42 The concepts of repetitive and non repetitive permutations may be useful in

42 the concepts of repetitive and non repetitive

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[42] The concepts of repetitive and non-repetitive permutations may be useful in tackling this question.
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Chapter 3 Tensor Operations There are various operations that can be performed on tensors to produce other tensors in general. Examples of these operations are addition/subtraction, multiplication by a scalar (rank-0 tensor), multiplication of tensors (each of rank > 0 ), contraction and permutation. Some of these operations, such as addition and multiplication, involve more than one tensor while others, such as contraction and permutation, are performed on a single tensor . In this chapter we provide a glimpse on the main elementary tensor operations of algebraic nature that permeate tensor algebra and calculus. First, we should remark that the last section of this chapter, which is about the quotient rule for tensor test, is added to this chapter because it is the most appropriate place for it in the present book considering the dependency of the definition of this rule on other tensor operations; otherwise the section is not about a tensor operation in the same sense as the operations presented in the other sections of this chapter. Another remark is that in tensor algebra division is allowed only for scalars, hence if the components of an indexed tensor should appear in a denominator, the tensor should be redefined to avoid this, e.g. B i = 1 A i . 83
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3.1 Addition and Subtraction 84 3.1 Addition and Subtraction Tensors of the same rank and type [43] can be added algebraically to produce a tensor of the same rank and type, e.g. a = b + c A i = B i - C i A i j = B i j + C i j (98) The added/subtracted terms should have the same indicial structure with regard to their free indices, as explained in § 2.3; hence A i jk and B j ik cannot be added or subtracted al- though they are of the same rank and type, but A mi mjk and B i jk can be added and subtracted. Addition of tensors is associative and commutative , that is: [44] ( A + B ) + C = A + ( B + C ) (99) A + B = B + A (100) 3.2 Multiplication of Tensor by Scalar A tensor can be multiplied by a scalar, which generally should not be zero , to produce a tensor of the same variance type, rank and indicial structure, e.g. A j ik = aB j ik (101) where a is a non-zero scalar. As indicated by the equation, multiplying a tensor by a scalar means multiplying each component of the tensor by that scalar. Multiplication [43] Here, “type” refers to variance type (covariant/contravariant/mixed) and true/pseudo type as well as other qualifications to which the tensors participating in an addition or subtraction operation should match such as having the same weight if they are relative tensors, as outlined previously (refer for example to § 2.6.3). [44] Associativity and commutativity can include subtraction if the minus sign is absorbed in the subtracted tensor; in which case the operation is converted to addition.
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3.3 Tensor Multiplication 85 by a scalar is commutative , and associative when more than two factors are involved.
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  • Summer '20
  • Rajendra Paramanik
  • Tensor, Coordinate system, Polar coordinate system, Coordinate systems

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