Poisson括弧の公式

(1) $${\{u,v\}=-\{v,u\}}$$
(2) $${\{u+v,w\}=\{u,w\}+\{v,w\}}$$
(3) $${\{u,vw\}=w\{u,v\}+v\{u,w\}}$$
(4) $${\{uv,w\}=v\{u,w\}+u\{v,w\}}$$
(5) $${\{uv,w\}+\{vw,u\}+\{wu,v\}=0}$$
(6) $${\{u,\{v,w\}\}+\{v,\{w,u\}\}+\{w,\{u,v\}\}=0}$$（Jacobi の恒等式）

簡単に証明できると思ったら(6)は結構やっかいだった。ケアレスミスの帝王は苦戦しました。

(1) $${\{u,v\}=-\{v,u\}}$$の証明
　左辺$${=\displaystyle\sum_{i=1}^f\Big(\dfrac{\partial u}{\partial q_i}\dfrac{\partial v}{\partial p_i}-\dfrac{\partial u}{\partial p_i}\dfrac{\partial v}{\partial q_i}\Big)=-\displaystyle\sum_{i=1}^f\Big(\dfrac{\partial u}{\partial p_i}\dfrac{\partial v}{\partial q_i}-\dfrac{\partial u}{\partial q_i}\dfrac{\partial v}{\partial p_i}\Big)}$$
　　$${=-\displaystyle\sum_{i=1}^f\Big(\dfrac{\partial v}{\partial q_i}\dfrac{\partial u}{\partial p_i}-\dfrac{\partial v}{\partial p_i}\dfrac{\partial u}{\partial q_i}\Big)=-\{v,u\}}$$

(2) $${\{u+v,w\}=\{u,w\}+\{v,w\}}$$の証明
　左辺$${=\displaystyle\sum_{i=1}^f\Big(\dfrac{\partial (u+v)}{\partial q_i}\dfrac{\partial w}{\partial p_i}-\dfrac{\partial (u+v)}{\partial p_i}\dfrac{\partial w}{\partial q_i}\Big)}$$
　　$${=\displaystyle\sum_{i=1}^f\Big\{\Big(\dfrac{\partial u}{\partial q_i}+\dfrac{\partial v}{\partial q_i}\Big)\dfrac{\partial w}{\partial p_i}-\Big(\dfrac{\partial u}{\partial p_i}+\dfrac{\partial v}{\partial p_i}\Big)\dfrac{\partial w}{\partial q_i}\Big\}}$$
　　$${=\displaystyle\sum_{i=1}^f\Big(\dfrac{\partial u}{\partial q_i}\dfrac{\partial w}{\partial p_i}+\dfrac{\partial v}{\partial q_i}\dfrac{\partial w}{\partial p_i}-\dfrac{\partial u}{\partial p_i}\dfrac{\partial w}{\partial q_i}-\dfrac{\partial v}{\partial p_i}\dfrac{\partial w}{\partial q_i}\Big)}$$
　　$${=\displaystyle\sum_{i=1}^f\Big(\dfrac{\partial u}{\partial q_i}\dfrac{\partial w}{\partial p_i}-\dfrac{\partial u}{\partial p_i}\dfrac{\partial w}{\partial q_i}\Big)+\sum_{i=1}^f\Big(\dfrac{\partial v}{\partial q_i}\dfrac{\partial w}{\partial p_i}-\dfrac{\partial v}{\partial p_i}\dfrac{\partial w}{\partial q_i}\Big)}$$
　　$${=\{u,w\}+\{v,w\}}$$

(3) $${\{u,vw\}=w\{u,v\}+v\{u,w\}}$$の証明
　左辺$${=\displaystyle\sum_{i=1}^f\Big(\dfrac{\partial u}{\partial q_i}\dfrac{\partial (vw)}{\partial p_i}-\dfrac{\partial u}{\partial p_i}\dfrac{\partial (vw)}{\partial q_i}\Big)}$$
　　$${=\displaystyle\sum_{i=1}^f\Big\{\dfrac{\partial u}{\partial q_i}\Big(\dfrac{\partial v}{\partial p_i}w+v\dfrac{\partial w}{\partial p_i}\Big)-\dfrac{\partial u}{\partial p_i}\Big(\dfrac{\partial v}{\partial q_i}w+v\dfrac{\partial w}{\partial q_i}\Big)\Big\}}$$
　　$${=\displaystyle\sum_{i=1}^f\Big(w\dfrac{\partial u}{\partial q_i}\dfrac{\partial v}{\partial p_i}+v\dfrac{\partial u}{\partial q_i}\dfrac{\partial w}{\partial p_i}-w\dfrac{\partial u}{\partial p_i}\dfrac{\partial v}{\partial q_i}+v\dfrac{\partial u}{\partial p_i}\dfrac{\partial w}{\partial q_i}\Big)}$$
　　$${=w\displaystyle\sum_{i=1}^f\Big(\dfrac{\partial u}{\partial q_i}\dfrac{\partial v}{\partial p_i}-\dfrac{\partial u}{\partial p_i}\dfrac{\partial v}{\partial q_i}\Big)+v\displaystyle\sum_{i=1}^f\Big(\dfrac{\partial u}{\partial q_i}\dfrac{\partial w}{\partial p_i}+v\dfrac{\partial u}{\partial p_i}\dfrac{\partial w}{\partial q_i}\Big)}$$
　　$${=w\{u,v\}+v\{uw\}}$$

(4) $${\{uv,w\}=\{v,w\}+v\{u,w\}}$$の証明
　左辺$${=-\{w,uv\}=-(v\{w,u\}+u\{w,v\})=-(-v\{u,w\}-u\{v,w\})}$$
　　$${=v\{u,w\}+u\{v,w\}}$$

(5) $${\{uv,w\}+\{vw,u\}+\{wu,v\}=0}$$の証明
　左辺$${=-\{w,uv\}-\{u,vw\}-\{v,wu\}}$$
　　$${=-u\{w,v\}-v\{w,u\}-v\{u,w\}-w\{u,v\}-w\{v,u\}-u\{v,w\}}$$
　　$${=u\{v,w\}-u\{v,w\}-v\{w,u\}+v\{w,u\}-w\{u,v\}+w\{u,v\}=0}$$

(6) $${\{u,\{v,w\}\}+\{v,\{w,u\}\}+\{w,\{u,v\}\}=0}$$（Jacobi の恒等式）の証明
　左辺$${=\displaystyle\sum_{i=1}^f\Big(\dfrac{\partial u}{\partial q_i}\dfrac{\partial\{v,w\}}{\partial p_i}-\dfrac{\partial u}{\partial p_i}\dfrac{\partial\{v,w\}}{\partial q_i}\Big)}$$
