Einstein convention

Relaxing the Einstein convention, sums over repeated Greek indices a = x,y,z are made explicit in this Section, to avoid misunderstanding whenever two couples of repeated indices a and / , with a < (3, appear in a formula, compare fo r (92) hereafter. Introducing a basis set x of atomic functions, for the second-order term one defines the expansion... 

The leading term P0(u) is the harmonic probability distribution, af = 1, 2, or 3, and Z)ai... Dtr is the rth partial derivative operator dr/(duxi. .. duXr). The Einstein convention of summation over repeated indices is implied. 

In this expression, the Einstein convention of summation over repeated indices has been followed p0 is the permanent dipole moment, while al7, fiijk, and yiJkl are the tensorial elements of the linear polarizability, and the second- and third-order hyperpolarizabilities of the molecule, respectively. 

P 61] The numerical simulations were based on the solution of the incompressible Navier-Stokes equation and a convection-diffusion equation for a concentration field by means of the finite-volume method [152], The Einstein convention of summation over repeated indices was used. For pressure-velocity coupling, the SIMPLEC algorithm and for discretization of the species concentration equation the QUICK differencing scheme were applied. Hybrid and the central differencing schemes referred to velocities and pressure, respectively (commercial flow solvers CFX4 and CFX5). 

This can be denoted as aaa because we are using the Einstein convention stipulating that repeated indices be summed. Thus23 Tr a = aSaa + P[uaua — 1 j3Saa] = 3a. The isotropic polarizability a is simply the trace of the polarizability tensor, that is a = Tra. It should be noted that the tensor a is symmetric (aafi = afa) and that the anisotropic part iPafi) is traceless (zero trace) because uaua — jSaa = 0. 

For brevity I have used in section 3.1 the system of natural units, in which the numerical value of the speed of light c and the numerical value of h = /i/(27t), with h being the Planck constant, are equal to 1. The Einstein convention of summing over repeated indices, of which one is covariant and one is contravariant, has been employed. 

Deformation is measured by a quantity known as strain (strain is a relative extension or contraction of dimension). Strain is similarly a tensor of the second rank having nine components (3x3 matrix). The relation between stress and strain in the elastic regime is given by the classical Hooke s law. It is therefore obvious that the Hooke s proportionality constant, known as the elastic modulus, is a tensor of 4 rank and is represented by a (9 x 9) matrix. Before further discussion we note the following. The stress tensor consists of 9 elements of which stability conditions require cjxy=(jyx, stress components in the symmetric stress matrix are only six. Similarly there are only six independent strain components. Therefore there can only be six stress and six strain components for an elastic body which has unequal elastic responses in x, y and z directions as in a completely anisotropic solid. The representation of elastic properties become simple by following the well known Einstein convention. The subscript xx, yy, zz, yz, zx and xy are respectively represented by 1, 2, 3, 4, 5 and 6. Therefore Hooke s law may now be written in a generalized form as. 

It can be shown that tensor components transform as follows (using the Einstein convention) ... 

As we have seen above, the tensor has components with three indices, For such a tensor it is possible to perform a mathematical operation, called conttaction, that gives a co-vector as the result. In other words, it can be shown that suitable linear combinations of components of the tensor behave as the components of a vector. For the tensor it is possible to contract over the conttavariant index and over one of the covariant indices, and the result is a (co)-vector, indicated as or , whose generic (i-nth) component is (Einstein convention). Therefore, the explicit form of the three components of p in the reference system R are ... 

Standard tensor formalism is employed throughout this paper, for example, the Einstein convention of implicit summation over two repeated Greek subscripts is in force. The notation adopted in previous references [15, 39,40] is used. The SI system of units has been adopted. 

Terminology and notation adopted in previous papers and reviews [2-5] are used, allowing for standard tensor formalism, e.g., smnmation over repeated Greek indices is implied according to the Einstein convention. 

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