Einstein notation

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. 

The free subindices notation was introduced by Einstein and Lorentz and is commonly called the Einstein notation. This notation is a useful way to collapse the information when dealing with equations in cartesian coordinates, and it is equivalent to subindices used when writing computer code. The Einstein notation has some basic rules that are as follows,... 

An alternative and simpler approach to deriving the result in equation (4.12) is to express the polarizability tensor as a general expansion in the two orthogonal unit vectors, u and p, embedded on the principal axes shown in Figure 4.4. Evidently, using Einstein notation, the polarizability can be written as... 

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. 

For small strains, it is assumed that the material behaves linearly, i.e. that every component of strain is linearly related to every component of stress. Using the Einstein notation for summation, this relationship can be written... 

In the above equations, 7 is the identity tensor and T is the Maxwell stress tensor Ty is the representation of this tensor with Einstein notation. 

The Einstein notation is used throughout this chapter and a single reaction of A is considered. In (12.3-la), the species concentrations are expressed in terms of molar concentrations, C. An alternative form in terms of the species mass fractions, Y, is ... 

A more compact way to write eq. (5-3) is through the use of Einstein notation ... 

Einstein s original treahnent [5] used a somewhat different notation, which is still in conmion use ... 

It can therefore be inferred that 0(3) electrodynamics is a theory of Rieman-nian curved spacetime, as is the homomorphic SU(2) theory of Barrett [50], Both 0(3) and SU(2) electrodynamics are substructures of general relativity as represented by the irreducible representations of the Einstein group, a continuous Lie group [117]. The Ba> field in vector notation is defined in curved spacetime by... 

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). 

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. 

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

The conventional Reynolds averaging procedure is deduced from the governing equations for incompressible fluid systems. In Cartesian coordinates the corresponding instantaneous equation of continuity takes the following form (i.e., written in a compact form by use Einstein s summation notation) ... 

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

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... 

Here +, B are the Einstein coefficients for absorptive transitions to the two exciton states, and +, are the incident light intensities at the respective transition wavelengths. (The Einstein coefficients are proportional to the extinction coefficients of the absorption bands). Throughout this chapter, the notation x indicates a unit vector that points in the same direction as x. 

Stokes-Einstein-Schmoluchowski (SSE) equation. This result, cast in the present notation, is given in equations (7) where the constant C incorporates the collision diameter of the collisional cage pair and the numerical constants. 

Einstein and Mayer (Bibl. 1931, 3) always use these general coordinate systems. In their work the mapping (21) plays a sinificant role that we hope to clarify in the following. Einstein and Mayer used the notation li instead... 

Returning to the conjugate problem, we see a more complex situation compared to the case of special relativity. As already pointed out, photons or particles of zero rest mass (mo = 0), exhibit a different gravitational law compared to particles with mo 0. The latter, i.e. the well-known prediction and the experimentally confirmed fact of the light deviation in the Sun s gravitational field, measured during a solar eclipse, instantly boosted Einstein to international fame. Therefore, we need to account for this inconsistency for zero rest mass particles, by introducing the notation /co(r) = Go M/(c r). Hence, one obtains (mo - 0) that... 

This notation is valid for concentrations expressed in any units. With mole fractions, in view of Equation 1.1.3-7 and Oi = Equation 43 gives Einstein s equation... 

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