Second-order tensor function

A frame indifferent second-order tensor function T is characterized by the frame indifference of a transformed vector v = Tu. That is, let be a frame indifferent vector (h = Q u), and v and v be the vectors corresponding to u and u transformed by T and T such that... 

Field variables identified by their magnitude and two associated directions are called second-order tensors (by analogy a scalar is said to be a zero-order tensor and a vector is a first-order tensor). An important example of a second-order tensor is the physical function stress which is a surface force identified by magnitude, direction and orientation of the surface upon which it is acting. Using a mathematical approach a second-order Cartesian tensor is defined as an entity having nine components T/j, i, j = 1, 2, 3, in the Cartesian coordinate system of ol23 which on rotation of the system to ol 2 3 become... 

In this section, well-known properties of second-order positive-definite symmetric tensors and functions involving them will be cited without proof. The principal values and principal vectors (m = 1, 2, 3) of a symmetric second-order tensor A are given by... 

A scalar-valued function/(/4) of one symmetric second-order tensor A is said to be symmetric if... 

A scalar-valued function f(A, B) of two symmetric second-order tensors A and B is said to be isotropic if... 

Equation (I-l) is the general representation of the dispersion model. The dispersion coefficient is a function of both the fluid properties and the flow situation the former have a major effect at low flow rates, but almost none at high rates. In this general representation, the dispersion coefficient and the fluid velocity are all functions of position. The dispersion coefficient, D, is also in general nonisotropic. In other words, it has different values in different directions. Thus, the coefficient may be represented by a second-order tensor, and if the principal axes are taken to correspond with the coordinate system, the tensor will consist of only diagonal elements. 

Measuring the surface tensions of solids poses a problem because it is almost impossible to extend a solid-fluid interface isothermally and reversibly. Stretching a solid-fluid interface is not performed against the interfacial tension, but against the interjacial stress r. The difference between r and y depends on the kinetics and history of the extension, so that generally the work w performed is not a sole characteristic of y or, for that matter, a function of state. Only when the extension can be ceirried out reversibly is it possible to relate r emd y. As r is a second order tensor (it has normal and shear components) and y is a scalar, this relation is complicated. For the very simple case that r does not depend on direction (a rather unrealistic situation for solids) and assuming reversibility the relation is ... 

After Taylor s work was reported, von Karman [178] noticed that the mean values of the products of the velocities at two (or more points) were tensors. The realization of van Karman [178] that the correlation is a tensor simplified the analysis considerably because a know tensor in one coordinate system can be transformed it into other coordinate systems simply by adopting the rules of such transformations for second order tensors. For the purposes of simplification, von Karman also introduced the assumption of self-preservation of the shape of the velocity product function during decay. 

Let h x) = hiCi be a vector-valued function, e.g., a displacement u x). The gradient of h x) is given as a second order tensor, and the components are written in a matrix form as... 

The Clausis-Duhem inequality (22) has established a relation between the stress and the strain through the Helmholtz free energy y/. Considering a viscoelastic material and constant and uniform temperature, it can be assumed that the Helmholtz free energy will only be function of the total strain and an undetermined number of second order tensors internal state variables [q ] (a = l,.,n). So, Eq. (22)becomes ... 

The experimentalist does not ordinarily measure a conformational state time correlation function. Usually a technique is sensitive to some first or second order tensor, an orientational quantity such as a dipole moment along or perpendicular to the chain, the direction of a C-H bond, a polarizability, or a transition moment of a chromophore. It is even more difficult to analyze these objects orientational time correlation functions. Furthermore they may exhibit both rapid relaxation due to conformational transitions and slow relaxation due to coupling to long wavelength modes. We have suggested fitting the short time part to the simplest function which accounts for single and pair transitions, ... 

The strength of the individual singlecouple and vector-dipole Green s function solutions (G j) is given by My which is a second-order tensor that is both real and symmetric (Mi2 = M21) enabling the decomposition into three orthogonal eigenvectors ... 

The second order tensor or Hessian matrix of the image function is defined in terms of the second partial derivatives of the image. 

A computer program for the theoretical determination of electric polarizabilities and hyperpolarizabilitieshas been implemented at the ab initio level using a computational scheme based on CHF perturbation theory [7-11]. Zero-order SCF, and first-and second-order CHF equations are solved to obtain the corresponding perturbed wavefunctions and density matrices, exploiting the entire molecular symmetry to reduce the number of matrix element which are to be stored in, and processed by, computer. Then a /j, and iap-iS tensors are evaluated. This method has been applied to evaluate the second hyperpolarizability of benzene using extended basis sets of Gaussian functions, see Sec. VI. 

In principle, one should solve the Boltzmann equation Eq. (65) in order to arrive at explicit expressions for the pressure tensor p and heat flux q, which proves not possible, not even for the simple BGK equation Eq. (11). However, one can arrive at an approximate expression via the Chapman Enskog expansion, in which the distribution function is expanded about the equilibrium distribution function fseq, where the expansion parameter is a measure of the variation of the hydrodynamic fields in time and space. To second order, one arrives at the familiar expression for p and q... 

The hfs (or quadrupole) tensors of geometrically (chemically) equivalent nuclei can be transformed into each other by symmetry operations of the point group of the paramagnetic metal complex. For an arbitrary orientation of B0 these nuclei may be considered as nonequivalent and the ENDOR spectra are described by the simple expressions in (B 4). If B0 is oriented in such a way that the corresponding symmetry group of the spin Hamiltonian is not the trivial one (Q symmetry), symmetry adapted base functions have to be used in the second order treatment for an accurate description of ENDOR spectra. We discuss the C2v and D4h covering symmetry in more detail. 

---