Vector notation, boldface

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. 

This and the next two footnotes are included for readers who are not familiar with vector notation. The quantity Fij is printed in boldface to indicate it is a vector having both magnitude and direction. 

We use either of the two notations in Eq. (9.2-11) to denote the magnitude of a vector the boldface letter within vertical bars or the letter in plain type. The magnitude of a vector is always non-negative (positive or zero). 

For vector notation see e.g. Margenau and Murphy (1956, Chap. 4) ordinary 3-dimensional vectors (not considered as elements of a general vector space) are set in boldface italic. 

In the following, vectors are boldface, scalars are not. Basis vectors are i, j, k in Cartesian coordinates and f, 6, in spherical coordinates. Alternative notations associated with the vector A and scalar A are listed in Table Al.l. 

We have introduced the boldface notation to underline that AF is a vector in the complex plane (see Fig. 5.9), because both Fobs and Fcalc are, in general, complex quantities, as is evident from Eq. (5.6). The phase angles

further discussed in section 5.2.5. In a different notation we may write, like Eq. (5.5),... 

A brief summary of the mathematical notation adopted throughout this text is in order. Scalar quantities, whether constants or variables, are represented by italic characters. Vectors and matrices are represented by boldface characters (individual matrix elements are scalar, however, and thus are represented by italic characters that are indexed by subscript(s) identifying the particular element). Quantum mechanical operators are represented by italic characters if diey have scalar expectation values and boldface characters if their expectation values are vectors or matrices (or if they are typically constructed as matrices for computational purposes). The only deliberate exception to the above rules is that quantities represented by Greek characters typically are made neither italic nor boldface, irrespective of their scalar or vector/matrix nature. 

In our notation, dt> designates the vector with the Cartesian components dnx dvr diiz. On the other hand, d3n = dvx dvy dv2 designates a volume element in velocity space boldface is not used for scalars. 

To further illustrate Dirac notation for some simple formulas in Euclidean 3-space, we can rewrite analogs of (9.20a-e) in Dirac notation, all in terms of underlying Dirac objects y) (using boldface symbols to stress the association with ordinary vectors) ... 

In the present study, matrix algebra (see [42] for a more detailed description) is used as a shorthand for otherwise tedious formulae. In matrix algebraic notation, small boldface letters denote vectors (for example, a) and capital boldface letters denote matrices (for example, X). Vectors can be either row vectors or column vectors (see Figure 6.31, in which some operations are... 

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. 

In this section the notation and symbols used throughout the book are listed alphabetically under their appropriate headings. Constants are usually given in nonitalic type and variables in italic. (This is only a general rule. By convention, Boltzmann s constant k, Planck s constant h, and other physical constants are in italic.) Boldfaced type indicates vectors and matrices. Except in section L2.4.A., the cgs (centimeter-gram-second) and the mks (meter-kilogram-second) systems of notation are used in parallel. Any symbols not listed in this section are defined where they are used or in the notation section of Level 3. 

In this chapter the following notation convention is used matrices and vectors are in boldface letters, for example, X and y thus uppercase letters are used when there is more than one variable, and lowercase letters when there is only one variable. Scalars are denoted by ordinary letters, for example, qa-... 

Notation. We will use boldface italic letters to denote vectors and tensors. We adopt the summation convention for repeated indices, imless stated otherwise. Most often, vectors are denoted by lowercase boldface italic letters, and second-order tensors, or 3x3 matrices, by lowercase boldface Greek letters. Fourth-order tensors are usually denoted by uppercase boldface italic letters. We will make use of a Cartesian coordinate system with an orthonormal basis ei, ej, e. Where it is necessary to show components of a vector or a tensor, these will always be relative to the orthonormal basis e, 2, 3. Throughout this work we will identify a second-order tensors r with a 3x3 matrix. We will always use 1 < / <3, to denote the components of the vector a, and the components of the... 

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