Cartesian component notation

This is identical to (2-8), though now expressed in Cartesian component notation.6 Time derivatives at fixed spatial position are often called the Eulerian time derivatives, whereas those taken at a fixed material point are known as Lagrangian. Although we have derived a simple relationship relating the convected or material derivative to the ordinary partial derivative at a fixed point, this cannot be applied directly to (2-6) without derivation of a general relationship, known as the Reynolds transport theorem. 

Here, for simplicity, we utilize Cartesian component notation. For the reader who is not familiar with this notation, a brief summary is given in Appendix B at the end of the book. 

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... 

Cartesian components of Q. We will also indicate the three scalar coordinates associated with each Q vector by the notation where i = 1,... and 5 = 1,..., 3. For the — 1 bond vectors we dehne... 

In our notation, dt> designates the vector with the Cartesian components dnx dvr diiz. On the other hand, d3n = dvx dvy dv2 designates a volume element in velocity space boldface is not used for scalars. 

These equations use Cartesian tensor notation in which a repeated Greek suffix denotes summation over the three components, and where ay7 is the third-rank antisymmetric unit tensor. For a molecule composed entirely of idealized axially symmetric bonds, for which [3 (G )2 = /3(A)2 and aG1 = 0 [13, 15], a simple bond polarizability theory shows that ROA is generated exclusively by anisotropic scattering, and the CID expressions then reduce to [13]... 

The commutation relations of the orbital angular momentum operators can be derived from those between the components of r and p. If we denote the Cartesian components by the subindices i, k, and /, we can use the short-hand notation... 

Polarizability of an Isolated Molecule.—In Cartesian tensor notation, the components of the molecular dipole moment p, induced by an electric field E, can be written as ... 

In the derivation of balance equations the tensor notation is used because it allows the equations to be written in a clearer and simpler fashion. We have restricted ourselves in this to cartesian coordinates. In the following, the essential features of cartesian tensor notation are only illustrated to an extent required for the derivation of the balance equations extensive publications are available for further reading. We will start with an example. The velocity w of a point of mass is known as a vector which can be set in a cartesian coordinate system using its components wxl wy, wz ... 

In component notation in Cartesian geometry and two dimensions, these equations are... 

Here, we call the attention of the reader that our Eq. (24) in the previous interlude correspond to Eqs. (32) in Ref. [3] for the cartesian components of the angular momentum in the body frame and the inertial frame, respectively, in terms of the Euler angles. Notice that the angles if and

commutation rules from Eq. (22) in Ref. [3], for the analysis of the rotations of asymmetric molecules are as follows ... 

Expressed in component form using Cartesian tensor notation, this equation is... 

The spatial part represents the two-electron field gradient integrals. Employing the notation for the Cartesian components... 

We will use the following notation Hnlm is an operator presenting an n th-order dependency on B, /-order in nN, and m-order in /ie, the subscript i refers to electron i, subscript N refers to nucleus N, and u and v represent Cartesian components of vectors and tensors. The resulting expression for the Hamiltonian including magnetic terms is as follows... 

The nine reference stress components, each of which depends on position x and time t, and referred to as the stress tensor components. In Cartesian tensor notation we may write... 

We are here and in the following using the dyad notation, such as fa, for nine component tensors, with each component being a product of cartesian components of the vectors. 

The simplest approach to the theory of light scattering in general and Raman scattering in particular is to introduce an oscillating electric dipole moment induced in the molecule by the electric field vector JE of the light wave. In cartesian tensor notation, the -component is written... 

Tensor notation is central to the theory of both linear and nonlinear light scattering, so a brief summary is appropriate here. A Greek subscript denotes a vector or tensor component and can be x,y or z. A repeated Greek subscript within a term denotes a sum over all three cartesian components this is the tensor equivalent of a scalar product so that, for example. 

