324 Assess the following statement considering the two meanings of the word

324 assess the following statement considering the

This preview shows page 96 - 100 out of 171 pages.

3.24 Assess the following statement considering the two meanings of the word “tensor” related to the rank: “Inner product operation of two tensors does not necessarily produce a tensor”. Can this statement be correct in a sense and wrong in another? 3.25 What is the operation of tensor permutation and how it is related to the operation of transposition of matrices? 3.26 Is it possible to permute scalars or vectors and why? 3.27 What is the meaning of the term “isomers”? 3.28 Describe in detail the quotient rule and how it is used as a test for tensors. 3.29 Why the quotient rule is used instead of the standard transformation equations of tensors?
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Chapter 4 δ and Tensors In this chapter, we conduct a preliminary investigation about the δ and tensors and their properties and functions as well as the relation between them. These tensors are of particular importance in tensor calculus due to their distinctive properties and unique transformation attributes. They are numerical tensors with fixed components in all coordinate systems. The first is called Kronecker delta or unit tensor , while the second is called Levi-Civita , [54] permutation , anti-symmetric and alternating tensor . 4.1 Kronecker δ The Kronecker δ is a rank-2 tensor in all dimensions. It is defined as: δ ij = 1 ( i = j ) 0 ( i 6 = j ) ( i, j = 1 , 2 , . . . n ) (118) where n is the space dimension, and hence it can be considered as the identity matrix . For example, in a 3D space the Kronecker δ tensor is given by: [ δ ij ] = δ 11 δ 12 δ 13 δ 21 δ 22 δ 23 δ 31 δ 32 δ 33 = 1 0 0 0 1 0 0 0 1 (119) [54] This name is usually used for the rank-3 tensor. Also some authors distinguish between the permu- tation tensor and the Levi-Civita tensor even for rank-3. Moreover, some of the common labels and descriptions of are more specific to rank-3. 96
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4.2 Permutation 97 The components of the covariant, contravariant and mixed types of this tensor are the same , that is: δ ij = δ ij = δ i j = δ j i (120) The Kronecker δ tensor is symmetric , that is: δ ij = δ ji δ ij = δ ji (121) where i, j = 1 , 2 , . . . , n . Moreover, it is conserved [55] under all proper and improper coordinate transformations. Since it is conserved under proper transformations, it is an isotropic tensor. [56] 4.2 Permutation The permutation tensor has a rank equal to the number of dimensions, and hence a rank- n permutation tensor has n n components. The rank-2 permutation tensor ij is defined by: 12 = 1 21 = - 1 11 = 22 = 0 (122) Similarly, the rank-3 permutation tensor ijk is defined by: ijk = 1 ( i, j, k is even permutation of 1,2,3 ) - 1 ( i, j, k is odd permutation of 1,2,3 ) 0 ( repeated index ) (123) Figure 17 is a graphical illustration of the rank-3 permutation tensor ijk while Figure [55] “Conserved” means that the tensor keeps the values of its components following a coordinate transfor- mation. [56] Here, being conserved under all transformations is stronger than being isotropic as the former applies even under improper coordinate transformations while isotropy is restricted to proper transformations.
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4.2 Permutation 98 18, which may be used as a mnemonic device, demonstrates the cyclic nature of the
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  • Summer '20
  • Rajendra Paramanik
  • Tensor, Coordinate system, Polar coordinate system, Coordinate systems

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