Tensor summation convention

The symbol M represents the masses of the nuclei in the molecule, which for simplicity are taken to be equal. The symbol is the Kionecker delta. The tensor notation is used in this section and the summation convention is assumed for all repeated indexes not placed in parentheses. In Eq. (91) the NACT appears (this being a matrix in the electronic Hilbert space, whose components are denoted by labels k, m, and a vector with respect to the b component of the nuclear coordinate R). It is given by an integral over the electron coordinates... 

In this equation ut should be interpreted as the volumetric flux density (directional flow rate per unit total area). The indexes range from 1 to 3, and repetition of an index indicates summation over that index according to the conventional summation convention for Cartesian tensors. The term superficial velocity is often used, but it is in our opinion that it is misleading because n, is neither equal to the average velocity of the flow front nor to the local velocity in the pores. The permeability Kg is a positive definite tensor quantity and it can be determined both from unidirectional and radial flow experiments [20], Darcy s law has to be supplemented by a continuity equation to form a complete set of equations. In terms of the flux density this becomes ... 

Although we are concerned with linear algebra it will be convenient to use a notation slightly different from the usual matrix notation and to adopt the summation convention from tensor analysis. Thus asr will denote the element in the rth row and 5 th column of the matrix a = (a ). The product a fi of a with a matrix ft = (, ) is the matrix (otsrfi s), where summation over s = 1, 2,. .., s is implied. The range of a particular affix (i.e., s = 1,2, s) is given in the table of nomenclature. In this notation the order of any matrix is apparent and the use of partitioned matrices can be represented rather easily. 

Since the publication of some prolegomena to the rational analysis of systems of chemical reaction [1] other cognate work has come to light and some earlier statements have been made more precise and comprehensive. I would like first to advert to an earlier work previously overlooked and to mention some recent publications that partially fill some of the undeveloped areas noticed before. Secondly, I wish to extend the theorem on the uniqueness of equilibrium to a more general case and to establish the conditions for the consistency of the kinetic and equilibrium expressions ( 2, 3). Thirdly, the conception of a reaction mechanism is to be reformulated in a more general way and the metrical connection between the kinetics of the mechanism and those of the ostensible reactions clarified. The notation of the earlier paper ([1], hereinafter referred to as P) will be followed and augmented where necessary. In particular the reader is reminded that the range of each affix is carefully specified and the summation convention of tensor analysis is employed. 

Each response is the sum of terms in that row (observing the summation convention). The coefficients are tensors, and the rank n of T(n) is shown by the number of subscripts. 

Tensors are denoted by bold-face symbols, 0 is the tensor product, and the scalar product. For example, with respect to a Cartesian basis e AB = AikBkjei ej,A B = AijBij, and C B = C hlBkiei 0 ej, with summation implied over repeated Latin indices. The summation convention is not used for repeated Greek indices. 

Tensors, from the same or different fields, can be combined by outer multiplication, denoted by juxtaposing indices with order preserved on the resultant tensor.33 It is possible that an index is present both in the covariant and contravariant index sets then with the repeated index summation convention, both are eliminated and a tensor of lower rank results. The elimination of pairs of indices is known as contraction, and outer multiplication followed by contraction is inner multiplication.33 In multiplication between tensors, contractions cannot take place entirely within one normal product (i.e., the generalized time-independent Wick theorem see Section IV) hence such tensors are called irreducible. 

We now show that the projection of the Schrodinger equation for the CCSDT wave function on the triply excited space [cf. Eq. (106)] can be written in terms of (at worst) products of unmodified rank 3 cluster coefficients and modified rank 2 integrals. Tensor notation with repeated index summation convention will be used, except, of course, for the permutation operator. 

Here we have used the fact that Fij = 5 + Uij. In addition, we have invoked the summation convention in which all repeated indices (in this case the index k) are summed over. For the case in which all the displacement gradient components satisfy Uij 1, the final term in the expression above may be neglected, resulting in the identification of the small strain (or infinitesimal strain) tensor,... 

According to the summation convention, we must sum over any repeated index over all possible values of that index. So the scalar product produces a scalar that is equal to A i Si -(- A2B2 + A3B3, whereas the vector product is a vector, the /th component of which is SijkAjBit (so, for example, the component in the 1 direction is A2B2 — A2B2), and the dyadic product is a second-order tensor with a typical component A, Bj (if we consider all possible combinations of i and j, there are clearly nine independent components). 

We begin by writing down the conservation or balance laws (Ericksen ). We shall employ the cartesian tensor notation, repeated tensor indices being subject to the usual summation convention. The comma denotes partial differentiation with respect to spatial coordinates and the superposed dot a material time derivative. For example,... 

The last definition of function / of 9 variables (allowed by symmetry of D) permits to employ the customary tensor (or matrix) descriptions, e.g. the summation convention in component form. This is the reason for using this definition of / in (3.146), (3.147) and other formulae in this book (similar definitions may be used for skew-symmetric tensor and vector and tensor functions [7, 14, 79]). As may be seen from the definition above, the main property of / is (when D is symmetrical and this is just such a case) that is indeed symmetrical, e.g. 

In this book Greek indices a, p, ft, v... are used to indicate the components of vectors and tensors, and the summation convention is used over the repeated indices.)... 

In relativistic theory one often encounters vector and tensor expressions in both three- and four-dimensional form. The most important of these expressions are briefly summarized in this section, where Einstein s summation convention (cf. section 3.1.2) is strictly applied for four-dimensional objects. 

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). 

A.4 Tensor operations and Einstein summation convention 453 and the value of the order below (4, B, ) Other typographic conventions... 

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