It is apparent that the 7 take the place in this formulation of the interaction tensors T of the conventional Cartesian formulation, but it should be emphasized once again that all the formulae given here refer to multipole moment components in the local, molecule-fixed frame of each molecule, whereas the corresponding Cartesian formulae deal in space-fixed components throughout and require a separate transformation between molecule-fixed and space-fixed frames. ( Space-fixed is perhaps a misleading term here, since the calculation is commonly carried out in a coordinate system with one of its axes along the intermolecular vector. However, the point is that in the Cartesian tensor notation there has to be a common set of axes for the system as a whole, and this can be the molecule-fixed frame for at most one of the molecules involved.)... 

By way of introduction, consider Ohm s law for isotropic media in standard form J = aS., which relates the current density via the conductivity to the electric field that is externally imposed. In standard component notation, this takes the form /, = crE,-, i= 1,2,3, where the numerals refer respectively to the mutually orthogonal Ox, Oy, and Oz Cartesian axes. For simplicity, the vector may be made to coincide with the Ox axis. In isotropic media, the current flow direction coincides with that of the electric field. 

The notation for the cartesian component indices of the various quadrupole (hyper) polarizabilities, e.g. differs slightly from the one for the pure dipole (hyper)polarizabilities, e.g. 0aP y<... 

The quantities in the two-dimensional list are called matrix elements. Each matrix element has two subscripts, one for the row and one for the column. The brackets written on the left and right are part of the notation. If a matrix has the same number of rows as columns m = n), it is a square matrix. A vector in ordinary space can be represented as a list of three Cartesian components, which is a matrix with one row and three columns. We call this a row vector. A vector can also be represented by a column vector with three rows and one column. We can also define row vectors and column vectors with more than three elements when they apply to something other than ordinary space. Just as there are types of algebra for scalars, vectors, and operators, there is a well-defined matrix algebra. 

Sometimes the notation (v V) is used for what is represented in Cartesian components by the operator Vidfdxi. Note that the operators defined in (4.5) and (4.6) may be applied to a scalar or vector quantity. Details on the physical interpretation of the material time derivative can be found in Aris [4, pp.77-79] where, in particular, it is shown that the velocity is given by v = x. 

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... 

In tensor notation the three Cartesian directions x, y, and z are designated by suffixed variables i,j, k, l, etc. (Landau and Lifshitz 1970 Auld 1973). Thus the force acting per unit area on a surface may be described as a traction vector with components rj j = x, y, z. The stress in an infinitesimal cube volume element may then be described by the tractions on three of the faces, giving nine elements of stress cry (i, j = x, y, z), where the first suffix denotes the normal to the plane on which a given traction operates, and the second suffix denotes the direction of a traction component. 

With this notation, the electric charge qo of a monopole equals Qoo-Cartesian dipole components px, py, pz, are related to the spherical tensor components as Ql0 = pz, Qi i = +(px ipy)/y/2, with i designating the imaginary unit. Similar relationships between Cartesian and spherical tensor components can be specified for the higher multipole moments (Gray and Gubbins 1984). 

Any Fock operator can be represented as a sum of the symmetric one and of a perturbation which includes both the dependence of the matrix elements on nuclear shifts from the equilibrium positions and the transition to a less symmetric environment due to the substitution. To pursue this, we first introduce some notations. Let hi be the supervector of the first derivatives of the matrix of the Fock operator with respect to nuclear shifts Sq counted from a symmetrical equilibrium configuration. By a supervector, we understand here a vector whose components numbered by the nuclear Cartesian shifts are themselves 10 x 10 matrices of the first derivatives of the Fock operator, with respect to the latter. Then the scalar product of the vector of all nuclear shifts 6q j and of the supervector hi yields a 10 x 10 matrix of the corrections to the Fockian linear in the nuclear shifts ... 

Here we introduce the notation ( . ..) for the scalar product of vectors whose components are numbered by the Cartesian shifts of the nuclei). Next, let h" be the supermatrix of the second derivatives of the matrix of the Fock operator with respect to the same shifts. As previously, we refer here to the supermatrix indexed by the pairs of nuclear shifts in order to stress that the elements of this matrix are themselves the 10 x 10 matrices of the corresponding second derivatives of the Fock operator with respect to the shifts. The contribution of the second order in the nuclear shifts can be given the form of the (super)matrix average over the vector of the nuclear shifts ... 

---