　　　　$${+\displaystyle\sum_{i=1}^f\Big(\dfrac{\partial v}{\partial q_i}\dfrac{\partial\{w,u\}}{\partial p_i}-\dfrac{\partial v}{\partial p_i}\dfrac{\partial\{w,u\}}{\partial q_i}\Big)}$$
　　　　$${+\displaystyle\sum_{i=1}^f\Big(\dfrac{\partial w}{\partial q_i}\dfrac{\partial\{u,v\}}{\partial p_i}-\dfrac{\partial w}{\partial p_i}\dfrac{\partial\{u,v\}}{\partial q_i}\Big)}$$
　　　$${=\displaystyle\sum_{i=1}^f\Big\{\dfrac{\partial u}{\partial q_i}\dfrac{\partial}{\partial p_i}\sum_{j=1}^f\Big(\dfrac{\partial v}{\partial q_j}\dfrac{\partial w}{\partial p_j}-\dfrac{\partial v}{\partial p_j}\dfrac{\partial w}{\partial q_j}\Big)}$$
　　　　　　　　　　　　　　　$${-\displaystyle\dfrac{\partial u}{\partial p_i}\sum_{j=1}^f\dfrac{\partial}{\partial q_i}\Big(\dfrac{\partial v}{\partial q_j}\dfrac{\partial w}{\partial p_j}-\dfrac{\partial v}{\partial p_j}\dfrac{\partial w}{\partial q_j}\Big)\Big\}}$$
　　　　$${+\displaystyle\sum_{i=1}^f\Big\{\dfrac{\partial v}{\partial q_i}\dfrac{\partial}{\partial p_i}\sum_{j=1}^f\Big(\dfrac{\partial w}{\partial q_j}\dfrac{\partial u}{\partial p_j}-\dfrac{\partial w}{\partial p_j}\dfrac{\partial u}{\partial q_j}\Big)}$$
　　　　　　　　　　　　　　　$${-\displaystyle\dfrac{\partial v}{\partial p_i}\dfrac{\partial}{\partial q_i}\sum_{j=1}^f\Big(\dfrac{\partial w}{\partial q_j}\dfrac{\partial u}{\partial p_j}-\dfrac{\partial w}{\partial p_j}\dfrac{\partial u}{\partial q_j}\Big)\Big\}}$$
　　　　$${+\displaystyle\sum_{i=1}^f\Big\{\dfrac{\partial w}{\partial q_i}\dfrac{\partial}{\partial p_i}\sum_{j=1}^f\Big(\dfrac{\partial u}{\partial q_j}\dfrac{\partial v}{\partial p_j}-\dfrac{\partial u}{\partial p_j}\dfrac{\partial v}{\partial q_j}\Big)}$$
　　　　　　　　　　　　　　　$${-\displaystyle\dfrac{\partial w}{\partial p_i}\dfrac{\partial}{\partial q_i}\sum_{j=1}^f\Big(\dfrac{\partial u}{\partial q_j}\dfrac{\partial v}{\partial p_j}-\dfrac{\partial u}{\partial p_j}\dfrac{\partial v}{\partial q_j}\Big)\Big\}}$$
　　　$${=\displaystyle\sum_{i=1}^f\sum_{j=1}^f\Big\{\dfrac{\partial u}{\partial q_i}\dfrac{\partial}{\partial p_i}\Big(\dfrac{\partial v}{\partial q_j}\dfrac{\partial w}{\partial p_j}\Big)-\dfrac{\partial u}{\partial q_i}\dfrac{\partial}{\partial p_i}\Big(\dfrac{\partial v}{\partial p_j}\dfrac{\partial w}{\partial q_j}\Big)}$$
　　　　　　$${-\dfrac{\partial u}{\partial p_i}\dfrac{\partial}{\partial q_i}\Big(\dfrac{\partial v}{\partial q_j}\dfrac{\partial w}{\partial p_j}\Big)+\dfrac{\partial u}{\partial p_i}\dfrac{\partial}{\partial q_i}\Big(\dfrac{\partial v}{\partial p_j}\dfrac{\partial w}{\partial q_j}\Big)}$$
　　　　　　$${+\dfrac{\partial v}{\partial q_i}\dfrac{\partial}{\partial p_i}\Big(\dfrac{\partial w}{\partial q_j}\dfrac{\partial u}{\partial p_j}\Big)-\dfrac{\partial v}{\partial q_i}\dfrac{\partial}{\partial p_i}\Big(\dfrac{\partial w}{\partial p_j}\dfrac{\partial u}{\partial q_j}\Big)}$$
　　　　　　$${-\dfrac{\partial v}{\partial p_i}\dfrac{\partial}{\partial q_i}\Big(\dfrac{\partial w}{\partial q_j}\dfrac{\partial u}{\partial p_j}\Big)+\dfrac{\partial v}{\partial p_i}\dfrac{\partial}{\partial q_i}\Big(\dfrac{\partial w}{\partial p_j}\dfrac{\partial u}{\partial q_j}\Big)}$$
　　　　　　$${+\dfrac{\partial w}{\partial q_i}\dfrac{\partial}{\partial p_i}\Big(\dfrac{\partial u}{\partial q_j}\dfrac{\partial v}{\partial p_j}\Big)-\dfrac{\partial w}{\partial q_i}\dfrac{\partial}{\partial p_i}\Big(\dfrac{\partial u}{\partial p_j}\dfrac{\partial v}{\partial q_j}\Big)}$$
　　　　　　$${-\dfrac{\partial w}{\partial p_i}\dfrac{\partial}{\partial q_i}\Big(\dfrac{\partial u}{\partial q_j}\dfrac{\partial v}{\partial p_j}\Big)+\dfrac{\partial w}{\partial p_i}\dfrac{\partial}{\partial q_i}\Big(\dfrac{\partial u}{\partial p_j}\dfrac{\partial v}{\partial q_j}\Big)\Big\}}$$
