424 Write the analytical expressions of the rank 3 and rank 4 permutation

424 write the analytical expressions of the rank 3

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4.24 Write the analytical expressions of the rank-3 and rank- 4 permutation tensors. 4.25 Collect from the Index all the terms from matrix algebra which have connection to the Kronecker δ . 4.26 Correct, if necessary, the following equations: i 1 ··· i n i 1 ··· i n = n ijk C j C k = 0 ijk D jk = 0 e i × e j = ijk e j ( e i × e j ) · e k = ijk where C is a vector, D is a symmetric rank-2 tensor, and the indexed e are orthonormal basis vectors in a 3D space with a right-handed coordinate system. 4.27 What is wrong with the following equations? ijk δ 1 i δ 2 j δ 3 k = - 1 ij kl = δ ik δ jl + δ il δ jk il kl = δ il ij ij = 3! 4.28 Write the following in their determinantal form describing the general pattern of the relation between the indices of and δ and the indices of the rows and columns of the determinant: ijk lmk ijk lmn i 1 ··· i n j 1 ··· j n δ i 1 ...i n j 1 ...j n
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4.5 Exercises 115 4.29 Give two mnemonic devices used to memorize the widely used epsilon-delta identity and make a simple graphic illustration for one of these. 4.30 Correct, if necessary, the following equations: rst rst = 3! pst qst = 2 δ pq rst δ rt = rst δ st 4.31 State the mathematical definition of the generalized Kronecker delta δ i 1 ...i n j 1 ...j n . 4.32 Write each one of i 1 ...i n and i 1 ...i n in terms of the generalized Kronecker δ . 4.33 Make a survey, based on the Index, about the general mathematical terms used in the operations conducted by using the Kronecker and permutation tensors. 4.34 Write the mathematical relation that links the covariant permutation tensor, the contravariant permutation tensor, and the generalized Kronecker delta. 4.35 State the widely used epsilon-delta identity in terms of the generalized and ordinary Kronecker deltas.
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Chapter 5 Applications of Tensor Notation and Tech- niques In this chapter, we provide common definitions in the language of tensor calculus for some basic concepts and operations from matrix and vector algebra. We also provide a preliminary investigation of the nabla based differential operators and operations using, in part, tensor notation. Common identities in vector calculus as well as the integral theorems of Gauss and Stokes are also presented from this perspective. Finally, we provide a rather extensive set of detailed examples about the use of tensor language and techniques in proving common mathematical identities from vector calculus. 5.1 Common Definitions in Tensor Notation In this section, we give some common definitions of concepts and operations from vector and matrix algebra in tensor notation. The trace of a matrix A representing a rank-2 tensor in an n D space is given by: tr ( A ) = A ii ( i = 1 , . . . , n ) (165) For a 3 × 3 matrix representing a rank-2 tensor in a 3D space, the determinant is given 116
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5.1 Common Definitions in Tensor Notation 117 by: det ( A ) = A 11 A 12 A 13 A 21 A 22 A 23 A 31 A 32 A 33 = ijk A 1 i A 2 j A 3 k = ijk A i 1 A j 2 A k 3 (166) where the last two equalities represent the expansion of the determinant by row and by column. Alternatively, the determinant of a 3 × 3 matrix can be given by: det ( A ) = 1 3!
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  • Summer '20
  • Rajendra Paramanik
  • Tensor, Coordinate system, Polar coordinate system, Coordinate systems

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