二項積
ベクトル同士の演算に基づいて、(結果がまた二項積となるように)ベクトルと二項積の間に四種類の積を構成できる。
{\displaystyle {\begin{aligned}\langle \mathbf {A} \rangle &=\quad A_{11}\,\mathbf {i} \times \mathbf {i} +A_{12}\,\mathbf {i} \times \mathbf {j} +A_{31}\,\mathbf {i} \times \mathbf {k} \\&\quad +A_{21}\,\mathbf {j} \times \mathbf {i} +A_{22}\,\mathbf {j} \times \mathbf {j} +A_{23}\,\mathbf {j} \times \mathbf {k} \\&\quad +A_{31}\mathbf {k} \times \mathbf {i} +A_{32}\mathbf {k} \times \mathbf {j} +A_{33}\mathbf {k} \times \mathbf {k} \\&=\quad A_{12}\mathbf {k} -A_{31}\mathbf {j} -A_{21}\mathbf {k} \\&\quad +A_{23}\,\mathbf {i} \,+A_{31}\mathbf {j} -A_{32}\mathbf {i} \\&=(A_{23}-A_{32})\mathbf {i} +(A_{31}-A_{13})\mathbf {j} +(A_{12}-A_{21})\mathbf {k} .\end{aligned}}}
A のこの縮約をレヴィ–チヴィタテンソルで書けば:
⟨ A ⟩ = ε i j k A j k . {\displaystyle \langle \mathbf {A} \rangle ={\epsilon _{i}}^{jk}A_{jk}.}