3階完全反対称tensorでVector3重積を展開する

$ \pmb{a}\times(\pmb{b}\times\pmb{c})={\Large\pmb{\epsilon}}:\pmb{a}\otimes(\pmb{b}\times\pmb{c})

$ ={\Large\pmb{\epsilon}}:\pmb{a}\otimes({\Large\pmb{\epsilon}}:\pmb{b}\otimes\pmb{c})

$ ={\Large\pmb{\epsilon}}{\Large\pmb{\epsilon}}\vdots(\pmb{a}\otimes\pmb{b}\otimes\pmb{c})

ここの変形は、成分表示して、和をとる添え字の対応を見つけるとわかる

$ \pmb{a}\times(\pmb{b}\times\pmb{c})=\def\r#1{\textcolor{orange}{#1}}\def\g#1{\textcolor{lime}{#1}}\def\b#1{\textcolor{skyblue}{#1}}\def\p#1{\textcolor{red}{#1}}\epsilon_{i\r{j}\g{k}}a_\r{j}\epsilon_{\g{k}\b{l}\p{m}}b_\b{l}c_\p{m}\pmb{e}_i

$ =\def\r#1{\textcolor{orange}{#1}}\def\g#1{\textcolor{lime}{#1}}\def\b#1{\textcolor{skyblue}{#1}}\def\p#1{\textcolor{red}{#1}}\epsilon_{i\r{j}\g{k}}\epsilon_{\g{k}\b{l}\p{m}}a_\r{j}b_\b{l}c_\p{m}\pmb{e}_i

$ =2{\cal\pmb{W}}\vdots(\pmb{a}\otimes\pmb{b}\otimes\pmb{c})

$ =(2{\cal\pmb{W}}:(\pmb{b}\otimes\pmb{c}))\pmb{a}

$ \underline{=(\pmb{b}\wedge\pmb{c})\pmb{a}\quad}

$ =(\pmb{b}\otimes\pmb{c}-\pmb{c}\otimes\pmb{b})\pmb{a}

$ =(\pmb{b}\otimes\pmb{c})\pmb{a}-(\pmb{c}\otimes\pmb{b})\pmb{a}

$ \underline{=(\pmb{c}\cdot\pmb{a})\pmb{b}-(\pmb{b}\cdot\pmb{a})\pmb{c}\quad}