number of pseudo tensors is a pseudo tensor The direct product of a mix of true

# Number of pseudo tensors is a pseudo tensor the

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objects in the reflected system. number of pseudo tensors is a pseudo tensor. The direct product of a mix of true and pseudo tensors is a true or pseudo tensor depending on the number of pseudo tensors involved in the product as being even or odd respectively. Similar rules to those of the direct product apply to the cross product , including the curl operation, involving tensors (which are usually of rank-1) with the addition of a pseudo factor for each cross product operation. This factor is contributed by the permutation tensor which is implicit in the definition of the cross product (see Eqs. 173 and 192). As we will see in § 4.2, the permutation tensor is a pseudo tensor. In summary, what determines the tensor type (true or pseudo) of the tensor terms involv- ing direct [34] and cross products is the parity of the multiplicative factors of pseudo type plus the number of cross product operations involved since each cross product operation contributes an factor. Examples of true scalars are temperature, mass and the dot product of two polar or [34] Inner product (see § 3.5) is the result of a direct product (see § 3.3) operation followed by a contraction (see § 3.4) and hence it is like a direct product in this context.
2.6.3 Absolute and Relative Tensors 68 two axial vectors, while examples of pseudo scalars are the dot product of an axial vector and a polar vector and the scalar triple product of polar vectors. Examples of polar vectors are displacement and acceleration, while examples of axial vectors are angular velocity and cross product of polar vectors in general, including the curl operation on polar vectors, due to the involvement of the permutation symbol which is a pseudo tensor as stated already. As indicated before, the essence of the distinction between true (i.e. polar) and pseudo (i.e. axial) vectors is that the direction of a pseudo vector depends on the observer choice of the handedness of the coordinate system whereas the direction of a true vector is independent of such a choice. Examples of true tensors of rank-2 are stress and rate of strain tensors, while examples of pseudo tensors of rank-2 are direct products of two vectors: one polar and one axial. Examples of true tensors of higher ranks are piezoelectric moduli tensor (rank-3) and elasticity tensor (rank-4), while examples of pseudo tensors of higher ranks are the permutation tensor of these ranks. 2.6.3 Absolute and Relative Tensors Considering an arbitrary transformation from a general coordinate system to another, a tensor of weight w is defined by the following general tensor transformation: ¯ A ij...k lm...n = ∂x ¯ x w ¯ x i ∂x a ¯ x j ∂x b · · · ¯ x k ∂x c ∂x d ¯ x l ∂x e ¯ x m · · · ∂x f ¯ x n A ab...c de...f (83)
2.6.3 Absolute and Relative Tensors 69 where ∂x ¯ x is the Jacobian of the transformation between the two systems. [35] When w = 0 the tensor is described as an absolute or true tensor, and when w 6 = 0 the tensor is described as a relative tensor. When w = - 1 the tensor may be described as a pseudo tensor, while when w = 1 the tensor may be described as a tensor density . [36]

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• Summer '20
• Rajendra Paramanik
• Tensor, Coordinate system, Polar coordinate system, Coordinate systems

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