Vector and Tensor Operations

In the following, a scalar is represented with italic type and a vector is denoted with boldface type. The vector can be represented in terms of its components in the directions represented as unit vectors, that is, 

The vector has a magnitude, which can be determined from its components 

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Following are some of the important vector and tensor operations which are used in the derivations in Sec. II of this review these relations are given for Cartesian coordinates. The quantities v and F are vectors, v is a second-order symmetric tensor, and T is a scalar. 

This appendix describes a few basic vector and tensor operations that may be useful in understanding the material presented in the book. The vectors and tensors are presented only in the Cartesian coordinate system [1,2],... 

Amplitudes of molecular optical events are proportional to off-diagonal matrix elements of interaction operators between the wavefunctions of the initial, final, and possibly also intermediate states of the molecule, 0), f), and /) respectively. These operators are projections of molecular transition vector and tensor operators onto the polarization directions of photons created or annihilated in the event (Table 1). The amplitudes depend on the wavenumber of the light used and can be real or complex. The probability of an optical event W is proportional to the square of the absolute value of its amplitude (Table 2). The proportionality constant is of no... 

The superscript refers to the rank of the tensor whereas the subscript distinguishes among its components. While represents the spherical transform of the parameter tensor, B 0 C yi represents the compound operator part constituted of the scalar, vector and tensor products of physical vectors. The important relationships are contained in Table 53. 

The vector differential operator, V, is the most widely used vector and tensor differential operator for the balance equations. In Cartesian coordinates it is defined as... 

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. 

If we subtract this zeroth order solution, fourier transform the x coordinates, convert the time coordinate to conformal time, r), defined by dr) = dt/a, and ignore vector and tensor perturbations (discussed in the lectures by J. Bartlett on polarization at this school), the Liouville operator becomes a first-order partial differential operator for /( (k, p, rj), depending also on the general-relativistic potentials, (I> and T. We further define the temperature fluctuation at a point, 0(jfc, p) = f( lj i lodf 0 1 /<9To) 1 where To is the average temperature and )i = cos 6 in the polar coordinates for wavevector k. 

In (1.25), the terms inside the brackets can be reformulated by use of vector and tensor notations. By comparing the terms inside the brackets with the mathematical definitions of the nabla or del operator, the vector product between this nabla operator and the mass flux vector we recognize that these... 

The vector and tensor differential operators that have been applied to the symmetrical dyad product pvv can be defined (e.g., [12], p. 574) ... 

We will use the following notation Hnlm is an operator presenting an n th-order dependency on B, /-order in nN, and m-order in /ie, the subscript i refers to electron i, subscript N refers to nucleus N, and u and v represent Cartesian components of vectors and tensors. The resulting expression for the Hamiltonian including magnetic terms is as follows... 

Here, nothing of this kind is used. Vectors and tensors are handled as mathematical beings representing a physical quantity independent of the stracture of space. No components and no coordinates or spatial frames are used. This means that there are no computations or exercises for j uggling with tensors, because physics must be first A bit of mathematics, however, is useful for understanding how operators are defined. As this not a prerequisite for using Formal Graphs, this mathematical part is discussed at the end of this chapter. 

Basic Principles. The Hamiltonian energy operator Tfspin for an electron spin is given by equation 1 (operators, vectors, and tensors are in bold). 

Appendix A Review of Operations with Vectors and Tensors... 

The physical theory which describes the forces and energies involved in the propagation of acoustic waves through solids and fluids is well established, albeit somewhat complex mathematically. The treatment below is intended to provide the reader with a brief overview in order to allow an understanding of how such properties as elasticity, density, and viscosity affect the operation of piezoelectric transducers and their application to chemical sensing. A detailed description of acoustic waves in solids may be found in several texts (1-3). To simplify notation, vectors and tensors are presented without subscripts, and it is assumed that mathematical operations are summed over the appropriate dimensions. 

For closed shell nuclei in the independent particle limit only the first four terms contribute. These are given in eqs. (3.29)-(3.34) in ref. [Tj 87b] in terms of the scalar, vector and tensor density form factors. The trace over nucleon 2 in eq. (4.66) leaves a Dirac operator acting on nucleon 1 (the projectile). Thus we may similarly expand as in eq. (4.69) to obtain... 

Matrix and tensor notation is useful when dealing with systems of equations. Matrix theory is a straightforward set of operations for linear algebra and is covered in Section A.I. Tensor notation, treated in Section A.2, is a classification scheme in which the complexity ranges upward from scalars (zero-order tensors) and vectors (first-order tensors) through second-order tensors and beyond. 

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