title: Antisymmetric tensor
text: # Antisymmetric tensor
  
  In mathematics and theoretical physics, a tensor is antisymmetric on (or with respect to) an index subset if it alternates sign (+/−) when any two indices of the subset are interchanged. The index subset must generally either be all covariant or all contravariant.
  
  For example,
  
  $T_{ijk\dots }=-T_{jik\dots }=T_{jki\dots }=-T_{kji\dots }=T_{kij\dots }=-T_{ikj\dots }$
  
  holds when the tensor is antisymmetric with respect to its first three indices.
  
  If a tensor changes sign under exchange of each pair of its indices, then the tensor is completely (or totally) antisymmetric. A completely antisymmetric covariant tensor field of order $k$ may be referred to as a differential $k$-form, and a completely antisymmetric contravariant tensor field may be referred to as a $k$-vector field.
  
  ## Antisymmetric and symmetric tensors
  
  A tensor A that is antisymmetric on indices $i$ and $j$ has the property that the contraction with a tensor B that is symmetric on indices $i$ and $j$ is identically 0.
  
  For a general tensor U with components $U_{ijk\dots }$ and a pair of indices $i$ and $j,$ U has symmetric and antisymmetric parts defined as:
  
      $U_{(ij)k\dots }={\frac {1}{2}}(U_{ijk\dots }+U_{jik\dots })$ (symmetric part) $U_{[ij]k\dots }={\frac {1}{2}}(U_{ijk\dots }-U_{jik\dots })$ (antisymmetric part).
  
  Similar definitions can be given for other pairs of indices. As the term "part" suggests, a tensor is the sum of its symmetric part and antisymmetric part for a given pair of indices, as in
  
  $U_{ijk\dots }=U_{(ij)k\dots }+U_{[ij]k\dots }.$
  
  ## Notation
  
  A shorthand notation for anti-symmetrization is denoted by a pair of square brackets. For example, in arbitrary dimensions, for an order 2 covariant tensor M,
  
  $M_{[ab]}={\frac {1}{2!}}(M_{ab}-M_{ba}),$
  
  and for an order 3 covariant tensor T,
  
  $T_{[abc]}={\frac {1}{3!}}(T_{abc}-T_{acb}+T_{bca}-T_{bac}+T_{cab}-T_{cba}).$
  
  In any 2 and 3 dimensions, these can be written as
  
  ${\begin{aligned}M_{[ab]}&={\frac {1}{2!}}\,\delta _{ab}^{cd}M_{cd},\\[2pt]T_{[abc]}&={\frac {1}{3!}}\,\delta _{abc}^{def}T_{def}.\end{aligned}}$
  
  where $\delta _{ab\dots }^{cd\dots }$ is the generalized Kronecker delta, and we use the Einstein notation to summation over like indices.
  
  More generally, irrespective of the number of dimensions, antisymmetrization over $p$ indices may be expressed as
  
  $T_{[a_{1}\dots a_{p}]}={\frac {1}{p!}}\delta _{a_{1}\dots a_{p}}^{b_{1}\dots b_{p}}T_{b_{1}\dots b_{p}}.$
  
  In general, every tensor of rank 2 can be decomposed into a symmetric and anti-symmetric pair as:
  
  $T_{ij}={\frac {1}{2}}(T_{ij}+T_{ji})+{\frac {1}{2}}(T_{ij}-T_{ji}).$
  
  This decomposition is not in general true for tensors of rank 3 or more, which have more complex symmetries.
  
  ## Examples
  
  Totally antisymmetric tensors include:
  
  * Trivially, all scalars and vectors (tensors of order 0 and 1) are totally antisymmetric (as well as being totally symmetric).
  * The electromagnetic tensor, $F_{\mu \nu }$ in electromagnetism.
  * The Riemannian volume form on a pseudo-Riemannian manifold.
  