Index notation

Cartesian tensors, i.e., tensors in a Cartesian coordinate system, will be discussed. Three Independent quantities are required to describe the position of a point in Cartesian coordinates. This set of quantities is X where X is (x, X2, X3). The index i in X has values 1,2, and 3 because of the three coordinates in three-dimensional space. The indices i and j in a j mean, therefore, that a j has nine components. Similarly, byi has 27 components, Cp has 81 components, etc. The indices are part of what is called index notation. The number of subscripts on the symboi denotes the order of the tensor. For example, a is a zero-order tensor... 

Thus, the transformation of coordinates can be written in index notation as... 

The generality of the index notation permits any specification of a chemical mechanism that the user desires. With the notation, the source term becomes ... 

The index notation (m +1) indicates the m +1 iteration for the value of y +i. The iteration matrix P is found by differentiating the right-hand side of Eq. 15.32 to yield... 

When a number is repeatedly multiplied by itself in an arithmetic expression, such as 3 x 3 x 3, or A x A x A x A, the power or Index notation (also often called the exponent) is used to write such products in the forms 33 and, respectively. Both numbers are in the general form an, where n is the index. If the index, n, is a positive integer, we define the number a " as a raised to the th power. 

Fig. 2. Stereographic triangle indicating various crystallographic orientations of fee solid surfaces using Miller index notations.

A single-index notation for symmetric 7(2)s introduced by Voigt is often very convenient. The pair of indices ij is contracted to the single indexp according to the following scheme ... 

Warnings (i) The Tpq do not form a second-rank tensor and so unitary transformations must be carried out using the four-index notation Tijki. (ii) The contraction of TiJki may be accompanied by the introduction of numerical factors, for example when 7(4) is the elastic stiffness (Nye (1957)). 

Find the 9 x 6 MR of the magnetothermoelectric power l,jki for a crystal of orthorhombic symmetry. Express your results in four-index notation, giving indices only. 

In the Miller index notation, the convention for denoting a direction perpendicnlar to a particnlar plane is to put square brackets around the numbers denoting that plane. The x-axis in Figure 2, for instance, is the [100] direction. 

In some structures, several planes and directions may be equivalent by symmetry. For example, this is the case for the (100), (010), (001), (100), (010), and (OOl) planes in the diamond cubic structure. Equivalent directions are denoted concisely as a group by using angular brackets. Thus, the (100) directions in a diamond cubic lattice include all of the directions that are perpendicular to the six planes noted above. The Miller index notation thus provides a concise designation for describing the surfaces of semiconductor crystals. 

Fig. 3.2. Surface structure of platinum single crystal catalysts with the corresponding Miller index notations given in brackets. (Reproduced, with permission, from Ref. 14.)...

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

In Cartesian index notation this term becomes ... 

To generalize the mass loss (or gain) rate term formulation, a staggered grid arrangement is used shifting the velocity index notation ... 

In these expressions, we have used the index notation i = (fi,...,/ ) such that = Note that, in the limit where PI 0, only the zeroth-order moment mi (0) = 1 is nonzero, and Eq. (3.126) reverts to the QMOM moments. The central Gaussian moments m-z(i) are known functions of the covariance matrix E. For a given moment order 7 = ki + + kM, Eq. (3.126) has a lower triangular form that can be inverted using forward substitution ... 

Problem 2-4. Index Notation. Write the following expressions using index notation ... 

Problem 2-13. Derivation of Transport Equations. Consider the arbitrary fluid element depicted in the figure. If we have a flow containing several species that are undergoing reaction (a source/sink per unit volume) and diffusion (a flux of each species in addition to convection), derive the equation that governs the conservation of each species. The source of species i that is due to reaction is denoted as Rt (units of mass of i per unit time per unit volume) and the total mass flux of species i (diffusion and convection) is given by (p u + ji), in which p, is the mass of species i per unit volume, u is the total mass average velocity of the fluid and j, is the diffusive flux of species i. Note that both u and j, are vectors. We are not using index notation in this problem ... 

The specific case of the third-order tensor K = Wu.x, was considered by Nadim and Stone.4 In index notation, the irreducible description of K is... 

One particularly useful feature of the Cartesian index notation is that it provides a very convenient framework for working out vector and tensor identities. Two simple examples follow ... 

It is important to note that vector relations such as the identity that we have proven in this example are invariant to the coordinate system. So, even though we have proven this result by using Cartesian index notation, the result is valid in all coordinate systems (e.g., Cartesian, cylindrical, spherical, etc.)... 

Repeated application of the creation operators generates a Fock space in which many-particle states are expanded. Each of the basis states of this Fock space is represented by its occupation vector n> given in the pair index notation... 

We showed in Chapter 18 that the molecular diffusion term on the RHS of (18.1) can be neglected in atmospheric flows. In Chapter 18 we used index notation, for example, S(ujCi)/Sxj. Here we use both vector and index notation. The three wind velocity components can be denoted ( i, u2,u ),(uxi uy, u7), or (u,v,w) in Cartesian coordinates, and each of these notations has been used at various points in the book. 

To investigate whether c, satisfies (25.52), we begin with the conservation equation using index notation... 

TABLE 2.2. Correspondence Between the Miller-Index Notation and Stepped-Surface Notation... 

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