The transformed systems are shown as dashed while the original system is shown

The transformed systems are shown as dashed while the

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coordinates. The transformed systems are shown as dashed while the original system is shown as solid. Transformations can be active , when they change the state of the observed object (e.g. translating the object in space), or passive when they are based on keeping the state of the object and changing the state of the coordinate system from which the object is observed. Such distinction is based on an implicit assumption of a more general frame of reference in the background. A permutation of a set of objects, which are normally numbers like (1 , 2 , . . . , n ) or symbols like ( i, j, k ) , is a particular ordering or arrangement of these objects. An even permutation is a permutation resulting from an even number of single-step exchanges (also known as transpositions ) of neighboring objects starting from a presumed original permutation of these objects. Similarly, an odd permutation is a permutation resulting
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2.3 General Rules 53 from an odd number of such exchanges. It has been shown that when a transformation from one permutation to another can be done in different ways, possibly with different numbers of exchanges, the parity of all these possible transformations is the same, i.e. all are even or all are odd, and hence there is no ambiguity in characterizing the transformation from one permutation to another by the parity alone. 2.3 General Rules In the following, we present some very general rules that apply to the mathematical ex- pressions and relations in tensor calculus. No index is allowed to occur more than twice in a legitimate tensor term. [24] A free index should be understood to vary over its range (e.g. 1 , . . . , n ) and hence it can be interpreted as saying “for all components represented by the index”. Therefore, a free index represents a number of terms or expressions or equalities equal to the number of allowed values of its range. For example, when i and j can vary over the range 1 , . . . , n the following expression: A i + B i (58) represents n separate expressions while the following equation: A j i = B j i (59) represents n × n separate equations. [24] We adopt this assertion, which is common in the literature of tensor calculus, as we think it is suitable for this level. However, there are many instances in the literature of tensor calculus where indices are repeated more than twice in a single term. The bottom line is that as long as the tensor expression makes sense and the intention is clear, such repetitions should be allowed with no need in our view to take special precaution like using parentheses. In particular, the forthcoming summation convention will not apply automatically in such cases although summation on such indices, if needed, can be carried out explicitly, by using the summation symbol or by a special declaration of such intention similar to the summation convention. Anyway, in the present book we will not use indices repeated more than twice in a single term.
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2.3 General Rules 54 According to the summation convention , which is widely used in the literature of tensor calculus including in the present book, dummy indices
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  • Summer '20
  • Rajendra Paramanik
  • Tensor, Coordinate system, Polar coordinate system, Coordinate systems

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