Finally the following identity can be obtained from the definition of the rank

Finally the following identity can be obtained from

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Finally, the following identity can be obtained from the definition of the rank-3 permuta- tion tensor (Eq. 123) and the use of the Kronecker delta as an index replacement operator (Eq. 134): ijk δ ij = ijk δ ik = ijk δ jk = 0 (157) 4.4 Generalized Kronecker δ The generalized Kronecker delta is defined inductively by: δ i 1 ...i n j 1 ...j n = 1 ( j 1 . . . j n is even permutation of i 1 . . . i n ) - 1 ( j 1 . . . j n is odd permutation of i 1 . . . i n ) 0 ( repeated j ’s ) (158)
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4.4 Generalized Kronecker δ 110 It can also be defined analytically by the following n × n determinant: δ i 1 ...i n j 1 ...j n = δ i 1 j 1 δ i 1 j 2 · · · δ i 1 j n δ i 2 j 1 δ i 2 j 2 · · · δ i 2 j n . . . . . . . . . . . . δ i n j 1 δ i n j 2 · · · δ i n j n (159) where the δ i j entries in the determinant are the ordinary Kronecker deltas as defined previously. In this equation, the pattern of the indices in the generalized Kronecker delta symbol δ i 1 ...i n j 1 ...j n in connection to the indices in the determinant is similar to the previous patterns, that is the upper indices in δ i 1 ...i n j 1 ...j n provide the upper indices in the ordinary deltas by indexing the rows of the determinant, while the lower indices in δ i 1 ...i n j 1 ...j n provide the lower indices in the ordinary deltas by indexing the columns of the determinant. From the above given identities, it can be shown that: i 1 ...i n j 1 ...j n = δ i 1 j 1 δ i 1 j 2 · · · δ i 1 j n δ i 2 j 1 δ i 2 j 2 · · · δ i 2 j n . . . . . . . . . . . . δ i n j 1 δ i n j 2 · · · δ i n j n (160) Now, on comparing the last equation with the definition of the generalized Kronecker delta, i.e. Eq. 159, we conclude that: i 1 ...i n j 1 ...j n = δ i 1 ...i n j 1 ...j n (161) As an instance of Eq. 161, the relation between the rank- n permutation tensor in its covariant and contravariant forms and the generalized Kronecker delta in an n D space is
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4.4 Generalized Kronecker δ 111 given by: i 1 ...i n = δ 1 ... n i 1 ...i n i 1 ...i n = δ i 1 ...i n 1 ... n (162) where the first of these equations can be obtained from Eq. 161 by substituting (1 . . . n ) for ( i 1 . . . i n ) in the two sides with relabeling j with i and noting that 1 ... n = 1 , while the second equation can be obtained from Eq. 161 by substituting (1 . . . n ) for ( j 1 . . . j n ) and noting that 1 ... n = 1 . Hence, the permutation tensor can be considered as an instance of the generalized Kronecker delta. Consequently, the rank- n permutation tensor can be written as an n × n determinant consisting of the ordinary Kronecker deltas. Moreover, Eq. 162 can provide another definition for the permutation tensor in its covariant and contravariant forms, in addition to the previous inductive and analytic definitions of this tensor as given by Eqs. 124 and 128. Returning to the widely used epsilon-delta identity of Eq. 151, if we define: [67] δ ij lm = δ ijk lmk (163) and consider the above identities which correlate the permutation tensor, the generalized Kronecker tensor and the ordinary Kronecker tensor, then an identity equivalent to Eq.
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  • Summer '20
  • Rajendra Paramanik
  • Tensor, Coordinate system, Polar coordinate system, Coordinate systems

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