26 Types of Tensors In the following subsections we introduce a number of

26 types of tensors in the following subsections we

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2.6 Types of Tensors In the following subsections we introduce a number of tensor types and categories and highlight their main characteristics and differences. These types and categories are not
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2.6.1 Covariant and Contravariant Tensors 59 x 1 x 2 x 3 e 1 e 1 x 1 x 2 x 3 e 1 e 2 x 1 x 2 x 3 e 1 e 3 x 1 x 2 x 3 e 2 e 1 x 1 x 2 x 3 e 2 e 2 x 1 x 2 x 3 e 2 e 3 x 1 x 2 x 3 e 3 e 1 x 1 x 2 x 3 e 3 e 2 x 1 x 2 x 3 e 3 e 3 Figure 13: The nine unit dyads associated with the double directions of rank-2 tensors in a 3D space with a rectangular Cartesian coordinate system. The vectors e i and e j ( i, j = 1 , 2 , 3 ) are unit vectors in the directions of coordinate axes where the first indexed e (blue) represents the first vector of the dyad while the second indexed e (red) represents the second vector of the dyad. In these nine frames, the first vector is fixed along each row while the second vector is fixed along each column. mutually exclusive and hence they overlap in general; moreover they may not be ex- haustive in their classes as some tensors may not instantiate any one of a complementary set of types such as being symmetric or anti-symmetric. 2.6.1 Covariant and Contravariant Tensors These are the main types of tensor with regard to the rules of their transformation between different coordinate systems. Covariant tensors are notated with subscript indices (e.g.
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2.6.1 Covariant and Contravariant Tensors 60 A i ) while contravariant tensors are notated with superscript indices (e.g. A ij ). A covariant tensor is transformed according to the following rule: ¯ A i = ∂x j ¯ x i A j (70) while a contravariant tensor is transformed according to the following rule: ¯ A i = ¯ x i ∂x j A j (71) where the barred and unbarred symbols represent the same mathematical object (tensor or coordinate) in the transformed and original coordinate systems respectively. An example of covariant tensors is the gradient of a scalar field while an example of contravariant tensors is the displacement vector. Some tensors of rank > 1 have mixed variance type, i.e. they are covariant in some indices and contravariant in others. In this case the covariant variables are indexed with subscripts while the contravariant variables are indexed with superscripts, e.g. A j i which is covariant in i and contravariant in j . A mixed type tensor transforms covariantly in its covariant indices and contravariantly in its contravariant indices, e.g. ¯ A l n m = ¯ x l ∂x i ∂x j ¯ x m ¯ x n ∂x k A i k j (72) To clarify the pattern of mathematical transformation of tensors, we explain step- by-step the practical rules to follow in writing tensor transformation equations between two coordinate systems, unbarred and barred, where for clarity we color the symbols of the tensor and the coordinates belonging to the unbarred system with blue while we use red to mark the symbols belonging to the barred system. Since there are three types of tensors: covariant, contravariant and mixed, we use three equations in each step.
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  • Summer '20
  • Rajendra Paramanik
  • Tensor, Coordinate system, Polar coordinate system, Coordinate systems

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