Of tensors

From the point of view of tensor properties, the surface nonlmear susceptibility /, J-(.is quite analogous to the... 

The electric moments are examples of tensor properties the charge is a rank 0 tensor (which i the same as a scalar quantity) the dipole is a rank 1 tensor (which is the same as a vectoi with three components along the x, y and z axes) the quadrupole is a rank 2 tensor witl nine components, which can be represented as a 3 x 3 matrix. In general, a tensor of ran] n has 3" components. 

The following three scalars remain independent of the choice of coordinate system in which the components of T are defined and hence are caUed the invariants of tensor T ... 

Vectors are commonly used for description of many physical quantities such as force, displacement, velocity, etc. However, vectors alone are not sufficient to represent all physical quantities of interest. For example, stress, strain, and the stress-strain iaws cannot be represented by vectors, but can be represented with tensors. Tensors are an especially useful generalization of vectors. The key feature of tensors is that they transform, on rotation of coordinates, in special manners. Tsai [A-1] gives a complete treatment of the tensor theory useful in composite materials analysis. What follows are the essential fundamentals. 

This type of transformation will be used to assist in the definition of various orders of tensors. Each tensor will be defined on the basis of the type of transformation it satisfies. Tensors transform according to the relations... 

The dipole polarizability, the field gradient and the quadrupole moment are all examples of tensor properties. A detailed treatment of tensors is outside the scope of the text, but you should be aware of the existence of such entities. 

These conditions show us immediately that in the case of the four-neighbor HPP lattice (V = 4) f is noni.sotropic, and the macroscopic equations therefore cannot yield a Navier-Stokes equation. For the hexagonal FHP lattice, on the other hand, we have V = 6 and P[. is isotropic through order Wolfram [wolf86c] predicts what models are conducive to f lavier-Stokes-like dynamics by using group theory to analyze the symmetry of tensor structures for polygons and polyhedra in d-dimensions. 

By this time Polya s Theorem had become a familiar combinatorial tool, and it was no longer necessary to explain it whenever it was used. Despite that, expositions of the theorem have continued to proliferate, to the extent that it would be futile to attempt to trace them any further. I take space, however, to mention the unusual exposition by Merris [MerRSl], who analyzes in detail the 4-bead 3-color necklace problem, and interprets it in terms of symmetry classes of tensors — an interpretation that he has used to good effect elsewhere (see [MerRSO, 80a]). 

MerRSOa Merris, R. Pattern inventories associated with symmetry classes of tensors. Lin. Alg. and Appl. 29 (1980) 225-230. MerR81 Merris, R. Polya s counting theorem via tensors. Amer. Math. Monthly 88 (1981) 179-185. 

Actually transversality in all the k variables already follows from transversality in any one of the k variables because of the symmetric character of the tensor alll...Un(k1, , kn). Again due to the freedom of gauge transformations an n photon configuration is not described by a unique amplitude but rather by an equivalence class of tensors. We define the notion of equivalence for these tensors, as follows a tensor rfUl. ..Bn( i, , kn) will be said to be equivalent to zero ... 

To obtain for 71 and jk compact dispersion formulas similar as Eq. (79) for 7, these hyperpolarizability components must be written as sums of tensor components which are irreducibel with respect to the permu-tational symmetry of the operator indices and frequency arguments ... 

For the example in Figure 2.14 it would be possible to perform the coordinate transformation analytically by introducing cylindrical coordinates. However, in general, geometries are too complex to be described by a simple analytical transformation. There are a variety of methods related to numerical curvilinear coordinate transformations relying on ideas of tensor calculus and differential geometry [94]. The fimdamental idea is to establish a numerical relationship between the physical space coordinates and the computational space curvilinear coordinates The local basis vectors of the curvilinear system are then given as... 

The eigenvalues of tensor (k) at the symmetric points of the first Brillouin zone that are determined by formula (3.1.7) take the form ... 