　　　$${=\displaystyle\sum_{i=1}^f\sum_{j=1}^f\Big(\dfrac{\partial u}{\partial q_i}\dfrac{\partial^2 v}{\partial p_i\partial q_j}\dfrac{\partial w}{\partial p_j}①+\dfrac{\partial u}{\partial q_i}\dfrac{\partial v}{\partial q_j}\dfrac{\partial^2 w}{\partial p_i\partial p_j}②}$$
　　　　　　$${-\dfrac{\partial u}{\partial q_i}\dfrac{\partial^2 v}{\partial p_i\partial p_j}\dfrac{\partial w}{\partial q_j}③-\dfrac{\partial u}{\partial q_i}\dfrac{\partial v}{\partial p_j}\dfrac{\partial^2 w}{\partial p_i\partial q_j}④}$$$${-\dfrac{\partial u}{\partial p_i}\dfrac{\partial^2 v}{\partial q_i\partial q_j}\dfrac{\partial w}{\partial p_j}⑤}$$
　　　　　　$${-\dfrac{\partial u}{\partial p_i}\dfrac{\partial v}{\partial q_j}\dfrac{\partial^2 w}{\partial q_i\partial p_j}⑥}$$$${+\dfrac{\partial u}{\partial p_i}\dfrac{\partial^2 v}{\partial q_i\partial p_j}\dfrac{\partial w}{\partial q_j}⑦+\dfrac{\partial u}{\partial p_i}\dfrac{\partial v}{\partial p_j}\dfrac{\partial^2 w}{\partial q_i\partial q_j}⑧}$$
　　　　　　$${+\dfrac{\partial v}{\partial q_i}\dfrac{\partial^2 w}{\partial p_i\partial q_j}\dfrac{\partial u}{\partial p_j}⑨+\dfrac{\partial v}{\partial q_i}\dfrac{\partial w}{\partial q_j}\dfrac{\partial^2 u}{\partial p_i\partial p_j}⑩}$$$${-\dfrac{\partial v}{\partial q_i}\dfrac{\partial^2 w}{\partial p_i\partial p_j}\dfrac{\partial u}{\partial q_j}⑪}$$
　　　　　　$${-\dfrac{\partial v}{\partial q_i}\dfrac{\partial w}{\partial p_j}\dfrac{\partial^2 u}{\partial p_i\partial q_j}⑫}$$$${-\dfrac{\partial v}{\partial p_i}\dfrac{\partial^2 w}{\partial q_i\partial q_j}\dfrac{\partial u}{\partial p_j}⑬-\dfrac{\partial v}{\partial p_i}\dfrac{\partial w}{\partial q_j}\dfrac{\partial^2 u}{\partial q_i\partial p_j}⑭}$$
　　　　　　$${+\dfrac{\partial v}{\partial p_i}\dfrac{\partial^2 w}{\partial q_i\partial p_j}\dfrac{\partial u}{\partial q_j}⑮+\dfrac{\partial v}{\partial p_i}\dfrac{\partial w}{\partial p_j}\dfrac{\partial^2 u}{\partial q_i\partial q_j}⑯}$$$${+\dfrac{\partial w}{\partial q_i}\dfrac{\partial^2 u}{\partial p_i\partial q_j}\dfrac{\partial v}{\partial p_j}⑰}$$
　　　　　　$${+\dfrac{\partial w}{\partial q_i}\dfrac{\partial u}{\partial q_j}\dfrac{\partial^2 v}{\partial p_i\partial p_j}⑱}$$$${-\dfrac{\partial w}{\partial q_i}\dfrac{\partial^2 u}{\partial p_i\partial p_j}\dfrac{\partial v}{\partial q_j}⑲-\dfrac{\partial w}{\partial q_i}\dfrac{\partial u}{\partial p_j}\dfrac{\partial^2 v}{\partial p_i\partial q_j}⑳}$$
　　　　　　$${-\dfrac{\partial w}{\partial p_i}\dfrac{\partial^2 u}{\partial q_i\partial q_j}\dfrac{\partial v}{\partial p_j}㉑-\dfrac{\partial w}{\partial p_i}\dfrac{\partial u}{\partial q_j}\dfrac{\partial^2 v}{\partial q_i\partial p_j}㉒}$$$${+\dfrac{\partial w}{\partial p_i}\dfrac{\partial^2 u}{\partial q_i\partial p_j}\dfrac{\partial v}{\partial q_j}㉓}$$
　　　　　　$${+\dfrac{\partial w}{\partial p_i}\dfrac{\partial u}{\partial p_j}\dfrac{\partial^2 v}{\partial q_i\partial q_j}㉔\Big)}$$
　　　　　　　　　↓　各項を$${\dfrac{\partial x}{\partial a_?}\dfrac{\partial^2 y}{\partial b_?\partial c_?}\dfrac{\partial z}{\partial d_?}}$$の形に並べ直す
　　　$${=\displaystyle\sum_{i=1}^f\sum_{j=1}^f\Big(\dfrac{\partial u}{\partial q_i}\dfrac{\partial^2 v}{\partial q_j\partial p_i}\dfrac{\partial w}{\partial p_j}①+\dfrac{\partial v}{\partial q_j}\dfrac{\partial^2 w}{\partial p_i\partial p_j}\dfrac{\partial u}{\partial q_i}②}$$
　　　　　　$${-\dfrac{\partial u}{\partial q_i}\dfrac{\partial^2 v}{\partial p_i\partial p_j}\dfrac{\partial w}{\partial q_j}③-\dfrac{\partial v}{\partial p_j}\dfrac{\partial^2 w}{\partial p_i\partial q_j}\dfrac{\partial u}{\partial q_i}④}$$$${-\dfrac{\partial u}{\partial p_i}\dfrac{\partial^2 v}{\partial q_i\partial q_j}\dfrac{\partial w}{\partial p_j}⑤}$$
