Tensor notations

In the tensor notation (Patterson 1959, Sands 1982) the basis vectors of the direct lattice are written as a( (i = 1, 2, 3), and the coordinates of a direct space vector as x . Thus, for a vector v, we can write 

The terms variant and covariant refer to the transformation properties of the quantities. A transformation may be defined by the transformation matrix T operating on the direct space basis a, such that 

As the vector v must be invariant under the transformation, the coordinates xJ must transform as 

Since a, af = S0, or in tensor notation, a, aj = S, the reciprocal axes are contravariant and are written as a . As the Miller indices are the coordinates in the reciprocal base system, they must be covariant and are written as ht. Thus, the Miller indices transform like the direct axes, both being covariant. 

By taking the scalar product of both sides of this equation with ak, we obtain 

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The symbol M represents the masses of the nuclei in the molecule, which for simplicity are taken to be equal. The symbol is the Kionecker delta. The tensor notation is used in this section and the summation convention is assumed for all repeated indexes not placed in parentheses. In Eq. (91) the NACT appears (this being a matrix in the electronic Hilbert space, whose components are denoted by labels k, m, and a vector with respect to the b component of the nuclear coordinate R). It is given by an integral over the electron coordinates... 

Dimensionless groups for a proeess model ean be easily obtained by inspeetion from Table 13-2. Eaeh of the three transport balanees is shown (in veetor/tensor notation) term-by-term under the deseription of the physieal meanings of the respeetive terms. The table shows how various well-known dimensionless groups are derived and gives the physieal interpretation of the various groups. Table 13-3 gives the symbols of the dimensions of the terms in Table 13-2. 

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. 

The relativistic invariance of the electromagnetic field is conveniently expressed in tensor notation. Factorized in Minkowski space the Maxwell equa-... 

In a similar fashion strain (deformation) y can be defined using a tensor notation ... 

In our formalism [5-9] excitation operators play a central role. Let an orthonormal basis p of spin orbitals be given. This basis has usually a finite dimension d, but it should be chosen such that in the limit —> cxd it becomes complete (in the so-called first Sobolev space [10]). We start from creation and annihilation operators for the ij/p in the usual way, but we use a tensor notation, in which subscripts refer to annihilation and superscripts to creation ... 

The Kronecker delta is written here in a tensor notation. One can define excitation operators as normal products (or products in normal order) of the same number of creation and annihilation operators normal order in the original sense means that all creation operators have to be on the left of all annihilation operators). 

This convention is in conflict with our tensor notation, that we do not want to abandon [8], However, what really matters is only the change of the definition of normal ordering, and this can easily be formulated in our language as well. 

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,... 

The three-dimensional Gaussian distribution function is, in tensor notation, given by... 

A complete treatment of interfacial boundary conditions in tensor notation is given by Scriven (S2). If surface viscosities are ignored, the normal stress condition reduces to... 

The use of these resistance tensors is developed in detail by Happel and Brenner (H3). While enabling compact formulation of fundamental problems, these tensors have limited application since their components are rarely available even for simple shapes. Here we discuss specific classes of particle shape without recourse to tensor notation, but some conclusions from the general treatment are of interest. Because the translation tensor is symmetric, it follows that every particle possesses at least three mutually perpendicular axes such that, if the particle is translating without rotation parallel to one of these axes, the total... 

In tensor notation the three Cartesian directions x, y, and z are designated by suffixed variables i,j, k, l, etc. (Landau and Lifshitz 1970 Auld 1973). Thus the force acting per unit area on a surface may be described as a traction vector with components rj j = x, y, z. The stress in an infinitesimal cube volume element may then be described by the tractions on three of the faces, giving nine elements of stress cry (i, j = x, y, z), where the first suffix denotes the normal to the plane on which a given traction operates, and the second suffix denotes the direction of a traction component. 

The expression for the stress tensor -t in a fluid mixture is the same as that for a pure substance. The tensor components given in Eqs. (7) and (8) may be summarized conveniently in tensor notation ... 

These equations use Cartesian tensor notation in which a repeated Greek suffix denotes summation over the three components, and where ay7 is the third-rank antisymmetric unit tensor. For a molecule composed entirely of idealized axially symmetric bonds, for which [3 (G )2 = /3(A)2 and aG1 = 0 [13, 15], a simple bond polarizability theory shows that ROA is generated exclusively by anisotropic scattering, and the CID expressions then reduce to [13]... 

We write creation and annihilation operators for a state 1/1) as a and aA, so that ) = a lO). We use the spin-orbital 2jm symbols of the relevant spin-orbital group G as the metric components to raise and lower indices gAA = (AA) and gAA = (/Li)3. If the group G is the symmetry group of an ion whose levels are split by ligand fields, the relevant irrep A of G (the main label within A) will contain precisely the states in the subshell, the degenerate set of partners. For example, in Ref. [10] G = O and A = f2. In the triple tensor notation X of Judd our notation corresponds to X = x( )k if G is a product spin-space group if spin-orbit interaction is included to couple these spaces, A will be an irrep appearing in the appropriate Kronecker decomposition of x( )k. 

There are many ways of writing equations that represent transport of mass, heat, and fluids trough a system, and the constitutive equations that model the behavior of the material under consideration. Within this book, tensor notation, Einstein notation, and the expanded differential form are considered. In the literature, many authors use their own variation of writing these equations. The notation commonly used in the polymer processing literature is used throughout this textbook. To familiarize the reader with the various notations, some common operations are presented in the following section. 

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