Editing Antisymmetric tensor (section)

{{Short description|Tensor equal to the negative of any of its transpositions}}In [[mathematics]] and [[theoretical physics]], a [[tensor]] is '''antisymmetric''' or '''alternating on''' (or '''with respect to''') '''an index subset''' if it alternates [[Sign (mathematics)|sign]] (+/−) when any two indices of the subset are interchanged.<ref>{{cite book|author1=K.F. Riley |author2=M.P. Hobson |author3=S.J. Bence | title=Mathematical methods for physics and engineering|url=https://archive.org/details/mathematicalmeth00rile |url-access=registration | publisher=Cambridge University Press| year=2010 | isbn=978-0-521-86153-3}}</ref><ref>{{cite book|author1=Juan Ramón Ruíz-Tolosa |author2=Enrique Castillo | title=From Vectors to Tensors | publisher=Springer| year=2005| isbn=978-3-540-22887-5 |url=https://books.google.com/books?id=vgGQUrQMzwYC&pg=PA225 |page=225}} section §7.</ref> The index subset must generally either be all ''covariant'' or all ''contravariant''.

For example,
<math display=block>T_{ijk\dots} = -T_{jik\dots} = T_{jki\dots} = -T_{kji\dots} = T_{kij\dots} = -T_{ikj\dots}</math>
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 [[Tensor order|order]] <math>k</math> may be referred to as a [[Differential form|differential <math>k</math>-form]], and a completely antisymmetric contravariant tensor field may be referred to as a [[Multivector|<math>k</math>-vector]] field.