Tensor linear

Dirac delta function in three dimensions extension rate extensional strain strain tensor (linear)... 

Now we come to the main theorem of this Appendix concerning the representation of scalar, vector, and tensor linear isotropic functions of vectors and tensors (scalars as independent variables play the role of parameters). 

Proof The most general forms of scalar, vector, and tensor functions depending on r vectors and s tensors linearly are... 

The states of stress and strain in a deformed crystal being idealized as a continuum are characterized by symmetric second-rank tensors and Cjj, respectively, each comprising six independent components. Hooke s law of linear elasticity for the most general anisotropic solid expresses each component of the stress tensor linearly in terms of all components of the strain tensor in the form... 

Let us consider investigation of stresses in a 3-D specimen. It has been shown [1] that in the case of weak birefringence a 3-D specimen can be investigated in a conventional transmission polariscope as if it were a two dimensional specimen. On every ray of light it is possible to determine the parameter of the isoclinic and the optical path difference A. The latter are related to the components of the stress tensor on the ray by linear integral relationships... 

The equation of momentum conservation, along with the linear transport law due to Newton, which relates the dissipative stress tensor to the rate of strain tensor = 1 (y. 4, and which introduces two... 

Flere, the linear polarizability, a (oip 2), corresponds to the doorway stage of the 4WM process while to the window stage. We also see the (complex) Raman resonant energy denominator exposed. Of the tliree energy denominator factors required at third order, the remaining two appear, one each, m the two Imear polarizability tensor elements. 

The polarization P is given in tenns of E by the constitutive relation of the material. For the present discussion, we assume that the polarization P r) depends only on the field E evaluated at the same position r. This is the so-called dipole approximation. In later discussions, however, we will consider, in some specific cases, the contribution of a polarization that has a non-local spatial dependence on the optical field. Once we have augmented the system of equation B 1.5.16. equation B 1.5.17. equation B 1.5.18. equation B 1.5.19 and equation B 1.5.20 with the constitutive relation for the dependence of Pon E, we may solve for the radiation fields. This relation is generally characterized tlirough the use of linear and nonlinear susceptibility tensors, the subject to which we now turn. 

In the simplest case of Newtonian fluids (linear Stokesian fluids) the extra stress tensor is expressed, using a constant fluid viscosity p, as... 

In Equation (1.28) function M(t - r ) is the time-dependent memory function of linear viscoelasticity, non-dimensional scalars 4>i and 4>2 and are the functions of the first invariant of Q(t - f ) and F, t t ), which are, respectively, the right Cauchy Green tensor and its inverse (called the Finger strain tensor) (Mitsoulis, 1990). The memory function is usually expressed as... 

For molecules that are non-linear and whose rotational wavefunctions are given in terms of the spherical or symmetric top functions D l,m,K, the dipole moment Pave can have components along any or all three of the molecule s internal coordinates (e.g., the three molecule-fixed coordinates that describe the orientation of the principal axes of the moment of inertia tensor). For a spherical top molecule, Pavel vanishes, so El transitions do not occur. 

The susceptibility tensors give the correct relationship for the macroscopic material. For individual molecules, the polarizability a, hyperpolarizability P, and second hyperpolarizability y, can be defined they are also tensor quantities. The susceptibility tensors are weighted averages of the molecular values, where the weight accounts for molecular orientation. The obvious correspondence is correct, meaning that is a linear combination of a values, is a linear combination of P values, and so on. 

These are linear equations which give the symmetry of the strain tensor Sij = Sji- In the general case, the strain tensor is nonlinear,... 

Let C i be a bounded domain with a smooth boundary L, and n = (ni,n2,n3) be a unit outward normal vector to L. Introduce the stress and strain tensors of linear elasticity (see Section 1.1.1),... 

Here c[-], which will be called the elastic modulus tensor, is a fourth-order linear mapping of its second-order tensor argument, while b[-], which will be called the inelastic modulus tensor, is a linear mapping of k whose form will depend on the specific properties assigned to k. They depend, in general, on and k. For example, if k consists of a single second-order tensor, then in component form... 

The linear piezoelectric tensor relating polarization and stress in x-cut quartz is... 

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. 

WilS73a Williamson, S. G. Tensor compositions and lists of combinatorial structures. Linear and Multilinear Algebra 1... 

It can readily be appreciated that in the absence of any knowledge regarding the differential polarizability tensor, it is a difficult exercise to obtain precise information from the Raman measurements. However, if r is known, Equations (19) are six linear simultaneous Equations in the six quantities (ot2I0N0) 1, P)0o> P220) PIoo> P420 and... 

Although Ki and Ki are defined by physical quantities of different nature, their time evolution is universally determined by orientational relaxation. This discussion is restricted to linear molecules and vibrations of spherical rotators for which / is a symmetric tensor / = fiki- In this case the following relation holds... 

The metric term Eq. (2.8) is important for all cases in which the manifold M has non-zero curvature and is thus nonlinear, e.g. in the cases of Time-Dependent Hartree-Fock (TDHF) and Time-Dependent Multi-Configurational Self-Consistent Field (TDMCSCF) c culations. In such situations the metric tensor varies from point to point and has a nontrivial effect on the time evolution. It plays the role of a time-dependent force (somewhat like the location-dependent gravitational force which arises in general relativity from the curvature of space-time). In the case of flat i.e. linear manifolds, as are found in Time-Dependent Configuration Interaction (TDCI) calculations, the metric is constant and does not have a significant effect on the dynamics. 

---