Einstein summation

This expression demonstrates use of the Einstein summation convention 6. The significance of r is made clear by examining a particle momentarily at rest in a Lorentz system. The components of the vector, transformed dx = (0, 0, 0, icdt ) and dr2 = —(1 /c dx dx = [dt )2. Thus dr is the time interval on a clock travelling with the particle and is therefore referred to as the interval of the particle s proper time or world time. The relationship between dr and an interval of time as measured in a given Lorentz system can be derived directly by expanding the equation... 

The electrostatic interaction between two nonoverlapping charge distributions A and B, consisting of NA and NB atoms, respectively, and each represented by their atom-centered multipole moments, is given by (using the Einstein summation convention for the indices a, / , y) (Buckingham 1978)... 

For notational convenience, the components of r are here r1 r2, r3 rather than x y z. The Einstein summation convention, an implied summation over repeated indices, is used ineq. (11). 

In Eq. (5.60), pi denotes the viscosity fluctuation which may result from the temperature fluctuation. In Eq. (5.60) the quantities with the same subscript m in the third and fourth terms on the right-hand side of the equation indicate the Einstein summation. 

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. 

These equations show the damped expression for the ath component of the electric field from the solute multipoles in ra on the solvent centre in rk where a polarizability is located observe that once more the Einstein summation convention has been used with respect to the Greek indices. The parameter c determines the range of the damply 1 Y... 

Using the Einstein summation convention the molecular electronic... 

Let a Slater determinant , built up from the spin-orbitals totally symmetric hermitean two-electron operator 0(1,2). We wish to determine pair functions of the type (if no summation sign is used, the Einstein summation convention over repeated indices is implied)... 

The use of bold denotes that all quantities are vectors. The velocity V on the left-hand side is the instantaneous velocity, which is comprised of the sum of the mean velocity at the point (denoted by the overbar) and the fluctuating element of the velocity (denoted by the lowercase m). To permit the use of the Einstein summation convention, the component of the instantaneous velocity in direction i at a point is written as the sum of the mean and fluctuating components of the velocity in that direction. 

Einstein summation convention A notation in which, when an index is used more than once in an equation, it is implied that the equation needs to be summed over the applicable range of indices. Also known as Einstein notation. 

Introducing the well known Einstein summation notation (e.g., described by [154]), the Cartesian form of the equation can be formulated... 

Or, after dividing by dr dc dt and adopting Einsteins summation index notation ... 

The Einstein summation convention has been used in Eq. (2.18) and will be employed in the remainder of this section. If the system has periodic boundary conditions, Eq. (2.18) can be integrated by parts to obtain a description in terms of the displacement field. 

Here, pa and ma are the electric and magnetic dipole moment functions, aap, Papy> YapyS 81 the polarizability and first and second hyperpolarizabilities, is the magnetizability, and, of the other terms, only the hypermagnetizability will be of interest (it relates to the Cotton-Mouton effect). The Greek subscripts a, p,... denote vector or tensor quantities and can be equal to the Cartesian coordinates x, y, or z. Einstein summation over these subscripts is implied both here and elsewhere. Differentiation of this expression with respect to F (or B) leads to an expression for the dipole moment (or polarization) of the species in the presence of the perturbing fields and it is clear that P, y, t), etc. will govern the non-linear terms in the induced electric (or magnetic) dipole moment - hence, non-linear optics. 

For the general case we will write as usual t = x° x/c = xl-,y/c = x2 z/c = x3, (these superscripts are not exponents) and, with the usual Einstein summation convention (summation on the indices that appear both up and down), we obtain ... 

An even more compact formulation is obtained in covariant notation. We will follow the advice of Sakurai [38, p.6] and not introduce the Minkowski space metric g p, since the distinction between covariant and contravariant 4-vectors is not needed at the level of special relativity. We shall, however, employ the Einstein summation convention in which repeated Greek indices implies summation over the components a = 1,2,3,4 of a 4-vector. From the 4-gradient... 

The particular merit of multipolar gauge is that is allows one to express the scalar and vector potentials directly in terms of the fields E and B, thus facilitating the identification of electric and magnetic multipoles for generally time-dependent fields. We will follow the three-vector derivation given by Bloch [68]. We will furthermore in this section make extensive use of the Einstein summation convention for coordinate indices. Consider a Taylor expansion of the scalar potential... 

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