Summation convention, Einstein

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). 

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. 

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. 

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... 

The Einstein summation convention [89] over repeated indices has been implied in (499) and will be used more often in this section. 

In the above expressions the Einstein summation convention is implied. 

The Einstein summation convention applies If an index occurs twice in one term of an expression, it is always to be summed unless the contrary is expressly stated. 

We obviously used the Einstein summation convention in this work on relativity theory. 

It is convenient to introduce the Einstein summation convention for products of covariant and contravariant vectors, whereby... 

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