{2} $x$ }$

$3$\displaystyle{=\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}}\hspace{1}{f}_{y} $ x,y+\hspace{1}{{\phi}}_{1}k $ }$

$3$\displaystyle{=\hspace{1}{f}_{ yx } $ x,y+\hspace{1}{{\phi}}_{1}k $ }$

よって

$3$\displaystyle{{\Delta}=\hspace{1}{f}_{ yx } $ x+\hspace{1}{{\phi}}_{2}h,y+\hspace{1}{{\phi}}_{1}k $ hk}$ ･･････(7)

(1)，(5)より

$3$\displaystyle{ { \lim }\limits_{\begin{array}{lll} h\rightarrow 0 & \vspace{6}\\ k\rightarrow 0 & \vspace{6}\\ \end{array}}\frac{\hspace{2}{\Delta}\hspace{2}}{\hspace{2} hk \hspace{2}}={ \lim }\limits_{\begin{array}{lll} h\rightarrow 0 & \vspace{6}\\ k\rightarrow 0 & \vspace{6}\\ \end{array}}\hspace{1}{f}_{ xy } $ x+\hspace{1}{{\theta}}_{1}h,y+\hspace{1}{{\theta}}_{2}k $ }$ $3$\displaystyle{ =\hspace{1}{f}_{ xy } $ x,y $ }$ ･･････(8)

(2)，(7)より

$3$\displaystyle{ { \lim }\limits_{\begin{array}{lll} h\rightarrow 0 & \vspace{6}\\ k\rightarrow 0 & \vspace{6}\\ \end{array}}\frac{\hspace{2}{\Delta}\hspace{2}}{\hspace{2} hk \hspace{2}}={ \lim }\limits_{\begin{array}{lll} h\rightarrow 0 & \vspace{6}\\ k\rightarrow 0 & \vspace{6}\\ \end{array}}\hspace{1}{f}_{ yx } $ x+\hspace{1}{{\phi}}_{2}h,y+\hspace{1}{{\phi}}_{1}k $ }$ $3$\displaystyle{ =\hspace{1}{f}_{ yx } $ x,y $ }$ ･･････(9)

(8)，(9)より

$3$\displaystyle{\hspace{1}{f}_{ xy } $ x,y $ =\hspace{1}{f}_{ yx } $ x,y $ }$

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最終更新日： 2025年4月22日