An important purpose of tensor analysis is to describe any physical or geometrical quantity in a form that remains invariant under a change of coordinate system. The simplest type of invariant is a scalar. The square of the line element ds of a space is an example of a scalar, or a tensor of rank zero. 

The interpretation of the HREELS spectrum and the structure belonging to the (2x2)-3CO LEED pattern has been the subject of some debate in the literature [57— 59], The CO stretch peak at the lower frequency had previously been assigned to a bridge-bonded CO [57], with obvious consequences for the way CO fills the (2x2) unit cell. A recent structural analysis from the same laboratory on the basis of tensor LEED has confirmed the structures of both the (V3xV3)R30° and the (2x2)-3CO as given in Fig. 8.14, i.e. with CO in linear and threefold positions in the (2x2)-3CO structure [58]. The assignments have also been supported by high-resolution XPS measurements [59],... 

The expansion (Equation 24.12) does not contain even powers of the field because of the spherical symmetry of an isolated atom. Indeed for an atom, the even derivatives in Equation 24.10 are zero as well as for any molecule having an inversion center. Note that a3 and a5 are, in fact, the components of tensors, respectively of the so-called second and fourth hyperpolarizabilities [4]. 

In this introduction, the viscoelastic properties of polymers are represented as the summation of mechanical analog responses to applied stress. This discussion is thus only intended to be very introductory. Any in-depth discussion of polymer viscoelasticity involves the use of tensors, and this high-level mathematics topic is beyond the scope of what will be presented in this book. Earlier in the chapter the concept of elastic and viscous properties of polymers was briefly introduced. A purely viscous response can be represented by a mechanical dash pot, as shown in Fig. 3.10(a). This purely viscous response is normally the response of interest in routine extruder calculations. For those familiar with the suspension of an automobile, this would represent the shock absorber in the front suspension. If a stress is applied to this element it will continue to elongate as long as the stress is applied. When the stress is removed there will be no recovery in the strain that has occurred. The next mechanical element is the spring (Fig. 3.10[b]), and it represents a purely elastic response of the polymer. If a stress is applied to this element, the element will elongate until the strain and the force are in equilibrium with the stress, and then the element will remain at that strain until the stress is removed. The strain is inversely proportional to the spring modulus. The initial strain and the total strain recovery upon removal of the stress are considered to be instantaneous. 

I returned to the University of Toronto in the summer of 1940, having completed a Master s degree at Princeton, to enroll in a Ph.D. program under Leopold Infeld for which I wrote a thesis entitled A Study in Relativistic Quantum Mechanics Based on Sir A.S. Eddington s Relativity Theory of Protons and Electrons. This book summarized his thought about the constants of Nature to which he had been led by his shock that Dirac s equation demonstrated that a theory which was invariant under Lorentz transformation need not be expressed in terms of tensors. 

As we have written it in Equation (8), the DMRG wave function contains redundant variational parameters. This means that the set of variational tensors f/ni... ij/Hk in the DMRG wave function is not unique, because we can find another set of tensors whose matrix product yields an identical state. This redundancy is analogous to the redundancy of the orbital parametrization of the Hartree-Fock determinant. In the case of the DMRG wave function, we can insert a matrix T and its inverse between any two variational tensors and leave the state invariant... 

Kay, D. C. (1988) Theory and problems of tensor calculus, Schaum s Outline Series. McGraw-Hill, New York. 

A trivariate normal distribution describes the probability distribution for anisotropic harmonic motion in three-dimensional space. In tensor notation (see appendix A for the notation, and appendix B for the treatment of symmetry and symmetry restrictions of tensor elements), with j and k (= 1, 3) indicating the axial directions,... 

Placement of indices as superscripts or subscripts follows the conventions of tensor analysis. Contravariant variables, which transform like coordinates, are indexed by superscripts, and coavariant quantities, which transform like derivatives, are indexed by subscripts. Cartesian and generalized velocities and 2 thus contravariant, while Cartesian and generalized forces, which transform like derivatives of a scalar potential energy, are covariant. 

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