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

Three even permutations of the indices of the rank 3

This preview shows page 99 - 103 out of 171 pages.

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 )
Image of page 99
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)
Image of page 100
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
Image of page 101
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.
Image of page 102
Image of page 103

You've reached the end of your free preview.

Want to read all 171 pages?

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

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern

Stuck? We have tutors online 24/7 who can help you get unstuck.
A+ icon
Ask Expert Tutors You can ask You can ask ( soon) You can ask (will expire )
Answers in as fast as 15 minutes
A+ icon
Ask Expert Tutors