As indicated earlier a tensor of m contravariant indices and n covariant

As indicated earlier a tensor of m contravariant

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As indicated earlier, a tensor of m contravariant indices and n covariant indices may be described as a tensor of type ( m, n ). This may be extended to include the weight w as a third entry and hence the type of the tensor is identified by ( m, n, w ). Relative tensors can be added and subtracted (see § 3.1) if they are of the same variance type and have the same weight; [37] the result is a tensor of the same type and weight. Also, relative tensors can be equated if they are of the same type and weight. Multiplication of relative tensors produces a relative tensor whose weight is the sum of the weights of the original tensors. Hence, if the weights are added up to a non-zero value the result is a relative tensor of that weight; otherwise it is an absolute tensor. [35] The Jacobian J is the determinant of the Jacobian matrix J of the transformation between the unbarred and barred systems, that is: J = det ( J ) = ∂x 1 ¯ x 1 ∂x 1 ¯ x 2 · · · ∂x 1 ¯ x n ∂x 2 ¯ x 1 ∂x 2 ¯ x 2 · · · ∂x 2 ¯ x n . . . . . . . . . . . . ∂x n ¯ x 1 ∂x n ¯ x 2 · · · ∂x n ¯ x n (84) For more details, the reader is advised to consult more advanced textbooks on this subject. [36] Some of these labels are used differently by different authors as the terminology of tensor calculus is not universally approved and hence the conventions of each author should be checked. Also, there is an obvious overlap between this classification (i.e. absolute and relative) and the previous classification (i.e. true and pseudo) at least according to some conventions. [37] This statement should be generalized by including w = 0 which corresponds to absolute tensors and hence “relative” in this statement is more general than being opposite to “absolute”. Accordingly, and from the perspective of relative tensors (i.e. assuming that other qualifications such as matching in the indicial structure are met), two absolute tensors can be added/subtracted but an absolute and a relative tensor (i.e. with w 6 = 0 ) cannot since they are “relative” tensors with different weights.
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2.6.4 Isotropic and Anisotropic Tensors 70 2.6.4 Isotropic and Anisotropic Tensors Isotropic tensors are characterized by the property that the values of their components are invariant under coordinate transformation by proper rotation of axes. In contrast, the values of the components of anisotropic tensors are dependent on the orientation of the coordinate axes. Notable examples of isotropic tensors are scalars (rank-0), the vector 0 (rank-1), Kronecker delta δ ij (rank-2) and Levi-Civita tensor ijk (rank-3). Many tensors describing physical properties of materials, such as stress and magnetic susceptibility, are anisotropic. Direct and inner products (see § 3.3 and 3.5) of isotropic tensors are isotropic tensors. The zero tensor of any rank is isotropic; therefore if the components of a tensor vanish in a particular coordinate system they will vanish in all properly and improperly rotated coordinate systems. [38] Consequently, if the components of two tensors are identical in a particular coordinate system they are identical in all transformed coordinate systems.
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  • Summer '20
  • Rajendra Paramanik
  • Tensor, Coordinate system, Polar coordinate system, Coordinate systems

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