二項積同士の乗法
出典: フリー百科事典『ウィキペディア（Wikipedia）』 (2018/10/19 10:45 UTC 版)
二項積同士の積がまた二項積となるように五種類の演算を以下のように定義する。a, b, c, d はベクトルとする。 点乗積交叉積点乗積点乗積: a b ⋅ c d := a ( b ⋅ c ) d = ( b ⋅ c ) a d {\displaystyle \mathbf {a} \mathbf {b} \cdot \mathbf {c} \mathbf {d} :=\mathbf {a} (\mathbf {b} \cdot \mathbf {c} )\mathbf {d} =(\mathbf {b} \cdot \mathbf {c} )\mathbf {a} \mathbf {d} } 二重点乗積: a b : c d := ( a ⋅ d ) ( b ⋅ c ) , or a b : c d := c ⋅ a b ⋅ d = ( a ⋅ c ) ( b ⋅ d ) . {\displaystyle {\begin{aligned}\mathbf {ab} :\mathbf {cd} &:=(\mathbf {a} \cdot \mathbf {d} )(\mathbf {b} \cdot \mathbf {c} ),\quad {\text{or}}\\\mathbf {ab} :\mathbf {cd} &:=\mathbf {c} \cdot \mathbf {ab} \cdot \mathbf {d} =(\mathbf {a} \cdot \mathbf {c} )(\mathbf {b} \cdot \mathbf {d} ).\end{aligned}}} 点交叉積: a b × ⋅ c d := ( a ⋅ c ) ( b × d ) {\displaystyle \mathbf {ab} \mathop {_{\textstyle \times }} ^{\textstyle \cdot }\mathbf {c} \mathbf {d} :=(\mathbf {a} \cdot \mathbf {c} )(\mathbf {b} \times \mathbf {d} )} 交叉積 交叉点乗積: a b ⋅ × c d := ( a × c ) ( b ⋅ d ) {\displaystyle \mathbf {ab} \mathop {_{\textstyle \cdot }} ^{\textstyle \times }\mathbf {cd} :=(\mathbf {a} \times \mathbf {c} )(\mathbf {b} \cdot \mathbf {d} )} 二重交叉積: a b × × c d := ( a × c ) ( b × d ) {\displaystyle \mathbf {ab} \mathop {_{\textstyle \times }} ^{\textstyle \times }\mathbf {cd} :=(\mathbf {a} \times \mathbf {c} )(\mathbf {b} \times \mathbf {d} )} 一般二項積テンソル A = ∑ i a i b i , B = ∑ i c i d i {\displaystyle \mathbf {A} =\sum _{i}\mathbf {a} _{i}\mathbf {b} _{i},\quad \mathbf {B} =\sum _{i}\mathbf {c} _{i}\mathbf {d} _{i}} に対しては: 点乗積交叉積点乗積点乗積: A ⋅ B := ∑ j ∑ i ( b i ⋅ c j ) a i d j {\displaystyle \mathbf {A} \cdot \mathbf {B} :=\sum _{j}\sum _{i}(\mathbf {b} _{i}\cdot \mathbf {c} _{j})\mathbf {a} _{i}\mathbf {d} _{j}} 二重点乗積: A : B := ∑ j ∑ i ( a i ⋅ d j ) ( b i ⋅ c j ) , or A : B := ∑ j ∑ i ( a i ⋅ c j ) ( b i ⋅ d j ) . {\displaystyle {\begin{aligned}\mathbf {A} :\mathbf {B} &:=\sum _{j}\sum _{i}(\mathbf {a} _{i}\cdot \mathbf {d} _{j})(\mathbf {b} _{i}\cdot \mathbf {c} _{j}),\quad {\text{or}}\\\mathbf {A} :\mathbf {B} &:=\sum _{j}\sum _{i}(\mathbf {a} _{i}\cdot \mathbf {c} _{j})(\mathbf {b} _{i}\cdot \mathbf {d} _{j}).\end{aligned}}} 点交叉積: A × ⋅ B := ∑ j ∑ i ( a i ⋅ c j ) ( b i × d j ) {\displaystyle \mathbf {A} \mathop {_{\textstyle \times }} ^{\textstyle \cdot }\mathbf {B} :=\sum _{j}\sum _{i}(\mathbf {a} _{i}\cdot \mathbf {c} _{j})(\mathbf {b} _{i}\times \mathbf {d} _{j})} 交叉積 交叉点乗積: A ⋅ × B := ∑ j ∑ i ( a i × c j ) ( b i ⋅ d j ) {\displaystyle \mathbf {A} \mathop {_{\textstyle \cdot }} ^{\textstyle \times }\mathbf {B} :=\sum _{j}\sum _{i}(\mathbf {a} _{i}\times \mathbf {c} _{j})(\mathbf {b} _{i}\cdot \mathbf {d} _{j})} 二重交叉積: A × × B := ∑ i , j ( a i × c j ) ( b i × d j ) {\displaystyle \mathbf {A} \mathop {_{\textstyle \times }} ^{\textstyle \times }\mathbf {B} :=\sum _{i,j}(\mathbf {a} _{i}\times \mathbf {c} _{j})(\mathbf {b} _{i}\times \mathbf {d} _{j})}
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