　　　　　　$${-\dfrac{\partial v}{\partial q_j}\dfrac{\partial^2 w}{\partial q_i\partial p_j}\dfrac{\partial u}{\partial p_i}⑥}$$$${+\dfrac{\partial u}{\partial p_i}\dfrac{\partial^2 v}{\partial p_j\partial q_i}\dfrac{\partial w}{\partial q_j}⑦+\dfrac{\partial v}{\partial p_j}\dfrac{\partial^2 w}{\partial q_i\partial q_j}\dfrac{\partial u}{\partial p_i}⑧}$$
　　　　　　$${+\dfrac{\partial v}{\partial q_i}\dfrac{\partial^2 w}{\partial q_j\partial p_i}\dfrac{\partial u}{\partial p_j}⑨+\dfrac{\partial w}{\partial q_j}\dfrac{\partial^2 u}{\partial p_i\partial p_j}\dfrac{\partial v}{\partial q_i}⑩}$$$${-\dfrac{\partial v}{\partial q_i}\dfrac{\partial^2 w}{\partial p_i\partial p_j}\dfrac{\partial u}{\partial q_j}⑪}$$
　　　　　　$${-\dfrac{\partial w}{\partial p_j}\dfrac{\partial^2 u}{\partial p_i\partial q_j}\dfrac{\partial v}{\partial q_i}⑫}$$$${-\dfrac{\partial v}{\partial p_i}\dfrac{\partial^2 w}{\partial q_i\partial q_j}\dfrac{\partial u}{\partial p_j}⑬-\dfrac{\partial w}{\partial q_j}\dfrac{\partial^2 u}{\partial q_i\partial p_j}\dfrac{\partial v}{\partial p_i}⑭}$$
　　　　　　$${+\dfrac{\partial v}{\partial p_i}\dfrac{\partial^2 w}{\partial p_j\partial q_i}\dfrac{\partial u}{\partial q_j}⑮+\dfrac{\partial w}{\partial p_j}\dfrac{\partial^2 u}{\partial q_i\partial q_j}\dfrac{\partial v}{\partial p_i}⑯}$$$${+\dfrac{\partial w}{\partial q_i}\dfrac{\partial^2 u}{\partial q_j\partial p_i}\dfrac{\partial v}{\partial p_j}⑰}$$
　　　　　　$${+\dfrac{\partial u}{\partial q_j}\dfrac{\partial^2 v}{\partial p_i\partial p_j}\dfrac{\partial w}{\partial q_i}⑱}$$$${-\dfrac{\partial w}{\partial q_i}\dfrac{\partial^2 u}{\partial p_i\partial p_j}\dfrac{\partial v}{\partial q_j}⑲-\dfrac{\partial u}{\partial p_j}\dfrac{\partial^2 v}{\partial p_i\partial q_j}\dfrac{\partial w}{\partial q_i}⑳}$$
　　　　　　$${-\dfrac{\partial w}{\partial p_i}\dfrac{\partial^2 u}{\partial q_i\partial q_j}\dfrac{\partial v}{\partial p_j}㉑-\dfrac{\partial u}{\partial q_j}\dfrac{\partial^2 v}{\partial q_i\partial p_j}\dfrac{\partial w}{\partial p_i}㉒}$$$${+\dfrac{\partial w}{\partial p_i}\dfrac{\partial^2 u}{\partial p_j\partial q_i}\dfrac{\partial v}{\partial q_j}㉓}$$
　　　　　　$${+\dfrac{\partial u}{\partial p_j}\dfrac{\partial^2 v}{\partial q_i\partial q_j}\dfrac{\partial w}{\partial p_i}㉔\Big)}$$
　　　　　　　　　↓　２階偏微分で整理する
　　$${=\displaystyle\sum_{i=1}^f\sum_{j=1}^f\Big(\dfrac{\partial w}{\partial q_j}\dfrac{\partial^2 u}{\partial p_i\partial p_j}\dfrac{\partial v}{\partial q_i}⑩-\dfrac{\partial w}{\partial q_i}\dfrac{\partial^2 u}{\partial p_i\partial p_j}\dfrac{\partial v}{\partial q_j}⑲}$$
　　　　　　$${+\dfrac{\partial w}{\partial p_j}\dfrac{\partial^2 u}{\partial q_i\partial q_j}\dfrac{\partial v}{\partial p_i}⑯-\dfrac{\partial w}{\partial p_i}\dfrac{\partial^2 u}{\partial q_i\partial q_j}\dfrac{\partial v}{\partial p_j}㉑}$$
　　　　　　$${+\dfrac{\partial w}{\partial p_i}\dfrac{\partial^2 u}{\partial p_j\partial q_i}\dfrac{\partial v}{\partial q_j}㉓-\dfrac{\partial w}{\partial p_j}\dfrac{\partial^2 u}{\partial p_i\partial q_j}\dfrac{\partial v}{\partial q_i}⑫}$$
　　　　　　$${+\dfrac{\partial w}{\partial q_i}\dfrac{\partial^2 u}{\partial q_j\partial p_i}\dfrac{\partial v}{\partial p_j}⑰-\dfrac{\partial w}{\partial q_j}\dfrac{\partial^2 u}{\partial q_i\partial p_j}\dfrac{\partial v}{\partial p_i}⑭}$$
　　　　　　$${+\dfrac{\partial u}{\partial q_j}\dfrac{\partial^2 v}{\partial p_i\partial p_j}\dfrac{\partial w}{\partial q_i}⑱-\dfrac{\partial u}{\partial q_i}\dfrac{\partial^2 v}{\partial p_i\partial p_j}\dfrac{\partial w}{\partial q_j}③}$$
　　　　　　$${+\dfrac{\partial u}{\partial p_j}\dfrac{\partial^2 v}{\partial q_i\partial q_j}\dfrac{\partial w}{\partial p_i}㉔-\dfrac{\partial u}{\partial p_i}\dfrac{\partial^2 v}{\partial q_i\partial q_j}\dfrac{\partial w}{\partial p_j}⑤}$$
