Vectors and Scalars

A scalar is a quantity that is completely described by its magnitude— a numerical value and usually a unit. Mass and temperature are scalars, with values, for example, like 10 kg and 300 K. A vector has, in addition, a direction. Velocity and force are vector quantities. A vector is usually printed as a boldface symbol, like A, while a scalar is printed in normal weight, usually in italics, like a. (Vectors are commonly handwritten by placing an arrow over the symbol, like A or V.) A vector in three-dimensional space can be considered as a sum of three components. Fig. 11.1 shows a vector A with its Cartesian components Ax, Ay, and A, alternatively written as Ai, A2, and A3. The vector A is represented by the sum 

FIGURE 11.1 Vector A with Cartesian components A, Ay, A. Unit vectors i, j, k 

A = A = yjAl + A + Al Newton s second law, written as a vector equation, 

Also Eq. (11.4) implies the corresponding relation betweeen the vector magnitudes 

A significant mathematical property of vector relationships is their invariance under translation and rotation. For example, if F and a are transformed to F and a by translation and/or rotation, the analog of Eq. (11.4), namely 

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Of course, k may be taken to be comprised of a number of such tensors, and it is not difficult to extend the theory to include a number of indifferent scalars and vectors, if desired. 

It is possible to assume other transformation properties for k. For example, for some purposes it may be more desirable to attribute strainlike properties obeying a transformation law like (A. 19), in which case the equations of this section will take a somewhat different form. Of course, k may be taken to be comprised of a number of such tensors, and it is not difficult to extend the theory to include a number of indifferent scalars and vectors, if desired. 

The complex product aj)pears to be made up of two terms, not too unlike the scalar and vector products, from which it seems to differ only because of a sign convention. 

These equations (14) and (15) determine the scalar and vector potentials in terms of p and J. When p and J are zero, these equations become wave equations with wave velocity c = y/l/pe. That is, A and are solutions of decoupled equations, where they are related by the wave operator... 

Here E(r, t) and B(r, t) denote the TD electric and magnetic fields defined by the scalar and vector potentials as... 

In Equation 6.19, which can be called TD Kohn-Sham-type equation, the effective scalar and vector potentials vcff(r, f) and Aeff(r, t), respectively, consist of contributions from the external potentials augmented by internal contributions determined by the density variables and can be expressed as... 

It is important to note once again that <, and A in equation (75) are the scalar and vector potential resulting from the external electromagnetic radiation field. Also note that here, , = V(, - and S, is the spin of the ith particle. We can identify each term in equation (75) as corresponding to a certain type of physical interaction of moving charged particles. The list of physical interpretations of terms follows in the same order as the terms in equation (75). 

The scalar and vector potentials created by one electron is related to its 4-current j1, ... 

The problem is treated as a two-dimensional one, with x-coordinate parallel to the surface and z-coordinate normal to the surface, with the z-axis negative into the solid. Since 3/3y and uy both vanish, the only non-zero component of the vector potential is fy. Suppose that there is a solution whose longitudinal and shear components each decay exponentially away from the surface, and that these are described by the scalar and vector potentials respectively. Then the potentials may be written... 

The fields can be derived from the well-known scalar and vector potentials in free space [8,14,16,17,23,24,26] as... 

R. M. Herman. Scalar and vector collisional interference in the vibration-rotation absorption spectra of H2 and HD. In R. J. Exton, ed., Spectral Line Shapes 4, p. 351, Deepak, Hampton, VA, 1987. 

Using scalar and vector products among the eigenvectors (direction cosines), demonstrate that the principal coordinates remain orthogonal. 

Equations (22), (24) are usually referred to as the equations defining the gauge transformations of the scalar and vector potentials for the fields. If we set... 

Grunewald, A.U., Gericke, K.-H., and Comes, F.J. (1988). Influence of H2O2 internal motion on scalar and vector properties of OH photofragments, J. Chem. Phys. 89, 345-354. 

To obtain the velocity of the relaxation mode, we shall transform vector equation (F.9). By multiplying it by the unit vector in two optional different ways (scalar and vector), we obtain the relations... 

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