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IEVref: | 102-03-43 | ID: | |

Language: | en | Status: Standard | |

Term: | antisymmetric tensor | ||

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Definition: | tensor of the second order defined by a bilinear form such that $f(U\text{,}\text{\hspace{0.17em}}V)=-f(V\text{,}\text{\hspace{0.17em}}U)$ Note 1 to entry: The components of an antisymmetric tensor are such that ${T}_{ij}=-{T}_{ji}$, and in particular ${T}_{ii}=0$. Note 2 to entry: An antisymmetric tensor defined on a three-dimensional space has three strict components which can be considered as the coordinates ${W}_{1}\text{,}{W}_{2}\text{,}{W}_{3}$ of an axial vector: $\left(\begin{array}{ccc}0& {W}_{3}& -{W}_{2}\\ -{W}_{3}& 0& {W}_{1}\\ {W}_{2}& -{W}_{1}& 0\end{array}\right)$ The axial vector associated with the antisymmetric tensor $U\otimes V-V\otimes U$ is the vector product of the two vectors. | ||

Publication date: | 2008-08 | ||

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Internal notes: | 2017-02-20: Editorial revisions in accordance with the information provided in C00019 (IEV 102) - evaluation. JGO | ||

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Note 1 to entry: The components of an antisymmetric tensor are such that ${T}_{ij}=-{T}_{ji}$, and in particular ${T}_{ii}=0$.

Note 2 to entry: An antisymmetric tensor defined on a three-dimensional space has three strict components which can be considered as the coordinates ${W}_{1}\text{,}{W}_{2}\text{,}{W}_{3}$ of an axial vector:

$\left(\begin{array}{ccc}0& {W}_{3}& -{W}_{2}\\ -{W}_{3}& 0& {W}_{1}\\ {W}_{2}& -{W}_{1}& 0\end{array}\right)$

The axial vector associated with the antisymmetric tensor $U\otimes V-V\otimes U$ is the vector product of the two vectors.