Scalars, vectors and tensors

We will deal with scalar (single valued) quantities A, Cartesian (three component) vectors 

Here ij, etc., represent diadic products of the elementary Cartesian vectors. The unit vector is 

When each component of one vector is linearly related to each component of another vector the coefficients of proportionality are the components of a second-rank tensor. Stress, defined as force per unit area, is the quotient of two vectors, and is an example of a second-rank tensor. Note that the condition defining positive in the direction of the outward normal means that a hydrostatic pressure must be negative. Strain is also a tensor, and in the most general case both stress and strain can be expressed in terms of nine tensor components. Stress and strain are both examples of second-rank tensors, which have nine components vectors, which have three components, are sometimes referred to as first-rank tensors single-valued scalars are zero rank tensors. 

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The superscript refers to the rank of the tensor whereas the subscript distinguishes among its components. While represents the spherical transform of the parameter tensor, B 0 C yi represents the compound operator part constituted of the scalar, vector and tensor products of physical vectors. The important relationships are contained in Table 53. 

The major notations of scalars, vectors, and tensors and their operations presented in the text are summarized in Tables A1 through A5. Table A1 gives the basic definitions of vector and second-order tensor. Table A2 describes the basic algebraic operations with vector and second-order tensor. Tables A3 through A5 present the differential operations with scalar, vector, and tensor in Cartesian, cylindrical, and spherical coordinates, respectively. It is noted that in these tables, the product of quantities with the same subscripts, e.g., a b, represents the Einstein summation and < jj refers to the Kronecker delta. The boldface symbols represent vectors and tensors. 

Appendix / Summary of Scalar, Vector, and Tensor Notations... 

By employing the Scalar, Vector and Tensor Harmonics, Stone derived the (7 n + 1) cluster valence MO count for deltahedral transition metal clusters and later developed the model to provide an alternative derivation of Mingos electron counting rules for condensed metal clusters1456. ... 

In this book, we confine ourselves only to the special case of fluids mixture (4.128) which is linear in vector and tensor variables.We denote it as the chemically reacting mixture of fluids with linear transport properties or simply the linear fluid mixture [56, 57, 64, 65]. Then (see Appendix A.2) the scalar, vector and tensor isotropic functions (4.129) linear in vectors and tensors (symmetrical or skew-symmetrical) have the forms ... 

Of course this important reduction (known also as the Curie principle roughly asserting that response of given tensor rank (scalar, vector and tensor) depends on variables of the same tensor rank [2-4, 119, 120]) is valid only in this linear case [12, 13]. The non-linear case is much more complicated [79, 121-123]. 

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). 

Representation theorem of linear isotropic functions Scalar, vector, and tensor (of second order) functions (with values t/,a, A) depending linearly on r vectors ya (a = 1,..., r) and s tensors (of second order) (/ = 1,..., s) are isotropic (relative to the full orthogonal group O) if and only if their (most general) forms are... 

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

GRAPH A3.2 Convention for representing nodes according to the mathematical structure of variables (i.e., scalars, vectors, and tensors). The circle is used as a generic drawing. 

Scalars, Vectors, and Tensors in Three-Dimensional Space... 

We are now in the position to precisely define scalars, vectors, and tensors under Galilean transformations. Each quantity, which features the same transformation property like the position vector X, ... 

In eq. (4.23) the three terms on the right hand side are scalar, vector, and tensor contributions, while s(9)> Pn,v( )> Pn,t( ) he scalar, vector, and tensor density form factors, respectively. These are given by ... 

The RIA optical potential model specified by eqs. (4.23)-(4.32) requires six separate target densities corresponding to scalar, vector and tensor distributions for both protons and neutrons. With the exception of the proton vector density, these must be obtained from theoretical models. The dependence of the RIA proton-nucleus scattering predictions on these theoretical nuclear structure models can be minimized, however, by the method discussed in the next two paragraphs. 

The one-meson exchange, relativistic optical potential was evaluated from eq. (4.8) using in eq. (4.37) and the kinematic factor in eq. (4.18) (except that the S ( )/S (0) factor was not includ ). For the direct term, p, only the scalar, vector and tensor invariants contribute to the optical potential for even-even nuclei, just as for the RIA potential of the previous section. For the exchange term it is clear from eq. (4.41) that all Lorentz components of contribute to the optical potential. For example, the exchange amplitude contribution to the scalar part of the optical potential involves the sum of amplitudes given by (using eq. (4.41) and the Fierz matrix in ref. [Ho 85])... 

For closed shell nuclei in the independent particle limit only the first four terms contribute. These are given in eqs. (3.29)-(3.34) in ref. [Tj 87b] in terms of the scalar, vector and tensor density form factors. The trace over nucleon 2 in eq. (4.66) leaves a Dirac operator acting on nucleon 1 (the projectile). Thus we may similarly expand as in eq. (4.69) to obtain... 

Here (following ref. [Lu87]) we reduce the single particle Dirac equation to a second-order differential equation which yields an equivalent upper component for the projectile wave function and consequently the same scattering observables. A more detailed account of the derivation of this so-called Schrodinger equivalent potential for local scalar, vector and tensor interactions is given in ref. [Cl 85],... 

In this work, the following mathematical notation is used for scalars, vectors and tensors ... 

Scalar, vector, and tensor fields. The gradient of a scalar field 0(x) is denoted by V0 and is the veetor defined by ... 

Continuum theories characterise the state of a physical system using scalar, vector and tensor valued functions of position x, a three dimensional point, and time t, a real number. These functions sometimes correspond to observable characteristics, such as velocity or position, and other times to more indirectly observable characteristics such as stress. Indirectly observable fields are theoretical constructs of a theory, and need the theory to determine their values. Note that in the following, the symbol x is used to represent a 3D point, an A -D point or the x-component of a 3D vector, x = (x, y, z) or a 2D vector, x = (x, y) the particular meaning of the symbol is always clear from the context. 

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