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| In [[mathematics]] and [[theoretical physics]], a [[tensor]] is '''antisymmetric 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| author=K.F. Riley, M.P. Hobson, S.J. Bence| title=Mathematical methods for physics and engineering| publisher=Cambridge University Press| year=2010 | isbn=978-0-521-86153-3}}</ref><ref>{{cite book| author=Juan Ramón Ruíz-Tolosa, Enrique Castillo| title=From Vectors to Tensors| other=§7| publisher=Springer| year=2005| isbn=978-3-540-22887-5}}, [http://books.google.co.za/books?id=vgGQUrQMzwYC&pg=PA225 google books]</ref> The index subset must generally either be all ''covariant'' or all ''contravariant''.
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| For example,
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| :<math>T_{ijk\dots} = -T_{jik\dots} = T_{jki\dots} = -T_{kji\dots} = T_{kij\dots} = -T_{ikj\dots}</math>
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| holds when the tensor is antisymmetric on it first three indices.
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| If a tensor changes sign under exchange of ''any'' pair of its indices, then the tensor is '''completely''' (or '''totally''') '''antisymmetric'''. A completely antisymmetric covariant tensor may be referred to as a [[differential form|''p''-form]], and a completely antisymmetric contravariant tensor may be referred to as a [[multivector|''p''-vector]].
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| ==Antisymmetric and symmetric tensors==
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| A tensor '''A''' that is antisymmetric on indices ''i'' and ''j'' has the property that the [[Tensor contraction|contraction]] with a tensor '''B''' that is symmetric on indices ''i'' and ''j'' is identically 0.
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| For a general tensor '''U''' with components <math>U_{ijk\dots}</math> and a pair of indices ''i'' and ''j'', '''U''' has symmetric and antisymmetric parts defined as:
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| :{|
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| | <math>U_{(ij)k\dots}=\frac{1}{2}(U_{ijk\dots}+U_{jik\dots})</math> || || (symmetric part)
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| |-
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| | <math>U_{[ij]k\dots}=\frac{1}{2}(U_{ijk\dots}-U_{jik\dots})</math> || ||(antisymmetric part).
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| |}
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| 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
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| :<math>U_{ijk\dots}=U_{(ij)k\dots}+U_{[ij]k\dots}.</math> | |
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| ==Notation==
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| 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''',
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| :<math>M_{[ab]} = \frac{1}{2!}(M_{ab} - M_{ba}) ,</math> | |
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| and for an order 3 covariant tensor '''T''',
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| :<math>T_{[abc]} = \frac{1}{3!}(T_{abc}-T_{acb}+T_{bca}-T_{bac}+T_{cab}-T_{cba}) .</math>
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| In any number of dimensions, these are equivalent to
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| :<math>M_{[ab]} = \frac{1}{2!} \, \delta_{ab}^{cd} M_{cd} ,</math>
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| :<math>T_{[abc]} = \frac{1}{3!} \, \delta_{abc}^{def} T_{def} .</math>
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| More generally, irrespective of the number of dimensions, antisymmetrization over ''p'' indices may be expressed as
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| :<math>S_{[a_1 \dots a_p]} = \frac{1}{p!} \delta_{a_1 \dots a_p}^{b_1 \dots b_p} S_{b_1 \dots b_p} .</math>
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| In the above,
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| :<math>\delta_{ab\dots}^{cd\dots}</math>
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| is the [[generalized Kronecker delta]] of the appropriate order.
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| ==Example==
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| An important antisymmetric tensor in physics is the [[electromagnetic tensor]] '''F''' in [[electromagnetism]].
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| == See also ==
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| *[[Levi-Civita symbol]]
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| *[[Symmetric tensor]]
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| *[[Antisymmetric matrix]]
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| *[[Exterior algebra]]
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| *[[Ricci calculus]]
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| ==References==
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| {{reflist}}
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| * {{cite book |pages=85–86, §3.5| author=J.A. Wheeler, C. Misner, K.S. Thorne| title=[[Gravitation (book)|Gravitation]]| publisher=W.H. Freeman & Co| year=1973 | isbn=0-7167-0344-0}}
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| * {{cite book |author=R. Penrose| title=[[The Road to Reality]]| publisher= Vintage books| year=2007 | isbn=0-679-77631-1}}
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| ==External links==
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| * [http://mathworld.wolfram.com/AntisymmetricTensor.html] - mathworld, wolfram
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| {{tensors}}
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| [[Category:Tensors]]
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