Einstein index convention

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

Here and further, we adopt the Einstein summation convention if an index appears twice in the same term, once as a subscript and once as a superscript, the sign 5Z will be omitted. 

Here and elsewhere in the present paper, we use a notation in which ( (cp) are the usual creation (annihilation) operators (c == Cp) associated with a given orthonormal spin-orbital basis set p. As usual, letters i and a designate the occupied and unoccupied spin-orbitals, respectively, while p is a generic index that runs over all (occupied and unoccupied) spin-orbitals. Whenever possible, the Einstein summation convention over repeated upper and lower indices is employed. 

The indices are underscored to denote that the Einstein summation convention is not to be used for the repeated index (see appendix A) i.e., they are not summed over. 

This notation is the so-called Einstein summation convention, stating that each index that occurs twice in an expression is summed over (with a range from 1 to 3). This index is called summation index. All other indices are called free indices. Thus, we can write... 

In the present research, the problem is formulated in a Cartesian coordinate system. For this, the index notation is used. The Einstein summation convention, i.e., the summation over identical indices within a term is performed. The spatial derivative Uij of the displacement vector u -... 

Similar to the distributed-multipole expansion of molecular electrostatic fields, one can derive a distributed-polarizability expansion of the molecular field response. We can start by including the multipole-expansion in the perturbing Hamiltonian term W = Qf(p, where we again use the Einstein sum convention for both superscripts a, referencing an expansion site, and subscripts t, which summarize the multipole components (/, k) in just one index. Using this approximation for the intermolecular electrostatic interaction, the second-order energy correction now reads ... 

The Greek indexes denote Cartesian components. Sap is the Kronecker delta, and the Einstein summation convention is implied. The viscous sttess has a Newtonian... 

It is assumed throughout that the reader has some familiarity with index notation (also called suffix notation in what is presented here) and the Einstein summation convention. Many suitable introductions are available such as those provided by, for example, the books of Aris [4], Leigh [161], Goodbody [115] or Spencer [256], or the introductory notes by Leslie [174]. 

We assume the familiar summation convention often ascribed to Einstein, so the repeated jS is understood as running through the standard index set for pairs. 

In all expressions the Einstein repeated index summation convention is used. Xi, x2 and x3 will be taken to be synonymous with x, y and z so that o-n = axx etc. The parameter B will be temperature-dependent through an activation energy expression and can be related to microstructural parameters such as grain size, diffusion coefficients, etc., on a case-by-case basis depending on the mechanism of creep involved.1 In addition, the index will depend on the mechanism which is active. In the linear case, n = 1 and B is equal to 1/3t/ where 17 is the linear shear viscosity of the material. Stresses, strains, and material parameters for the fibers will be denoted with a subscript or superscript/, and those for the matrix with a subscript or superscript m. 

RBEE). It is necessary at this point to note that for the above equation and for all equations, Einstein s summation convention is in place which implies (if not explicitly defined otherwise) a summation over repeated index. In RBEE, C,y(A) = 5y if A e E, dyU if A is on a smooth boundary, and 0 if A V. The 2D first and second fundamental solutions of the Stokes flow, e.g., G(A, P) and //(A, P), that appear as kernels in the integrals in RBEE... 

The mechanics of a deformable body treated here is based on Newton s laws of motion and the laws of thermodynamics. In this Chapter we present the fundamental concepts of continuum mechanics, and, for conciseness, the material is presented in Cartesian tensor formulation with the implicit assumption of Einstein s summation convention. Where this convention is exempted we shall denote the index thus ( a). 

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 ... 

The use of the more general tensor of rank 4, Cijki, simplifies the constitutive equations when the Einstein repeated-index summation convention is employed this convention consists of summing over all the allowed values of indices which appear more than once on one side of an equation. For example, the above mentioned linear elastic constitutive... 

---