ホッジ双対
A i = ( A 1 e 2 e 3 + A 2 e 3 e 1 + A 3 e 1 e 2 ) e 1 e 2 e 3 = A 1 e 1 ( e 2 e 3 ) 2 + A 2 e 2 ( e 3 e 1 ) 2 + A 3 e 3 ( e 1 e 2 ) 2 = − ( A 1 e 1 + A 2 e 2 + A 3 e 3 ) = − ( ⋆ A ) {\displaystyle {\begin{aligned}{\boldsymbol {A}}i&=(A_{1}{\boldsymbol {e}}_{2}{\boldsymbol {e}}_{3}+A_{2}{\boldsymbol {e}}_{3}{\boldsymbol {e}}_{1}+A_{3}{\boldsymbol {e}}_{1}{\boldsymbol {e}}_{2}){\boldsymbol {e}}_{1}{\boldsymbol {e}}_{2}{\boldsymbol {e}}_{3}\\&=A_{1}{\boldsymbol {e}}_{1}({\boldsymbol {e}}_{2}{\boldsymbol {e}}_{3})^{2}+A_{2}{\boldsymbol {e}}_{2}({\boldsymbol {e}}_{3}{\boldsymbol {e}}_{1})^{2}+A_{3}{\boldsymbol {e}}_{3}({\boldsymbol {e}}_{1}{\boldsymbol {e}}_{2})^{2}\\&=-(A_{1}{\boldsymbol {e}}_{1}+A_{2}{\boldsymbol {e}}_{2}+A_{3}{\boldsymbol {e}}_{3})\\&=-(\star {\boldsymbol {A}})\end{aligned}}}