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.^[1]^[2] 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.

^ K.F. Riley; M.P. Hobson; S.J. Bence (2010). Mathematical methods for physics and engineering. Cambridge University Press. ISBN 978-0-521-86153-3.
^ Juan Ramón Ruíz-Tolosa; Enrique Castillo (2005). From Vectors to Tensors. Springer. p. 225. ISBN 978-3-540-22887-5. section §7.

[1] K.F. Riley; M.P. Hobson; S.J. Bence (2010). Mathematical methods for physics and engineering. Cambridge University Press. ISBN 978-0-521-86153-3.

[2] Juan Ramón Ruíz-Tolosa; Enrique Castillo (2005). From Vectors to Tensors. Springer. p. 225. ISBN 978-3-540-22887-5. section §7.

[1]

[2]