　　　　　　$${+\dfrac{\partial u}{\partial p_i}\dfrac{\partial^2 v}{\partial p_j\partial q_i}\dfrac{\partial w}{\partial q_j}⑦-\dfrac{\partial u}{\partial p_j}\dfrac{\partial^2 v}{\partial p_i\partial q_j}\dfrac{\partial w}{\partial q_i}⑳}$$
　　　　　　$${+\dfrac{\partial u}{\partial q_i}\dfrac{\partial^2 v}{\partial q_j\partial p_i}\dfrac{\partial w}{\partial p_j}①-\dfrac{\partial u}{\partial q_j}\dfrac{\partial^2 v}{\partial q_i\partial p_j}\dfrac{\partial w}{\partial p_i}㉒}$$
　　　　　　$${+\dfrac{\partial v}{\partial q_j}\dfrac{\partial^2 w}{\partial p_i\partial p_j}\dfrac{\partial u}{\partial q_i}②-\dfrac{\partial v}{\partial q_i}\dfrac{\partial^2 w}{\partial p_i\partial p_j}\dfrac{\partial u}{\partial q_j}⑪}$$
　　　　　　$${+\dfrac{\partial v}{\partial p_j}\dfrac{\partial^2 w}{\partial q_i\partial q_j}\dfrac{\partial u}{\partial p_i}⑧-\dfrac{\partial v}{\partial p_i}\dfrac{\partial^2 w}{\partial q_i\partial q_j}\dfrac{\partial u}{\partial p_j}⑬}$$
　　　　　　$${+\dfrac{\partial v}{\partial p_i}\dfrac{\partial^2 w}{\partial p_j\partial q_i}\dfrac{\partial u}{\partial q_j}⑮-\dfrac{\partial v}{\partial p_j}\dfrac{\partial^2 w}{\partial p_i\partial q_j}\dfrac{\partial u}{\partial q_i}④}$$
　　　　　　$${+\dfrac{\partial v}{\partial q_i}\dfrac{\partial^2 w}{\partial q_j\partial p_i}\dfrac{\partial u}{\partial p_j}⑨-\dfrac{\partial v}{\partial q_j}\dfrac{\partial^2 w}{\partial q_i\partial p_j}\dfrac{\partial u}{\partial p_i}⑥\Big)}$$
　　　　　　　↓　※１ $${\displaystyle\sum_{i=1}^f\sum_{j=1}^f\dfrac{\partial^2x}{\partial a_i\partial a_j}\Big(\dfrac{\partial y}{\partial b_i}\dfrac{\partial z}{\partial b_j}-\dfrac{\partial y}{\partial b_j}\dfrac{\partial z}{\partial b_i}\Big)=0}$$
  　　　　　　　　※２ $${\displaystyle\sum_{i=1}^f\sum_{j=1}^f\Big(\dfrac{\partial x}{\partial a_i}\dfrac{\partial^2y}{\partial a_j\partial b_i}\dfrac{\partial z}{\partial b_j}-\dfrac{\partial x}{\partial a_j}\dfrac{\partial^2y}{\partial a_i\partial b_j}\dfrac{\partial z}{\partial b_i}\Big)=0}$$ 
　　　　　　　　　より各項＝０である。（※１､２の証明は後述）
　　$${=\displaystyle\sum_{i=1}^f\sum_{j=1}^f\dfrac{\partial^2 u}{\partial p_i\partial p_j}\Big(\dfrac{\partial w}{\partial q_j}\dfrac{\partial v}{\partial q_i}⑩-\dfrac{\partial w}{\partial q_i}\dfrac{\partial v}{\partial q_j}⑲\Big)}$$
　　　$${+\displaystyle\sum_{i=1}^f\sum_{j=1}^f\dfrac{\partial^2 u}{\partial q_i\partial q_j}\Big(\dfrac{\partial w}{\partial p_j}\dfrac{\partial v}{\partial p_i}⑯-\dfrac{\partial w}{\partial p_i}\dfrac{\partial v}{\partial p_j}㉑\Big)}$$
　　　$${+\displaystyle\sum_{i=1}^f\sum_{j=1}^f\Big(\dfrac{\partial w}{\partial p_i}\dfrac{\partial^2 u}{\partial p_j\partial q_i}\dfrac{\partial v}{\partial q_j}㉓-\dfrac{\partial w}{\partial p_j}\dfrac{\partial^2 u}{\partial p_i\partial q_j}\dfrac{\partial v}{\partial q_i}⑫\Big)}$$
　　　$${+\displaystyle\sum_{i=1}^f\sum_{j=1}^f\Big(\dfrac{\partial w}{\partial q_i}\dfrac{\partial^2 u}{\partial q_j\partial p_i}\dfrac{\partial v}{\partial p_j}⑰-\dfrac{\partial w}{\partial q_j}\dfrac{\partial^2 u}{\partial q_i\partial p_j}\dfrac{\partial v}{\partial p_i}⑭\Big)}$$
　　　$${+\displaystyle\sum_{i=1}^f\sum_{j=1}^f\dfrac{\partial^2 v}{\partial p_i\partial p_j}\Big(\dfrac{\partial u}{\partial q_j}\dfrac{\partial w}{\partial q_i}⑱-\dfrac{\partial u}{\partial q_i}\dfrac{\partial w}{\partial q_j}③\Big)}$$
　　　$${+\displaystyle\sum_{i=1}^f\sum_{j=1}^f\dfrac{\partial^2 v}{\partial q_i\partial q_j}\Big(\dfrac{\partial u}{\partial p_j}\dfrac{\partial w}{\partial p_i}㉔-\dfrac{\partial u}{\partial p_i}\dfrac{\partial w}{\partial p_j}⑤\Big)}$$
