173 and the scalar triple product see Eq 174 in tensor notation plus the fact

173 and the scalar triple product see eq 174 in

This preview shows page 106 - 109 out of 171 pages.

(see Eq. 173) and the scalar triple product (see Eq. 174) in tensor notation plus the fact that these vectors are unit vectors.
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4.3.3 Identities Involving δ and 106 4.3.3 Identities Involving δ and For the rank-2 permutation tensor, we have the following identity which involves the Kronecker delta in 2D: ij kl = δ ik δ il δ jk δ jl = δ ik δ jl - δ il δ jk (148) This identity can simply be proved inductively by building a table for the values on the left and right hand sides as the indices are varied. The pattern of the indices in the determinant of this identity is simple, that is the indices of the first provide the indices for the rows while the indices of the second provide the indices for the columns. [62] Another useful identity involving the rank-2 permutation tensor with the Kronecker delta in 2D is the following: il kl = δ ik (149) This can be obtained from the previous identity by replacing j with l followed by a minimal algebraic manipulation using tensor calculus rules. [63] Similarly, we have the following identity which correlates the rank-3 permutation tensor to the Kronecker delta in 3D: ijk lmn = δ il δ im δ in δ jl δ jm δ jn δ kl δ km δ kn = ( δ il δ jm δ kn + δ im δ jn δ kl + δ in δ jl δ km ) - (150) ( δ il δ jn δ km + δ im δ jl δ kn + δ in δ jm δ kl ) [62] The role of these indices in indexing the rows and columns can be shifted. This can be explained by the fact that the positions of the two epsilons can be exchanged, since ordinary multiplication is commutative, and hence the role of the epsilons in providing the indices for the rows and columns will be shifted. This can also be done by taking the transposition of the array of the determinant, which does not change the value of the determinant since det ( A ) = det ( A T ) , with an exchange of the indices of the Kronecker symbols since the Kronecker symbol is symmetric in its two indices. [63] That is: il kl = δ ik δ ll - δ il δ lk = 2 δ ik - δ il δ lk = 2 δ ik - δ ik = δ ik
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4.3.3 Identities Involving δ and 107 Again, the indices in the determinant of this identity follow the same pattern as that of Eq. 148. Another useful identity in this category is the following: ijk lmk = δ il δ im δ jl δ jm = δ il δ jm - δ im δ jl (151) This identity can be obtained from the identity of Eq. 150 by replacing n with k . [64] The pattern of the indices in this identity is as before if we exclude the repetitive indices. More generally, the determinantal form of Eqs. 148 and 150, which link the rank-2 and rank-3 permutation tensors to the Kronecker tensors in 2D and 3D spaces, can be extended to link the rank- n permutation tensor to the Kronecker tensor in an n D space, that is: i 1 i 2 ··· i n j 1 j 2 ··· 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 (152) Again, the pattern of the indices in the determinant of this identity in their relation to the indices of the two epsilons follow the same rules as those of Eqs. 148 and 150.
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  • Summer '20
  • Rajendra Paramanik
  • Tensor, Coordinate system, Polar coordinate system, Coordinate systems

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