Tensor index notation

Switching over from tensor index notation to direct tensor notation this corresponds to... 

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... 

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)). 

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 ... 

If we want to specify a component of a tensor in a certain coordinate system, we use the index notation. Each tensor order requires its own index. Usually, lowercase letters are used as indices, starting with i (uj, Aij, Algebraic rules are frequently written down in index notation. Although the components themselves depend on the coordinate system, the rules are nevertheless valid in all systems. 

If all components of a tensor are to be described, this is done by adding parentheses to the tensor written in index notation, for example, (a ), Aij), Cijki). Implicitly, it is assumed that each index runs from 1 to 3. In second-order tensors, the first index denotes the row and the second index denotes the column of the component in the matrix notation. The components of the tensor... 

Because a tensor product is a scalar quantity in index notation, the commutative law can be used within the sums ... 

In this list, the matrices and tensors are printed in the index notation. The corresponding symbol notation can be built using the scheme (Aij) = A and (Aijki) = A, respectively. Unless stated otherwise, the indices run from 1 to 3. 

These tangents are generally non-orthogonal and are not normalized. Greek indices take values in 1,2, and repeated indices are summed according to index notation. Equation (5.1) defines the basis for the tangent plane at X, which is characterized by the metric tensor... 

Indicial notation and compact form of generalized Hooke s Law Because of the cumbersome form of the generalized Hooke s Law for material constitutive response in three dimensions (Eq, 2,28), a shorthand notation referred to as indicial or index notation is extensively used. Here we provide a brief summary of indicial notation and further details may be found in many books on continuum mechanics (e.g., Fliigge, 1972). In indicial notation, the subscripts on tensors are used with very precise rules and conventions and provide a compact way to relate and manipulate ten-sorial expressions. 

With the chosen type of coordinates and for the sake of simplicity, it can be abstained from the index notation. The classification of tensors with the applied typesetting conventions is given in Table 3.1. 

Many students with engineering or physics backgrounds are already familiar with the stress tensor. They may skip ahead to the next section. The key concepts in this section are understanding (1) that tensors can operate on vectors (eq. 1.2.10), (2) standard index notation (eq. 1.2.21), (3) symmetry of the stress tensor (eq. 1.2.37), (4) the concept of pressure (eq. 1.2.44), and(5)normalstressdifferences(eq. 1.2.45). 

This numbering of components leads to a convenient index notation. As indicated the nine scalar components of the stress tensor can be represented by 7,7, where i and j can take the values from 1 to 3 and the unit vectors Xi, X2. Xa become x,. Thus, we can write the stress tensor with its unit dyads as... 

To summarize, three types of notation are used in vector and tensor manipulations. The simplest to write is the Gibbs form (e.g., n T), which is convenient for writing equations and seeing the physics of things quickly. The index notation in its expanded form (e-g-. Yli Hy or abbreviated (e.g., njTj,), indicates... 

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... 

The multi-index in parentheses, used as superscript on the tensor, is actually the norm k however, since it also specifies the rank of the tensor, and therefore we use a special notation. 

Even though we do not invoke the full machinery of tensor analysis (Butkov, 1968), it is useful to keep the distinction between contravariant and covariant components clear. To avoid conflicting notation we do not use upper and lower indices to denote contravariant and covariant indices. Instead, we will use the suffix ao (lower case letters) on tensors whose indices are all contravariant, and AO (capital letters) on tensors whose indices are all covariant. No special suffix is used in the MO basis. For example, using the two- and four-index trace operators the energy is... 

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. 

The coupling tensors Q, D, o, and J are tensors P of rank 2 [Hael, Mehl, Spil]. For each of the corresponding interactions, A, the tensor, can be separated into an isotropic part Pf, an antisymmetric part P[ and a traceless symmetric part P . For simplicity of notation, the index A is not carried along in the next six equations. In the principal axes system XYZ of the symmetric part of the coupling tensor, the generic coupling tensor P = Pij, where i,j = X, Y, Z, is written as... 

