Tensor symbolic notation

There are different ways to write down tensors and their components, whose usefulness depends on the context. If we talk about the tensor itself, independent of a coordinate system, we use the symbolic notation. Different typographical styles can be found in the literature. In this book, a first-order tensor is underscored once a, b,. ..), a, second-order tensor twice (and usually denoted with a capital letter. A, B,. ..). Higher-order tensors get a tilde... 

The number of indices in the result of a contraction of tensors of arbitrary order is equal to the number of indices in the contraction that are not doubled, the free indices. Here are a few examples, whose symbolic notation is, in some cases, explained later ... 

In this list, the matrices and tensors are printed in the index notation. The corresponding symbol notation can be built using the scheme (Aij) = A and (Aijki) = A, respectively. Unless stated otherwise, the indices run from 1 to 3. 

The symbol M represents the masses of the nuclei in the molecule, which for simplicity are taken to be equal. The symbol is the Kionecker delta. The tensor notation is used in this section and the summation convention is assumed for all repeated indexes not placed in parentheses. In Eq. (91) the NACT appears (this being a matrix in the electronic Hilbert space, whose components are denoted by labels k, m, and a vector with respect to the b component of the nuclear coordinate R). It is given by an integral over the electron coordinates... 

Dimensionless groups for a proeess model ean be easily obtained by inspeetion from Table 13-2. Eaeh of the three transport balanees is shown (in veetor/tensor notation) term-by-term under the deseription of the physieal meanings of the respeetive terms. The table shows how various well-known dimensionless groups are derived and gives the physieal interpretation of the various groups. Table 13-3 gives the symbols of the dimensions of the terms in Table 13-2. 

We write creation and annihilation operators for a state 1/1) as a and aA, so that ) = a lO). We use the spin-orbital 2jm symbols of the relevant spin-orbital group G as the metric components to raise and lower indices gAA = (AA) and gAA = (/Li)3. If the group G is the symmetry group of an ion whose levels are split by ligand fields, the relevant irrep A of G (the main label within A) will contain precisely the states in the subshell, the degenerate set of partners. For example, in Ref. [10] G = O and A = f2. In the triple tensor notation X of Judd our notation corresponds to X = x( )k if G is a product spin-space group if spin-orbit interaction is included to couple these spaces, A will be an irrep appearing in the appropriate Kronecker decomposition of x( )k. 

The major notations of scalars, vectors, and tensors and their operations presented in the text are summarized in Tables A1 through A5. Table A1 gives the basic definitions of vector and second-order tensor. Table A2 describes the basic algebraic operations with vector and second-order tensor. Tables A3 through A5 present the differential operations with scalar, vector, and tensor in Cartesian, cylindrical, and spherical coordinates, respectively. It is noted that in these tables, the product of quantities with the same subscripts, e.g., a b, represents the Einstein summation and < jj refers to the Kronecker delta. The boldface symbols represent vectors and tensors. 

The operator denotes the tensor product of the electric fields. The tildes above the electric field and polarization symbols portray the wave character of the fields and are used for better distinction with the field amplitudes E(r) and P(r) as defined below. For the notation of the polarization components follows... 

Figure 2.7-6 A Assignment of the Cartesian coordinate axes and the symmetry operations of a planar molecule of point group C2,.. B Character table, 1 symbol of the point group after Schoen-flies 2 international notation of the point group 3 symmetry species (irreducible representations) 4 symmetry operations 5 characters of the symmetry operations in the symmetry species +1 means symmetric, -1 antisymmetric 6 x, y, z assignment of the normal coordinates of the translations, direction of the change of the dipole moment by the infrared active vibrations, R, Ry, and R stand for rotations about the axes specified in the subscript 7 x, xy,. .. assign the Raman active species by the change of the components of the tensor of polarizability, aw, (Xxy,. ...

In equations (l)-(7), vector notation is employed Vis the gradient operator, U is the unit tensor, two dots ( ) imply that the tensors are to be contracted twice, and the superscript T denotes the transpose of the tensor. The symbols appearing here are summarized in Table 1.1. 

The -th order operator is defined by analogy to equation (41). In equation (45), the symbol [n] in Jansen s notation denotes -fold scalar contraction of the product of two n-th rank tensors and T , with... 