　　　$${+\displaystyle\sum_{i=1}^f\sum_{j=1}^f\Big(\dfrac{\partial u}{\partial p_i}\dfrac{\partial^2 v}{\partial p_j\partial q_i}\dfrac{\partial w}{\partial q_j}⑦-\dfrac{\partial u}{\partial p_j}\dfrac{\partial^2 v}{\partial p_i\partial q_j}\dfrac{\partial w}{\partial q_i}⑳\Big)}$$
　　　$${+\displaystyle\sum_{i=1}^f\sum_{j=1}^f\Big(\dfrac{\partial u}{\partial q_i}\dfrac{\partial^2 v}{\partial q_j\partial p_i}\dfrac{\partial w}{\partial p_j}①-\dfrac{\partial u}{\partial q_j}\dfrac{\partial^2 v}{\partial q_i\partial p_j}\dfrac{\partial w}{\partial p_i}㉒\Big)}$$
　　　$${-\displaystyle\sum_{i=1}^f\sum_{j=1}^f\dfrac{\partial^2 w}{\partial p_i\partial p_j}\Big(\dfrac{\partial v}{\partial q_i}\dfrac{\partial u}{\partial q_j}⑪-\dfrac{\partial v}{\partial q_j}\dfrac{\partial u}{\partial q_i}②\Big)}$$
　　　$${-\displaystyle\sum_{i=1}^f\sum_{j=1}^f\dfrac{\partial^2 w}{\partial q_i\partial q_j}\Big(\dfrac{\partial v}{\partial p_i}\dfrac{\partial u}{\partial p_j}⑬-\dfrac{\partial v}{\partial p_j}\dfrac{\partial u}{\partial p_i}⑧\Big)}$$
　　　$${+\displaystyle\sum_{i=1}^f\sum_{j=1}^f\Big(\dfrac{\partial v}{\partial p_i}\dfrac{\partial^2 w}{\partial p_j\partial q_i}\dfrac{\partial u}{\partial q_j}⑮-\dfrac{\partial v}{\partial p_j}\dfrac{\partial^2 w}{\partial p_i\partial q_j}\dfrac{\partial u}{\partial q_i}④\Big)}$$
　　　$${+\displaystyle\sum_{i=1}^f\sum_{j=1}^f\Big(\dfrac{\partial v}{\partial q_i}\dfrac{\partial^2 w}{\partial q_j\partial p_i}\dfrac{\partial u}{\partial p_j}⑨-\dfrac{\partial v}{\partial q_j}\dfrac{\partial^2 w}{\partial q_i\partial p_j}\dfrac{\partial u}{\partial p_i}⑥\Big)}$$
　　$${=0}$$

※１ $${\displaystyle\sum_{i=1}^f\sum_{j=1}^f\dfrac{\partial^2x}{\partial a_i\partial a_j}\Big(\dfrac{\partial y}{\partial b_i}\dfrac{\partial z}{\partial b_j}-\dfrac{\partial y}{\partial b_j}\dfrac{\partial z}{\partial b_i}\Big)=0}$$  の証明
　(I)$${f=1}$$のとき
　　左辺$${=\dfrac{\partial^2x}{\partial a_1\partial a_1}\Big(\dfrac{\partial y}{\partial b_1}\dfrac{\partial z}{\partial b_1}-\dfrac{\partial y}{\partial b_1}\dfrac{\partial z}{\partial b_1}\Big)=0}$$　成り立つ
　(II)$${f=n}$$のとき､成り立つとする。
　　$${\displaystyle\sum_{i=1}^n\sum_{j=1}^n\dfrac{\partial^2x}{\partial a_i\partial a_j}\Big(\dfrac{\partial y}{\partial b_i}\dfrac{\partial z}{\partial b_j}-\dfrac{\partial y}{\partial b_j}\dfrac{\partial z}{\partial b_i}\Big)=0}$$  (A)
　(II)$${f=n+1}$$のとき
　　左辺$${=\displaystyle\sum_{i=1}^{n+1}\sum_{j=1}^{n+1}\dfrac{\partial^2x}{\partial a_i\partial a_j}\Big(\dfrac{\partial y}{\partial b_i}\dfrac{\partial z}{\partial b_j}-\dfrac{\partial y}{\partial b_j}\dfrac{\partial z}{\partial b_i}\Big)}$$
　　　$${=\displaystyle\sum_{i=1}^{n+1}\Big\{\sum_{j=1}^{n}\dfrac{\partial^2x}{\partial a_i\partial a_j}\Big(\dfrac{\partial y}{\partial b_i}\dfrac{\partial z}{\partial b_j}-\dfrac{\partial y}{\partial b_j}\dfrac{\partial z}{\partial b_i}\Big)}$$
　　　　　　　　　　　　　　$${+\dfrac{\partial^2x}{\partial a_i\partial a_{n+1}}\Big(\dfrac{\partial y}{\partial b_i}\dfrac{\partial z}{\partial b_{n+1}}-\dfrac{\partial y}{\partial b_{n+1}}\dfrac{\partial z}{\partial b_i}\Big)\Big\}}$$
　　　$${=\displaystyle\sum_{j=1}^{n}\sum_{i=1}^{n+1}\dfrac{\partial^2x}{\partial a_i\partial a_j}\Big(\dfrac{\partial y}{\partial b_i}\dfrac{\partial z}{\partial b_j}-\dfrac{\partial y}{\partial b_j}\dfrac{\partial z}{\partial b_i}\Big)}$$
　　　　　　　　　　　　　　$${+\displaystyle\sum_{i=1}^{n+1}\dfrac{\partial^2x}{\partial a_i\partial a_{n+1}}\Big(\dfrac{\partial y}{\partial b_i}\dfrac{\partial z}{\partial b_{n+1}}-\dfrac{\partial y}{\partial b_{n+1}}\dfrac{\partial z}{\partial b_i}\Big)}$$
　　　$${=\underline{\displaystyle\sum_{j=1}^{n}\sum_{i=1}^{n}\dfrac{\partial^2x}{\partial a_i\partial a_j}\Big(\dfrac{\partial y}{\partial b_i}\dfrac{\partial z}{\partial b_j}-\dfrac{\partial y}{\partial b_j}\dfrac{\partial z}{\partial b_i}\Big)}}$$　← (A)より 下線部$${=0}$$
