Levi-Civita Symbol

Abstract

The Levi-Civita symbol ijk is a tensor of rank three and is defined by ijk =    0, if any two labels are the same 1, if i, j, k is an even permutation of 1,2,3 −1, if i, j, k is an odd permutation of 1,2,3 (1) The Levi-Civita symbol ijk is anti-symmetric on each pair of indexes. The determinant of a matrix A with elements a ij can be written in term of ijk as det a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 = 3 i=1 3 j=1 3 k=1 ijk a 1i a 2j a 3k = ijk a 1i a 2j a 3k (2) Note the compact notation where the summation over the spatial directions is dropped. It is this one that is in use. (4) orforeachcoordinate(a×b)i=ijkajbk(5)12Properties • TheLevi-Civitatensorijkhas3×3×3=27components. • 3×(6+1)=21componentsareequalto0. • 3componentsareequalto1. • 3componentsareequalto−1.3IdentitiesTheproductoftwoLevi-CivitasymbolscanbeexpressedasafunctionoftheKronecker'ssym-bolδijijklmn=+δilδjmδkn+δimδjnδkl+δinδjlδkm−δimδjlδkn−δilδjnδkm−δinδjmδkl(6)Settingi=lgivesijkimn=δjmδkn−δjnδkm(7)proofijkimn=δii(δjmδkn−δjnδkm)+δimδjnδki+δinδjiδkm−δimδjiδkn−δinδjmδki=3(δjmδkn−δjnδkm)+δkmδjn+δjnδkm−δjmδkn−δknδjm=δjmδkn−δjnδkmSettingi=landj=mgivesijkijn=2δkn(8)