# Definition The Levi-Civita or volume tensor is a rank-3 [[tensor]] which gives the oriented volume of a parallelepiped. It is defined by $ \overline{\epsilon}(\pmb{u},\pmb{v},\pmb{w}) = \pmb{u} \cdot (\pmb{v} \times \pmb{w}) $ where $\pmb{u}, \pmb{v}, \pmb{w}$ are three [[vector|vectors]] in $\mathbb{R}^3$ spanning the parallelepiped. Its components on an [[Orthogonality|orthonormal]] basis $\{\pmb{e}_1, \pmb{e}_2, \pmb{e}_3\}$ are $ \epsilon{\pmb{e}_i, \pmb{e}_j, \pmb{e}_k} = = \left\{ \begin{array}{ll} 0 & \text{ unless } i \neq j \neq k\\ +1 & \{i,j,k\} = \{1,2,3\}, \{2,3,1\}, \text{ or } \{3,1,2\}\\ -1 & \{i,j,k\} = \{3,2,1\}, \{1,3,2\}, \text{ or } \{2,1,3\}\\ \end{array} \right. $ which is just the Levi-Civita Symbol, i.e., the volume of the oriented unit parallelepiped spanned by $\pmb{e}_1, \pmb{e}_2$ and $\pmb{e}_3$ # Applications The Levi-Civita symbol allows us to write many quantities more easily. For example, the [[Angular Momentum|angular momentum]] commutation relations $ [L_i, L_j] = i \epsilon_{ijk} L_k $ where we used the [[Einstein summation convention]].