three even permutations of the indices of the rank 3 permutation tensor and the

# Three even permutations of the indices of the rank 3

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three even permutations of the indices of the rank-3 permutation tensor and the three odd permutations of these indices assuming no repetition in indices. The three permutations in each case are obtained by starting from a given number in the cycle and rotating in the given direction to obtain the other two numbers in the permutation. 111 113 112 121 133 123 122 132 131 211 213 212 221 233 223 222 232 231 311 313 312 321 333 323 322 332 331 Figure 17: Graphical illustration of the rank-3 permutation tensor ijk where circular nodes represent 0 , square nodes represent 1 and triangular nodes represent - 1 . The definition of the rank - n permutation tensor (i.e. i 1 i 2 ...i n ) is similar to the definition of the rank-3 permutation tensor with regard to the repetition in its indices ( i 1 , i 2 , · · · , i n )
4.2 Permutation 99 Even 1 3 2 +1 Odd 1 3 2 - 1 Figure 18: Graphical demonstration of the cyclic nature of the even and odd permutations of the indices of the rank-3 permutation tensor assuming no repetition in indices. and being even or odd permutations in their correspondence to (1 , 2 , · · · , n ) , that is: i 1 i 2 ...i n = 1 ( i 1 , i 2 , . . . , i n is even permutation of 1 , 2 , . . . , n ) - 1 ( i 1 , i 2 , . . . , i n is odd permutation of 1 , 2 , . . . , n ) 0 ( repeated index ) (124) As well as the inductive definition of the permutation tensor (as given by Eqs. 122, 123 and 124), the permutation tensor of any rank can also be defined analytically where the entries of the tensor are calculated from closed form formulae. The entries of the rank-2 permutation tensor can be calculated from the following closed form equation: ij = ( j - i ) (125) Similarly, for the rank-3 permutation tensor we have: ijk = 1 2 ( j - i ) ( k - i ) ( k - j ) (126)
4.2 Permutation 100 while for the rank-4 permutation tensor we have: ijkl = 1 12 ( j - i ) ( k - i ) ( l - i ) ( k - j ) ( l - j ) ( l - k ) (127) More generally, the entries of the rank- n permutation tensor can be obtained from the following identity: a 1 a 2 ··· a n = n - 1 Y i =1 " 1 i ! n Y j = i +1 ( a j - a i ) # = 1 S ( n - 1) Y 1 i<j n ( a j - a i ) (128) where S ( n - 1) is the super factorial function of the argument ( n - 1) which is defined by: S ( k ) = k Y i =1 i ! = 1! · 2! · . . . · k ! (129) A simpler formula for calculating the entries of the rank- n permutation tensor can be obtained from the previous one by dropping the magnitude of the multiplication factors and taking their signs only , that is: a 1 a 2 ··· a n = Y 1 i<j n sgn ( a j - a i ) = sgn Y 1 i<j n ( a j - a i ) ! (130) where sgn( k ) is the sign function of the argument k which is defined by: sgn( k ) = +1 ( k > 0) - 1 ( k < 0) 0 ( k = 0) (131) The permutation tensor is totally anti-symmetric (see § 2.6.5) in each pair of its
4.3 Useful Identities Involving δ or/and 101 indices, i.e. it changes sign on swapping any two of its indices, that is: i 1 ...i k ...i l ...i n = - i 1 ...i l ...i k ...i n (132) The reason is that any exchange of two indices requires an even/odd number of single- step shifts to the right of the first index plus an odd/even number of single-step shifts to the left of the second index, so the total number of shifts is odd and hence it is an odd permutation of the original arrangement.

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• Summer '20
• Rajendra Paramanik
• Tensor, Coordinate system, Polar coordinate system, Coordinate systems

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