Note that we use the same symbols for Green s tensors in time and frequency domains to simplify the notations. One can eeisily recognize the corresponding tensor by checking for arguments t or u in the corresponding equations. 

Where the relationship of our conventional symbol, co, is given in respect of the common spectroscopic symbol v. ) Substituting for the fundamental constants, see below for the tensor notation B. 

We use bold-face italic symbols for vectors and bold-face Greek symbols for second-order tensors dot-product operations enclosed in () are scalars, those in [ ] are vectors, and those in are second-order tensors. The vector-tensor notation and conventions are identical with those used by Bird, Stewart, and Lightfoot (8) unless otherwise indicated. 

In the notation of tensor components we are using some symbols taken from the classical work of Landau Lifschitz (1953), especially... 

All standard mathematical and chemical symbols are taken to have their usual meaning. Both Cartesian tensor and boldface vector notation have been employed in the book. In the following list only the boldface form is given for vector quantities to avoid confusion with the use of the subscripts i and / in Cartesian tensor notation and the use of a subscript i or / to denote a species. 

The recent development of tensorial schemes for characterizing the intrinsic hydrodynamic resistance of particles of arbitrary shape, and the application of singular perturbation techniques to obtain asymptotic solutions of the Navier-Stokes equations at small Reynolds numbers constitute significant contributions to oim understanding of slow viscous flow around bodies. It is with these topics that this review is primarily concerned. In presenting this material we have elected to use Gibbs polyadics in preference to conventional tensor notation. For in our view, the former symbolism— dealing as it does with direction as a primitive concept—is more closely related to the physical world in which we live than is the latter notation. 

Moreover, polyadic symbols, being free from extraneous indices which detract from the invariant significance of the single entities they represent, are more suggestive and possess greater aesthetic appeal than do their tensor counterparts. Our polyadic notation is essentially equivalent to that employed in Milne s book (Ml 6). Other useful discussions of this notation will be found in References B6, D6, G3, M15. 

After arranging our notation to accommodate spherical tensors we turn to a powerful tool for the calculation of matrix elements. The Wigner-Eckart theorem allows the factorisation of a matrix element into a part containing the dependence of the magnetic quantum numbers, essentially a 3j symbol, and a part independent of these, called the reduced matrix element and already employed in sect. 2 ... 

Notation. The symbols a, f, y are used throughout to denote the electric field polarizability, first and second hyperpolarizabilities respectively, suitably qualified by frequency factors where necessary. The magnetizability is denoted by y and the nuclear screening tensor by a. The numerous but well-known acronyms specifying the computational procedures are used without definition. The possibly rather less well-known acronyms for the principal gauge invariant procedures are given in Table 1. 

It was shown earlier that there are only six independent tensor coordinates for strain and stress. The matrix notation allows us to assign them just six different symbols. Following Table 3.1 the assignment Ty = 7 is chosen and... 

This formulation is reasonably simple to understand, but it has a number of disadvantages which make it cumbersome to use. Firstly, the cartesian tensor for rank n is an object with n suffixes. This in itself introduces some awkwardness of notation while it is possible to use the same symbol above) for all ranks, distinguishing between the ranks by the number of suffixes, it is necessary in general expressions such as (5) to include the superscript (n) to indicate the rank. Moreover it has become conventional to use different symbols (q, 0) for the moments of ranks 0... 

The simplest tensorial or matricial notations are used. Where there is no ambiguity, bold letters are used for all non-scalar quantities. However, if the same s3mibol is used for a 1 - or 2" -order tensor, this is specified by a single arrow or bar below the S5mibol for a l -order tensor, and a double arrow or double bar below the symbol for a 2 < -order tensor. Thus, electrical polarization is represented by a vector P or a column matrix P, whereas the pressure tensor, which is a 2" -order tensor, corresponds to the tensorial notation P or matricial notation P. ... 

A convenient notation for describing polarization data from single crystals is due to S. Porto. This involves four symbols (usually chosen from x, y, z axis designations), which define the propagation direction of the incident radiation, the direction of the E of the incident radiation, the direction of the E of the scattered radiation being examined, and the direction of propagation of the scattered radiation. The second and third symbols are placed in parentheses, for example, z(xy)x, and define the components of the scattering tensor. 

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