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Antisymmetric tensor
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<h2 id="antisymmetric-and-symmetric-tensors">Antisymmetric and symmetric tensors</h2> <p>A tensor A that is antisymmetric on indices i {\displaystyle i} and j {\displaystyle j} has the property that the <a href="/facts/Tensor_contraction/fqz0zvWp">contraction</a> with a tensor B that is symmetric on indices i {\displaystyle i} and j {\displaystyle j} is identically 0. </p><p>For a general tensor U with components U i j k … {\displaystyle U_{ijk\dots }} and a pair of indices i {\displaystyle i} and j , {\displaystyle j,} U has symmetric and antisymmetric parts defined as: </p> <table><tbody><tr><td> U ( i j ) k … = 1 2 ( U i j k … + U j i k … ) {\displaystyle U_{(ij)k\dots }={\frac {1}{2}}(U_{ijk\dots }+U_{jik\dots })} </td><td> </td><td>(symmetric part)</td></tr><tr><td> U [ i j ] k … = 1 2 ( U i j k … − U j i k … ) {\displaystyle U_{[ij]k\dots }={\frac {1}{2}}(U_{ijk\dots }-U_{jik\dots })} </td><td> </td><td>(antisymmetric part).</td></tr></tbody></table> <p>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 i j k … = U ( i j ) k … + U [ i j ] k … . {\displaystyle U_{ijk\dots }=U_{(ij)k\dots }+U_{[ij]k\dots }.} </p> <h2 id="notation">Notation</h2> <p>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 [ a b ] = 1 2 ! ( M a b − M b a ) , {\displaystyle M_{[ab]}={\frac {1}{2!}}(M_{ab}-M_{ba}),} and for an order 3 covariant tensor T, T [ a b c ] = 1 3 ! ( T a b c − T a c b + T b c a − T b a c + T c a b − T c b a ) . {\displaystyle T_{[abc]}={\frac {1}{3!}}(T_{abc}-T_{acb}+T_{bca}-T_{bac}+T_{cab}-T_{cba}).} </p><p>In any 2 and 3 dimensions, these can be written as M [ a b ] = 1 2 ! δ a b c d M c d , T [ a b c ] = 1 3 ! δ a b c d e f T d e f . {\displaystyle {\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 δ a b … c d … {\displaystyle \delta _{ab\dots }^{cd\dots }} is the <a href="/facts/Generalized_Kronecker_delta/gcGAy0bm">generalized Kronecker delta</a>, and the <a href="/facts/Einstein_summation_convention/UXwS43wU">Einstein summation convention</a> is in use. </p><p>More generally, irrespective of the number of dimensions, antisymmetrization over p {\displaystyle p} indices may be expressed as T [ a 1 … a p ] = 1 p ! δ a 1 … a p b 1 … b p T b 1 … b p . {\displaystyle 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}}.} </p><p>In general, every tensor of rank 2 can be decomposed into a symmetric and anti-symmetric pair as: T i j = 1 2 ( T i j + T j i ) + 1 2 ( T i j − T j i ) . {\displaystyle T_{ij}={\frac {1}{2}}(T_{ij}+T_{ji})+{\frac {1}{2}}(T_{ij}-T_{ji}).} </p><p>This decomposition is not in general true for tensors of rank 3 or more, which have more complex symmetries. </p> <h2 id="examples">Examples</h2> <p>Totally antisymmetric tensors include: </p> <ul><li>Trivially, all scalars and vectors (tensors of order 0 and 1) are totally antisymmetric (as well as being totally symmetric).</li> <li>The <a href="/facts/Electromagnetic_tensor/MWYuKLPy">electromagnetic tensor</a>, F μ ν {\displaystyle F_{\mu \nu }} in <a href="/facts/Electromagnetism/oAkmALHT">electromagnetism</a>.</li> <li>The <a href="/facts/Riemannian_volume_form/2x37pDLg">Riemannian volume form</a> on a <a href="/facts/Pseudo-Riemannian_manifold/VFXsVjqb">pseudo-Riemannian manifold</a>.</li></ul> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Antisymmetric_matrix/nw5CQeLN">Antisymmetric matrix</a> – Form of a matrixPages displaying short descriptions of redirect targets</li> <li><a href="/facts/Exterior_algebra/rdN4TvpR">Exterior algebra</a> – Algebra associated to any vector space</li> <li><a href="/facts/Levi-Civita_symbol/WBKUA3Tw">Levi-Civita symbol</a> – Antisymmetric permutation object acting on tensors</li> <li><a href="/facts/Ricci_calculus/We4yudeq">Ricci calculus</a> – Tensor index notation for tensor-based calculations</li> <li><a href="/facts/Symmetric_tensor/CWKBfgnv">Symmetric tensor</a> – Tensor invariant under permutations of vectors it acts on</li> <li><a href="/facts/Symmetrization/6XbIgHGx">Symmetrization</a> – process that converts any function in n variables to a symmetric function in n variablesPages displaying wikidata descriptions as a fallback</li></ul> <h2 id="notes">Notes</h2> <ul><li><a href="/facts/Roger_Penrose/r4mlrFUH">Penrose, Roger</a> (2007). <i><a href="/facts/The_Road_to_Reality/306chfRy">The Road to Reality</a></i>. Vintage books. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-679-77631-4.</li> <li>J.A. Wheeler; C. Misner; K.S. Thorne (1973). <i><a href="/facts/Gravitation_(book)/1KAfFThq">Gravitation</a></i>. W.H. Freeman & Co. pp. 85–86, §3.5. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-7167-0344-0.</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="http://mathworld.wolfram.com/AntisymmetricTensor.html">Antisymmetric Tensor – mathworld.wolfram.com</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>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. <a href="978-0-521-86153-3" target="_blank">978-0-521-86153-3</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Juan Ramón Ruíz-Tolosa; Enrique Castillo (2005). From Vectors to Tensors. Springer. p. 225. ISBN 978-3-540-22887-5. section §7. <a href="978-3-540-22887-5" target="_blank">978-3-540-22887-5</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> </ol>