The permutation tensor is a pseudo tensor since it acquires a minus sign under

The permutation tensor is a pseudo tensor since it

This preview shows page 102 - 105 out of 171 pages.

The permutation tensor is a pseudo tensor since it acquires a minus sign under an improper orthogonal transformation of coordinates, i.e. inversion of axes with possible superposition of rotation (see § 2.2). However, it is an isotropic tensor since it is conserved under proper coordinate transformations. The permutation tensor may be considered as a contravariant relative tensor of weight +1 or a covariant relative tensor of weight - 1 . Hence, in 2D, 3D and n D spaces we have the following identities for the components of the permutation tensor: [57] ij = ij ijk = ijk i 1 i 2 ...i n = i 1 i 2 ...i n (133) 4.3 Useful Identities Involving δ or/and In the following subsections we introduce and discuss a number of common identities which involve the Kronecker and permutations tensors. Some of these identities involve only one of these tensors while others involve both. [57] We note that in equalities like this we are equating the components, as indicated above.
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4.3.1 Identities Involving δ 102 4.3.1 Identities Involving δ When an index of the Kronecker delta is involved in a contraction operation by repeating an index in another tensor in its own term, the effect of this is to replace the shared index in the other tensor by the other index of the Kronecker delta, that is: δ ij A j = A i (134) In such cases the Kronecker delta is described as an index replacement or substitution operator . Hence, we have: δ ij δ jk = δ ik (135) Similarly: δ ij δ jk δ ki = δ ik δ ki = δ ii = n (136) where n is the space dimension. The last part of this equation (i.e. δ ii = n ) can be easily justified by the fact that δ ii is the trace of the identity tensor considering the summation convention. Due to the fact that the coordinates are independent of each other (see § 2.2), we also have the following identity: [58] ∂x i ∂x j = j x i = x i,j = δ ij (137) Hence, in an n D space we obtain the following identity from the last two identities: i x i = δ ii = n (138) [58] This identity, like many other identities in this chapter and in the book in general, is valid even for general coordinate systems although we use Cartesian notation to avoid unnecessary distraction at this level. The alert reader should be able to notate such identities in their general forms.
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4.3.1 Identities Involving δ 103 Based on the above identities and facts, the following identity can be shown to apply in orthonormal Cartesian coordinate systems: ∂x i ∂x j = ∂x j ∂x i = δ ij = δ ij (139) This identity is based on the two facts that the coordinates are independent, and the covariant and contravariant types are the same in orthonormal Cartesian coordinate sys- tems. Similarly, for a coordinate system with a set of orthonormal [59] basis vectors, such as the orthonormal Cartesian system, the following identity can be easily proved: e i · e j = δ ij (140) where the indexed e are the basis vectors. This identity is no more than a mathematical statement of the fact that the basis vectors in orthonormal systems are mutually orthogonal and of unit length.
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  • Summer '20
  • Rajendra Paramanik
  • Tensor, Coordinate system, Polar coordinate system, Coordinate systems

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