Cartesian component notation offers an extremely convenient shorthand representation for vectors, tensors, and vector calculus operations. In this formalism, we represent vectors or tensors in terms of their typical components. For example, we can represent a vector A in terms of its typical component Ait where the index i has possible values 1, 2, or 3. Hence we represent the position vector x as x, and the (vector) gradient operator V as Note that there is nothing special about the letter that is chosen to represent the index. We could equally well write x . x/ . or x , as long as we remember that, whatever letter we choose, its possible values are 1, 2, or 3. The second-order identity tensor I is represented by its components %, defined to be equal to 1 when i = j and to be equal to 0 if z j. The third-order alternating tensor e is represented by its components ,< , defined as... 

At this point it is useful to pause and describe the notation we have used for various states. A state of A is given by the tensor product of a state of A with quantum number q and an index i, and a state a of the additional site. Thus,... 

The tensors and 7 constitute the molecular origin of the second-and third-order nonlinear optical phenomena such as electro-optic Pock-els effect (EOPE), optical rectification (OR), third harmonic generation (THG), electric field induced second harmonic generation (EFI-SHG), intensity dependent refractive index (IDRI), optical Kerr effect (OKE), electric field induced optical rectification (EFI-OR). To save space we do not indicate the full expressions for and 7 related to the different second and third order processes but we introduce the notations —(Ajy,ui,cj2) and 7(—a , o i,W2,W3), where the frequency relations to be used for the various non-linear optical processes which can be obtained in the case of both static and oscillating monochromatic fields are reported in Table 1.7. 

Because of the symmetry of the indices j and k one can replace these two by a single index (subscript) m. Consequently the notation for the SHG nonlinear coefficient in reduced form is where m takes the values 1 to 6. Only noncentrosymmetric crystals can possess a nonvanishing tensor (third rank). The unit of the SHG coefficients is m/V (in the MKSQ/SI system). 

Notice that the terms r j of the submatrix R are functions of the angles between the axes of two reference frames, so fhe acfual independenf variables are three and not nine. The notation of Equafion 1.1 means fhat T is the tensor transforming poses from reference frame PI to reference frame P2. Tensors can be inverted to reverse the transformation and chain multiplied to combine different transformations. In the multiplications the right upper index of the first matrix must equal the lower right index of the second. 

Since Ci and 8 are symmetric tensors, each of them has 6 independent components, the fourth order stiffness tensor Cyia contains at most 36 independent constants, such that it can be displayed as a 6 x 6 matrix of components using contracted notation, Cto), where m,n — 1,2,3,4,5,6. There is a unique correspondence between of and Cijki- The index m is related to ij, and n is related to Id, as shown in Table B.l. For instance, Cu22 = C12, C1323 = C54- Since C = C the number of independent constants is generally 21. For orthotropic materials, the the number of independent constants further reduces to 9. When the fourth order tensor is transformed to the principal axes, all Qj — 0, except for Cn, C22, C33, C12. 13. 23. 44, 55, and... 

The independerrt componerrt s stmctrrre of the piezoelectric coefficient is more clearly seen in rrratrix than in tensor notation. Symmetry of the piezoelectric tensor reflects syrrrmetry of mechanical stress/strain (they are secorrd-rarrk syrrrmet-rical tensors). Piezoelectric coefficient is therefore third-rarrk terrsor syrrrmetrical with respect to the permutation of two indexes. Piezoelectric coeffiderrts satisfy following relations between tensor dyk and matrix di coefficients... 

In reality, the rate of dissipation is a local physical notion that varies in both space and time. As will be seen hereinafter, turbulence dramatically increases the rate of energy dissipation, and hence its paramount importance in fluid mechanics. The rate of energy dissipation can be defined rigorously on the basis of the Navier-Stokes equations, by estabhshing the evolution equation for kinetic energy. For this derivation, it is assumed that the fluid is incompressible. A tensor notation is used (Einstein s notation) where the velocity vector is expressed in the form Ui with index i varying from 1 to 3 to designate the three space dimensions (namely u =... 

While the tensor notation is corrverrierrt for srrbsequent calcrrlatiorts, it should be remembered that, in equatiorrs [2.34] artd [2.35], the terrrrs involving index j are summed for all three values ofy, i.e. in eqrration [2.35],... 

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