　　　　　　　　　　　　　　$${+\displaystyle\sum_{j=1}^{n}\dfrac{\partial^2x}{\partial a_{n+1}\partial a_j}\Big(\dfrac{\partial y}{\partial b_{n+1}}\dfrac{\partial z}{\partial b_j}-\dfrac{\partial y}{\partial b_j}\dfrac{\partial z}{\partial b_{n+1}}\Big)}$$
　　　　$${+\displaystyle\sum_{i=1}^{n}\dfrac{\partial^2x}{\partial a_i\partial a_{n+1}}\Big(\dfrac{\partial y}{\partial b_i}\dfrac{\partial z}{\partial b_{n+1}}-\dfrac{\partial y}{\partial b_{n+1}}\dfrac{\partial z}{\partial b_i}\Big)}$$
　　　　　　$${+\underline{\dfrac{\partial^2x}{\partial a_{n+1}\partial a_{n+1}}\Big(\dfrac{\partial y}{\partial b_{n+1}}\dfrac{\partial z}{\partial b_{n+1}}-\dfrac{\partial y}{\partial b_{n+1}}\dfrac{\partial z}{\partial b_{n+1}}\Big)}}$$　← 下線部$${=0}$$
　　　$${=\displaystyle\sum_{j=1}^{n}\dfrac{\partial^2x}{\partial a_{n+1}\partial a_j}\Big(\dfrac{\partial y}{\partial b_{n+1}}\dfrac{\partial z}{\partial b_j}-\dfrac{\partial y}{\partial b_j}\dfrac{\partial z}{\partial b_{n+1}}\Big)}$$
　　　　$${+\displaystyle\sum_{i=1}^{n}\dfrac{\partial^2x}{\partial a_i\partial a_{n+1}}\Big(\dfrac{\partial y}{\partial b_i}\dfrac{\partial z}{\partial b_{n+1}}-\dfrac{\partial y}{\partial b_{n+1}}\dfrac{\partial z}{\partial b_i}\Big)}$$
　　　$${=\displaystyle\sum_{j=1}^{n}\dfrac{\partial^2x}{\partial a_{n+1}\partial a_j}\dfrac{\partial y}{\partial b_{n+1}}\dfrac{\partial z}{\partial b_j}-\displaystyle\sum_{j=1}^{n}\dfrac{\partial^2x}{\partial a_{n+1}\partial a_j}\dfrac{\partial y}{\partial b_j}\dfrac{\partial z}{\partial b_{n+1}}}$$
　　　　$${+\displaystyle\sum_{i=1}^{n}\dfrac{\partial^2x}{\partial a_i\partial a_{n+1}}\dfrac{\partial y}{\partial b_i}\dfrac{\partial z}{\partial b_{n+1}}-\displaystyle\sum_{i=1}^{n}\dfrac{\partial^2x}{\partial a_i\partial a_{n+1}}\dfrac{\partial y}{\partial b_{n+1}}\dfrac{\partial z}{\partial b_i}}$$
　　　$${=\underline{\displaystyle\sum_{j=1}^{n}\dfrac{\partial^2x}{\partial a_{n+1}\partial a_j}\dfrac{\partial y}{\partial b_{n+1}}\dfrac{\partial z}{\partial b_j}-\displaystyle\sum_{i=1}^{n}\dfrac{\partial^2x}{\partial a_{n+1}\partial a_i}\dfrac{\partial y}{\partial b_{n+1}}\dfrac{\partial z}{\partial b_i}}}$$　←下線部$${=0}$$
　　　　$${+\underline{\displaystyle\sum_{i=1}^{n}\dfrac{\partial^2x}{\partial a_{n+1}\partial a_i}\dfrac{\partial y}{\partial b_i}\dfrac{\partial z}{\partial b_{n+1}}-\displaystyle\sum_{j=1}^{n}\dfrac{\partial^2x}{\partial a_{n+1}\partial a_j}\dfrac{\partial y}{\partial b_j}\dfrac{\partial z}{\partial b_{n+1}}}}$$ ←下線部$${=0}$$
　　　$${=0}$$
　よって､すべての自然数$${n}$$について成り立つ。

※２ $${\displaystyle\sum_{i=1}^f\sum_{j=1}^f\Big(\dfrac{\partial x}{\partial a_i}\dfrac{\partial^2y}{\partial a_j\partial b_i}\dfrac{\partial z}{\partial b_j}-\dfrac{\partial x}{\partial a_j}\dfrac{\partial^2y}{\partial a_i\partial b_j}\dfrac{\partial z}{\partial b_i}\Big)=0}$$  の証明
　(I)$${f=1}$$のとき
　　左辺$${=\dfrac{\partial x}{\partial a_1}\dfrac{\partial^2y}{\partial a_1\partial b_1}\dfrac{\partial z}{\partial b_1}-\dfrac{\partial x}{\partial a_1}\dfrac{\partial^2y}{\partial a_1\partial b_1}\dfrac{\partial z}{\partial b_1}=0}$$　成り立つ。
　(II)$${f=1}$$のとき成り立つとする。
　　$${\displaystyle\sum_{i=1}^n\sum_{j=1}^n\Big(\dfrac{\partial x}{\partial a_i}\dfrac{\partial^2y}{\partial a_j\partial b_i}\dfrac{\partial z}{\partial b_j}-\dfrac{\partial x}{\partial a_j}\dfrac{\partial^2y}{\partial a_i\partial b_j}\dfrac{\partial z}{\partial b_i}\Big)=0}$$  (B)
　(III)$${f=n+1}$$のとき
　　左辺$${=\displaystyle\sum_{i=1}^{n+1}\sum_{j=1}^{n+1}\Big(\dfrac{\partial x}{\partial a_i}\dfrac{\partial^2y}{\partial a_j\partial b_i}\dfrac{\partial z}{\partial b_j}-\dfrac{\partial x}{\partial a_j}\dfrac{\partial^2y}{\partial a_i\partial b_j}\dfrac{\partial z}{\partial b_i}\Big)}$$
