An Introduction to Tensors

Single crystals are generally not isotropic. It can be expected then, that the physical properties of single crystals will be anisotropic or dependent on the direction in which they are measured. Table 6.1 reveals that the magnitudes of the fluxes and driving 

Principles of Inorganic Materials Design, Second Edition. By John N. Lalena and David A. Cleary Copyright 2010 John Wiley Sons, Inc. 

TABLE 6.1. Physical Properties Relating Vector Fluxes and Their Driving Forces 

Flux Driving Force Physical Property Tensor Equation 

Heat flow, q Temperature gradient, VJ Thermal conductivity, k q = -kWT 

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Spiegel, Murray R., Schaum s Outline of Theory and Problems of Vector Analysis and an Introduction to Tensor Analysis, McGraw-Hill Book Company, New York (1959). 

Burdick, D.S., An introduction to tensor products with applications to multiway... 

A note on terminology operations that survive the equivalence are sometimes called well defined on equivalence classes. A function on the original set S taking the same value on every element of an equivalence class is called an invariant of the equivalence relation. We will see an example of an invariant of an equivalence class in our introduction to tensor products in Section 2.6. 

This was an introduction to rheology. If you want to go further, you will find that it is a difficult subject. For three-dimensional problems you need sophisticated mathematics to get anywhere. You will need tensor equations the three stresses Xxy, Xyx and Xyy that you have seen are three of the nine components of the stress tensor (Figure C4-17). And that is just... 

Chapter 3 provides an introduction to Gaussian quadrature and the moment-inversion algorithms used in quadrature-based moment methods (QBMM). In this chapter, the product-difference (PD) and Wheeler algorithms employed for the classical univariate quadrature method of moments (QMOM) are discussed, together with the brute-force, tensor-product, and conditional QMOM developed for multivariate problems. The chapter concludes with a discussion of the extended quadrature method of moments (EQMOM) and the direct quadrature method of moments (DQMOM). 

A knowledge of tensors will not be required for the understanding of this book, but the interested reader will find an introduction to this subject in H. Margenau and G. M. Murphy, The Mathematics of Physios and Chemistry, Van Nostrand, New York, 1943. 

For an introduction to Laguerre polynomials, see pages 93-97 of Courant and Hilbert (1956), referenced in Chapter 4. The functions Ajirj, r]") may be regarded as from the tensor product space of L2[0, oo with itself, having an inner product as defined in (6.2.11). For a definition of the tensor product of function spaces, see Ramkrishna and Amundson (1985), referenced in Chapter 4. 

We give in this section an introduction to the construction of CSFs and more generally to the construction of spin tensor operators. We shall employ the genealogical coupling scheme, where the final CSF for N electrons is arrived at in a sequence of N steps [2]. At each step, a new electron is introduced and coupled to those already present. We thus arrive at the final CSF through a sequence of N —I intermediate CSFs, each of which represents a spin eigenfunction. 

RDCs belong to the so-called anisotropic NMR parameters which cannot be observed in isotropically averaged samples as, for example, is the case in liquids. Besides RDCs, a number of other anisotropic parameters can be used for structure elucidation, like residual chemical shift anisotropy, residual quad-rupolar couplings for spin-1 nuclei, or pseudo-contact shifts in paramagnetic samples. Here, we will focus on RDCs where we give a brief introduction into the dipolar interaction, then into the averaging effects with the description by the alignment tensor and concepts to deal with the flexibility of molecules. For the other anisotropic NMR parameters, we refer the reader to ref 19 for an introduction and to refs. 6-8 for a detailed description. 

Physical Properties of Crystals Their Representation by Tensors and Matrices, by J. F. Nye, Revised Edition, Clarendon Press, Oxford, 1985. This book gives an excellent introduction to the use of tensors in physical science. 

Theory and Physics of Piezoelectricity. The discussion that follows constitutes a very brief introduction to the theoretical formulation of the physical properties of crystals. If a solid is piezoelectric (and therefore also anisotropic), acoustic displacement and strain will result in electrical polarization of the solid material along certain of its dimensions. The nature and extent of the changes are related to the relationships between the electric field (E) and electric polarization (P). which are treated as vectors, and such elastic factors as stress Tand strain (S), which are treated as tensors. In piezoelectric crystals an applied stress produces an electric polarization. Assuming Ihe dependence is linear, the direct piezoelectric effect can be described by the equation ... 

The hydrodynamic theory for uniaxial nematic liquid crystals was developed around 1968 by Leslie [10, 11] and Ericksen [12, 13] (Leslie-Ericksen theory, LE theory). An introduction into this theory is presented by F. M. Leslie (see Chap. Ill, Sec. 1 of this Volume). In 1970 Parodi [14] showed that there are only five independent coefficients among the six coefficients of the original LE theory. This LEP theory has been tested in numerous experiments and has been proved to be valid between the same limits as the Navier-Stokes theory. An alternative derivation of the stress tensor was given by Vertogen [15]. 

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. 

The Laplace-Young equation refers to a spherical phase boundary known as the surface of tension which is located a distance from the center of the drop. Here the surface tension is a minimum and additional, curvature dependent, terms vanish (j ). The molecular origin of the difficulties, discussed in the introduction, associated with R can be seen in the definition of the local pressure. The pressure tensor of a spherically symmetric inhomogeneous fluid may be computed through an integration of the one and two particle density distributions. 

Thus, the non-relativistic wave function (1.14) of an electron is a two-component spinor (tensor having half-integer rank) whereas its relativistic counterpart is already, due to the presence of large (/) and small (g) components, a four-component spinor. The choice of / in the form (1 + l — l ) is conditioned by the requirement of a standard phase system for the wave functions (see Introduction, Eq. (4)). 

Various methods have been developed for dealing with the anomalous commutation relationships in molecular quantum mechanics, chief among them being Van Vleck s reversed angular momentum method [10]. Most of these methods are rather complicated and require the introduction of an array of new symbols. Brown and Howard [15], however, have pointed out that it is quite possible to handle these difficulties within the standard framework of spherical tensor algebra. If matrix elements are evaluated directly in laboratory-fixed coordinates and components are referred to axes mounted on the molecule only when necessary, it is possible to avoid the anomalous commutation relationships completely. Only the standard equations given earlier in this chapter are used to derive the required results it is just necessary to keep a cool head in the process ... 

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