いくつか有用な等式
出典: フリー百科事典『ウィキペディア（Wikipedia）』 (2017/09/09 01:26 UTC 版)
主変数 z は副変数 x, y の函数かつ x, y は u, v の函数とし、各々に関する微分は完全微分とする。連鎖律により、 d z = ( ∂ z ∂ x ) y d x + ( ∂ z ∂ y ) x d y = ( ∂ z ∂ u ) v d u + ( ∂ z ∂ v ) u d v {\displaystyle {\mathit {dz}}={\Bigl (}{\frac {\partial z}{\partial x}}{\Big )}_{y}{\mathit {dx}}+{\Bigl (}{\frac {\partial z}{\partial y}}{\Big )}_{x}{\mathit {dy}}={\Bigl (}{\frac {\partial z}{\partial u}}{\Big )}_{v}{\mathit {du}}+{\Bigl (}{\frac {\partial z}{\partial v}}{\Big )}_{u}{\mathit {dv}}} (1) となるが、やはり連鎖律により d x = ( ∂ x ∂ u ) v d u + ( ∂ x ∂ v ) u d v {\displaystyle {\mathit {dx}}={\Bigl (}{\frac {\partial x}{\partial u}}{\Big )}_{v}{\mathit {du}}+{\Bigl (}{\frac {\partial x}{\partial v}}{\Big )}_{u}{\mathit {dv}}} (2) および d y = ( ∂ y ∂ u ) v d u + ( ∂ y ∂ v ) u d v {\displaystyle {\mathit {dy}}={\Bigl (}{\frac {\partial y}{\partial u}}{\Big )}_{v}{\mathit {du}}+{\Bigl (}{\frac {\partial y}{\partial v}}{\Big )}_{u}{\mathit {dv}}} (3) により d z = [ ( ∂ z ∂ x ) y ( ∂ x ∂ u ) v + ( ∂ z ∂ y ) x ( ∂ y ∂ u ) v ] d u + [ ( ∂ z ∂ x ) y ( ∂ x ∂ v ) u + ( ∂ z ∂ y ) x ( ∂ y ∂ v ) u ] d v {\displaystyle dz={\Bigl [}{\Bigl (}{\frac {\partial z}{\partial x}}{\Big )}_{y}{\Bigl (}{\frac {\partial x}{\partial u}}{\Big )}_{v}+{\Bigl (}{\frac {\partial z}{\partial y}}{\Big )}_{x}{\Bigl (}{\frac {\partial y}{\partial u}}{\Big )}_{v}{\Bigr ]}du+{\Bigl [}{\Bigl (}{\frac {\partial z}{\partial x}}{\Big )}_{y}{\Bigl (}{\frac {\partial x}{\partial v}}{\Big )}_{u}+{\Bigl (}{\frac {\partial z}{\partial y}}{\Big )}_{x}{\Bigl (}{\frac {\partial y}{\partial v}}{\Big )}_{u}{\Bigr ]}dv} (4) となり、さらに ( ∂ z ∂ u ) v = ( ∂ z ∂ x ) y ( ∂ x ∂ u ) v + ( ∂ z ∂ y ) x ( ∂ y ∂ u ) v {\displaystyle {\Bigl (}{\frac {\partial z}{\partial u}}{\Big )}_{v}={\Bigl (}{\frac {\partial z}{\partial x}}{\Big )}_{y}{\Bigl (}{\frac {\partial x}{\partial u}}{\Big )}_{v}+{\Bigl (}{\frac {\partial z}{\partial y}}{\Big )}_{x}{\Bigl (}{\frac {\partial y}{\partial u}}{\Big )}_{v}} (5) を導く。v = y と置けば ( ∂ z ∂ u ) y = ( ∂ z ∂ x ) y ( ∂ x ∂ u ) y , {\displaystyle {\Bigl (}{\frac {\partial z}{\partial u}}{\Big )}_{y}={\Bigl (}{\frac {\partial z}{\partial x}}{\Big )}_{y}{\Bigl (}{\frac {\partial x}{\partial u}}{\Big )}_{y},} (6) および u = y と置けば ( ∂ z ∂ y ) v = ( ∂ z ∂ y ) x + ( ∂ z ∂ x ) y ( ∂ x ∂ y ) v , {\displaystyle {\Bigl (}{\frac {\partial z}{\partial y}}{\Big )}_{v}={\Bigl (}{\frac {\partial z}{\partial y}}{\Big )}_{x}+{\Bigl (}{\frac {\partial z}{\partial x}}{\Big )}_{y}{\Bigl (}{\frac {\partial x}{\partial y}}{\Big )}_{v},} (7) あるいは、u = y, v = z と置いて ( ∂ z ∂ y ) x = − ( ∂ z ∂ x ) y ( ∂ x ∂ y ) z , {\displaystyle {\Bigl (}{\frac {\partial z}{\partial y}}{\Big )}_{x}=-{\Bigl (}{\frac {\partial z}{\partial x}}{\Big )}_{y}{\Bigl (}{\frac {\partial x}{\partial y}}{\Big )}_{z},} (8) また相反関係式により三重積の微分法則 ( ∂ z ∂ x ) y ( ∂ x ∂ y ) z ( ∂ y ∂ z ) x = − 1 {\displaystyle {\Bigl (}{\frac {\partial z}{\partial x}}{\Big )}_{y}{\Bigl (}{\frac {\partial x}{\partial y}}{\Big )}_{z}{\Bigl (}{\frac {\partial y}{\partial z}}{\Big )}_{x}=-1} (9) を得る。
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