　　　$${=\displaystyle\sum_{i=1}^{n+1}\Big\{\sum_{j=1}^{n}\Big(\dfrac{\partial x}{\partial a_i}\dfrac{\partial^2y}{\partial a_j\partial b_i}\dfrac{\partial z}{\partial b_j}-\dfrac{\partial x}{\partial a_j}\dfrac{\partial^2y}{\partial a_i\partial b_j}\dfrac{\partial z}{\partial b_i}\Big)}$$
　　　　　　　　　　　　$${+\Big(\dfrac{\partial x}{\partial a_i}\dfrac{\partial^2y}{\partial a_{n+1}\partial b_i}\dfrac{\partial z}{\partial b_{n+1}}-\dfrac{\partial x}{\partial a_{n+1}}\dfrac{\partial^2y}{\partial a_i\partial b_{n+1}}\dfrac{\partial z}{\partial b_i}\Big)\Big\}}$$
　　　$${=\displaystyle\sum_{j=1}^{n}\sum_{i=1}^{n+1}\Big(\dfrac{\partial x}{\partial a_i}\dfrac{\partial^2y}{\partial a_j\partial b_i}\dfrac{\partial z}{\partial b_j}-\dfrac{\partial x}{\partial a_j}\dfrac{\partial^2y}{\partial a_i\partial b_j}\dfrac{\partial z}{\partial b_i}\Big)}$$
　　　　　　　　　　$${+\displaystyle\sum_{i=1}^{n+1}\Big(\dfrac{\partial x}{\partial a_i}\dfrac{\partial^2y}{\partial a_{n+1}\partial b_i}\dfrac{\partial z}{\partial b_{n+1}}-\dfrac{\partial x}{\partial a_{n+1}}\dfrac{\partial^2y}{\partial a_i\partial b_{n+1}}\dfrac{\partial z}{\partial b_i}\Big)}$$
　　　$${=\underline{\displaystyle\sum_{j=1}^{n}\sum_{i=1}^{n}\Big(\dfrac{\partial x}{\partial a_i}\dfrac{\partial^2y}{\partial a_j\partial b_i}\dfrac{\partial z}{\partial b_j}-\dfrac{\partial x}{\partial a_j}\dfrac{\partial^2y}{\partial a_i\partial b_j}\dfrac{\partial z}{\partial b_i}\Big)}}$$ ← (B)より下線部$${=0}$$
　　　　　　$${+\displaystyle\sum_{j=1}^{n}\Big(\dfrac{\partial x}{\partial a_{n+1}}\dfrac{\partial^2y}{\partial a_j\partial b_{n+1}}\dfrac{\partial z}{\partial b_j}-\dfrac{\partial x}{\partial a_j}\dfrac{\partial^2y}{\partial a_{n+1}\partial b_j}\dfrac{\partial z}{\partial b_{n+1}}\Big)}$$
　　　　$${+\displaystyle\sum_{i=1}^{n}\Big(\dfrac{\partial x}{\partial a_i}\dfrac{\partial^2y}{\partial a_{n+1}\partial b_i}\dfrac{\partial z}{\partial b_{n+1}}-\dfrac{\partial x}{\partial a_{n+1}}\dfrac{\partial^2y}{\partial a_i\partial b_{n+1}}\dfrac{\partial z}{\partial b_i}\Big)}$$
　　　　　　$${+\underline{\Big(\dfrac{\partial x}{\partial a_{n+1}}\dfrac{\partial^2y}{\partial a_{n+1}\partial b_{n+1}}\dfrac{\partial z}{\partial b_{n+1}}-\dfrac{\partial x}{\partial a_{n+1}}\dfrac{\partial^2y}{\partial a_{n+1}\partial b_{n+1}}\dfrac{\partial z}{\partial b_{n+1}}\Big)}}$$
　　　　　　　　　　　　　　　　　　　　　　　　　　　　← 下線部$${=0}$$
　　　$${=\displaystyle\sum_{j=1}^{n}\dfrac{\partial x}{\partial a_{n+1}}\dfrac{\partial^2y}{\partial a_j\partial b_{n+1}}\dfrac{\partial z}{\partial b_j}-\displaystyle\sum_{j=1}^{n}\dfrac{\partial x}{\partial a_j}\dfrac{\partial^2y}{\partial a_{n+1}\partial b_j}\dfrac{\partial z}{\partial b_{n+1}}}$$
　　　　$${+\displaystyle\sum_{i=1}^{n}\dfrac{\partial x}{\partial a_i}\dfrac{\partial^2y}{\partial a_{n+1}\partial b_i}\dfrac{\partial z}{\partial b_{n+1}}-\displaystyle\sum_{i=1}^{n}\dfrac{\partial x}{\partial a_{n+1}}\dfrac{\partial^2y}{\partial a_i\partial b_{n+1}}\dfrac{\partial z}{\partial b_i}}$$
　　　$${=\underline{\displaystyle\sum_{j=1}^{n}\dfrac{\partial x}{\partial a_{n+1}}\dfrac{\partial^2y}{\partial a_j\partial b_{n+1}}\dfrac{\partial z}{\partial b_j}-\displaystyle\sum_{i=1}^{n}\dfrac{\partial x}{\partial a_{n+1}}\dfrac{\partial^2y}{\partial a_i\partial b_{n+1}}\dfrac{\partial z}{\partial b_i}}}$$ ← 下線部$${=0}$$
　　　　$${+\underline{\displaystyle\sum_{i=1}^{n}\dfrac{\partial x}{\partial a_i}\dfrac{\partial^2y}{\partial a_{n+1}\partial b_i}\dfrac{\partial z}{\partial b_{n+1}}-\displaystyle\sum_{j=1}^{n}\dfrac{\partial x}{\partial a_j}\dfrac{\partial^2y}{\partial a_{n+1}\partial b_j}\dfrac{\partial z}{\partial b_{n+1}}}}$$ ← 下線部$${=0}$$
　　　$${=0}$$
　よって､すべての自然数$${n}$$について成り立つ。

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