form based on components identified by indices and hence the notation is

Form based on components identified by indices and

This preview shows page 50 - 52 out of 171 pages.

form based on components identified by indices and hence the notation is suggestive of an underlying coordinate system, although being a tensor makes it form-invariant under certain coordinate transformations and therefore it possesses certain invariant properties. The index-free notation is usually identified by using bold face non-italic symbols, like a and B , while the indicial notation is identified by using light face indexed italic symbols such as a i and B ij . It is noteworthy that although rank-0 and rank-1 tensors are, respectively, scalars and vectors, not all scalars and vectors (in their generic sense) are tensors of these ranks. Similarly, rank-2 tensors are normally represented by square matrices but not all square matrices represent rank-2 tensors. 2.2 General Terms and Concepts In the following, we introduce and define a number of essential concepts and terms which form a principal part of the technical and conceptual structure of tensor calculus. These concepts and terms are needed in the development of the forthcoming sections and chap-
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2.2 General Terms and Concepts 50 ters. Tensor term is a product of tensors including scalars and vectors. Tensor expression is an algebraic sum (or more generally a linear combination) of tensor terms which may be a trivial sum in the case of a single term. Tensor equality (which is symbolized by = ”) is an equality of two tensor terms and/or expressions. A special case of this is tensor identity which is an equality of general validity. [22] An index that occurs once in a tensor term is a free index while an index that occurs twice in a tensor term is a dummy or bound index . The order of a tensor is identified by the number of its indices (e.g. A i jk is a tensor of order 3) which normally identifies the tensor rank as well. However, when contraction (see § 3.4) of indices operation takes place once or more, the order of the tensor is not affected but its rank is reduced by two for each contraction operation. [23] Hence, the order of a tensor is equal to the number of all of its indices including the dummy indices, while the rank is equal to the number of its free indices only. Tensors with subscript indices, like A ij , are called covariant , while tensors with su- perscript indices, like A k , are called contravariant . Tensors with both types of indices, like A lmn lk , are called mixed type. More details about this classification will follow in § 2.6.1. Subscript indices, rather than subscripted tensors, are also dubbed covariant and superscript indices are dubbed contravariant. The Zero tensor is a tensor whose all components are zero. The Unit tensor or unity tensor, which is usually defined for rank-2 tensors, is a tensor whose all elements are zero except the ones with identical values of all indices which are assigned the value 1.
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  • Summer '20
  • Rajendra Paramanik
  • Tensor, Coordinate system, Polar coordinate system, Coordinate systems

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