Components of vectors and tensors

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... 

In this book Greek indices a, p, ft, v... are used to indicate the components of vectors and tensors, and the summation convention is used over the repeated indices.)... 

What makes vector analysis difficult is the quasisystanatic use in most textbooks of space coordinates and components of vectors and tensors. This requires the acquisition of certain skills in matrix and tensor calculus before being acquainted with the physics behind it. 

The allocation of vector and tensor components to IRs and similarity of orientation have already been determined for the point group C3v in Exercise 15.3-1. Therefore the linear equations relating the vector xy z and the symmetric T(2) o are... 

If we consider time-differentiation of vectors and tensors, not only the components but also the basis must be differentiated. Thus even if the original vectors and tensors are frame indifferent, their time-derivatives are not in general frame indifferent. In order to avoid this difficulty several time-differential forms are considered. 

Throughout the book, the subscripts 1,2, 3 will denote x,y z Cartesian components, respectively, of vectors and tensors. 

The components of the pressure tensor and of the heat-flow vector... 

In the display of the field in Section V, we apply the above separation into parts generated by the various components of the shielding tensor. The secular part of the shielding vector, eqs.(23,24), and the intensity quantities / (R us) and 7 (R ujg), obtained from eq.(21) as discussed above, lend themselves immediately to the response graph technique described in ref. [14], as illustrated in Figures (1-3,5), while the antisymmetry vector is illustrated in Figure 4. 

The components of the translation and rotation vectors are given as Tx> Ty, T and RX Ry, Rz, respectively. The components of the polarizability tensor appear as linear combinations such as axx + (xyy> etc, that have the symmetry of the indicated irreducible representation. 

In this book, vector quantities such as x and y above are normally column vectors. When necessary, row vectors are indicated by use of the transpose (e.g., r). If the components of x and y refer to coordinate axes [e.g., orthogonal coordinate axes ( i, 2, 3) aligned with a particular choice of right, forward, and up in a laboratory], the square matrix M is a rank-two tensor.9 In this book we denote tensors of rank two and higher using boldface symbols (i.e., M). If x is an applied force and y is the material response to the force (such as a flux), M is a rank-two material-property tensor. For example, the full anisotropic form of Ohm s law gives a charge flux Jq in terms of an applied electric field E as... 

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. 

For an element in equilibrium with no body forces, the equations of equilibrium were obtained by Lame and Clapeyron (1831). Consider the stresses in a cubic element in equilibrium as shown in Fig. 2.3. Denote 7y as a component of the stress tensor T acting on a plane whose normal is in the direction of e and the resulting force is in the direction of ej. In the Cartesian coordinates in Fig. 2.3, the total force on the pair of element surfaces whose normal vectors are in the direction of ex can be given by... 

Generally, the five independent components of the alignment tensor A can be derived by mathematical methods like the singular value decomposition (SVD)23 as long as a minimum set of five RDCs has been measured in which no two internuclear vectors for the RDCs are oriented parallel to each other and no more than three RDC vectors lie in a plane. Any further measured RDC directly... 

A first-rank tensor operator 3 V) is also called a vector operator. It has three components, 2T and jH j. Operators of this type are the angular momentum operators, for instance. Relations between spherical and Cartesian components of first-rank tensor operators are given in Eqs. [36] and [37], Operating with the components of an arbitrary vector operator ( 11 on an eigenfunction u1fF) of the corresponding operators and 3 yields... 

Formally, if one has the experimental values of the dielectric tensor e, the magnetic permeability tensor /jl, and the optical rotation tensors p and p for the substrate, one can construct first the optical matrix M, then the differential propagation matrix A, and C, which, to repeat, is the x component of the wavevector of the incident wave. Once A is known, the law of propagation (wave equation) for the generalized field vector ift (the components of E and H parallel to the x and y axes) is specified by Eq. (2.15.18). Experimentally, one travels this path backwards. 

Since a second-rank cartesian tensor Tap transforms in the same way as the set of products uaVfj, it can also be expressed in terms of a scalar (which is the trace T,y(y), a vector (the three components of the antisymmetric tensor (1 /2 ) Tap — Tpaj), and a second-rank spherical tensor (the five components of the traceless, symmetric tensor, (I /2)(Ta/= + Tpa) - (1/3)J2Taa). The explicit irreducible spherical tensor components can be obtained from equations (5.114) to (5.118) simply by replacing u vp by T,/ . These results are collected in table 5.2. It often happens that these three spherical tensors with k = 0, 1 and 2 occur in real, physical situations. In any given situation, one or more of them may vanish for example, all the components of T1 are zero if the tensor is symmetric, Yap = Tpa. A well-known example of a second-rank spherical tensor is the electric quadrupole moment. Its components are defined by... 

The components C2qi (9, spherical harmonics, with the angles 9 and < /> defined in figure 8.52, shown in appendix 8.1. Equation (8.229) is similar to (8.10), except that we have chosen to couple the vectors differently because of the basis set used in the present problem. Clearly the components of the cartesian tensor T are related to those of the spherical tensor T2(C) these relationships are derived in appendix 8.2. 

Four thiourea molecules at sites are the building blocks of the unit cell of the crystal of the space group >2. The point group which is relevant for the selection rules is found by deleting the superscript, which yields >2a- Table 2.7-4 lists the details and results of the application of Eqs. I and II from Table 2.7-1. As shown in Fig. 2.7-8, the results are assigned to the components of the polarizability tensor and the dipole moment vector of the crystal, with x and y = a, b, c, which explains the Raman and infrared activity. We see that in addition to the translations of the whole crystal most... 

These results require further that w, v, R, 0, and Yi be continuous in the first approximation and rely on the assumptions that fi2, and 6 are continuous. The conditions obtained from equations (89), (90), and (93) in effect involve only the longitudinal components of the stress tensor and of the heat-flux vector. The first of the conditions quoted from equations (89) and (90) expresses a pressure discontinuity that balances the discontinuity in the viscous stress tensor, and the second states that the streamwise gradients of the components of velocity tangent to the sheet are continuous. 

This equation can be written for scalar components of the stress tensor and deformation vector as follows (Udias, 1999)... 

---