...Another important multiple index quantity is the permutation tensor or epsilon tensor, defined by: 1 if ijk is an even permutation of 123 -1 if ijk is an odd permutation of 123 0 otherwise, ie if ijk are not all distinct. Most of the basic identities of vector algebra and vector...

...with components (uxv)i = -(vxu)< = eijkujVk , (1.2.4) where e^ is the alternating symbol with values {+1 if ijk is an even permutation of 123, -1 if ijk is an odd permutation of 123, (1.2.5) 0 otherwise. The array of components of the second-order unit tensor I in the rectangular frame...

...way to write this concisely is by using the alternating symbol e^ 425 which is defined to be equal to 1 if ijk is an even permutation of 123, -1 if ijk is an odd permutation of 123, and zero if any two of ijk are equal. Using this notation, and the Einstein summation convention (where...

...to define the vector cross product of the unit vectors, e,, that is e, xe; = e0kek (4.1.5) where + 1 if ijk is an even permutation of 123 - 1 if ijk is an odd permutation of 123 0 if ijk has a repeated number axb = (amej x (6nej = aj7&m x e.) = a^e^e, The triple scalar product...

...commutation relations (2.62), expressed in the form [a,-, aI] = 2ieiikak, (2.123) where + 1 if (//A:) is an even permutation of (123), - 1 if (ijk) is an odd permutation of (123), 0 otherwise, (2.124) equation (2.27) reads V'x = Vx + 6Vy, V'y = Vy- 6Vx, V'z = Vz, which is the z...

...\-iooy \ o oo and a simple calculation shows that [T^Tj] = 52k=i fijkTk, where Cyfc is the Levi-Civita symbol (1 if ijk is an even permutation of 123, —1 if ijk is an odd permutation of 123, and 0 otherwise). The usual basis for su(2) consists of Tfc = —\ia^ where are the Pauli spin matrices...

...simple calculation shows that [r,,^] = £]*=1 e »;* r *i wn ere eiik is the Levi-Civita symbol (l if ijk is an even permutation of 123, -1 if ijk is an odd permutation of 123, and 0 otherwise). The usual basis for su(2) consists of Tit — — i^k, where are the Paul! spin matrices...

...referred to as the LeviCivita tensor, is the completely antisymmetric tensor of rank 3, €ijk, defined by 1 if ijk is an even permutation of 123 — 1 if ijk is an odd permutation of 123 (4.6) 0 if any two of ijk are equal That the transformed 6^ is completely antisymmetric is easy to...

...introduce the permutation symbol e¿¿b also called the alternating symbol or the alternator, defined by {1 if (ijk) is an even permutation of (123). -1 if (ijk) is an odd permutation of (123), 0 otherwise, ie, if two subscripts are repeated. From this definition it follows that €jjk = -Cijy...

...3 (UX v)/ — y~^ 2^. ZijkUjVk y=l *=1 where 0 if any pair of indices /', /. k are equal, 1 if ij k is an even permutation of 123, - 1 if ijk is an odd permutation of 123. The vector space R3 with this law of composition is a non-commutative, non-associative algebra. The...