Unit tensors

Equations (8.20) are not sufficiently specific for practical purposes, so it is important to consider special cases leading to simpler relations. When the pore orientations are isotropically distributed, the second order tensors k, 3 and y are isotropic and are therefore scalar multiples of the unit tensor. Thus equation (8.20) simplifies to... 

If we assume that the body force is only due to gravity and use the definition r = —prL +1 where pr is the isotropic resin pressure, / is the unit tensor, and r is the deviatoric stress as well as assuming a constant density and define a new pressure Pr —pr + prgh, Equation 5.21 simplifies to,... 

Here D, D , and Dr are, respectively, the longitudinal, transverse, and rotational diffusion coefficients of the chain averaged over the internal degree of freedom, h an external field, and v and angular velocity of the chain induced by a flow field in the solution. Furthermore, I is the unit tensor and 91 is the rotational operator defined by... 

Here < >means the average with respect to f(a t), and I is the unit tensor. 

In this equation, the unit tensor, 1, appears. This equation is a set of (vector) equations which would have to be solved simultaneously to get the velocity tj for all the particles. It can be inverted and put in a more obvious physical form by writing the inverse tensor element... 

The viscosity of the medium is t, and 1 is the unit tensor. (The Oseen tensor is the Green s function for the Navier-Stokes equation under the conditions that the fluid is incompressible, convective effects can be neglected, and inertial effects coming from the time derivative can be neglected.)... 

Angular momentum and tensorial algebra 5.4 Unit tensors... 

Unit tensors play, together with spherical functions, a very important role in theoretical atomic spectroscopy, particularly when dealing with the many-electron aspect of this problem. Unit tensorial operator uk is defined via its one-electron submatrix element [22]... 

To conclude this chapter, let us present the main formulas for sums of unit tensors, necessary for evaluation of matrix elements df the energy operator. They will be necessary in Part 5. The matrix element of any irreducible tensorial operator may be written as follows ... 

The analogue of unit tensor (5.23) in j-space may be defined as follows ... 

The need to have only one sort of unit tensor (7.2) and its sum (7.4) is conditioned by the fact that, unlike a non-relativistic case, where we... 

When constructing many-electron wave functions it is necessary to ensure their antisymmetry under permutation of any pair of coordinates. Having introduced the concepts of the CFP and unit tensors, Racah [22, 23] laid the foundations of the tensorial approach to the problem of constructing antisymmetric wave functions and finding matrix elements of operators corresponding to physical quantities. 

Apart from phase system (13.42) some of the tensorial operators (e.g. unit tensors and their sums (5.27), (5.28)) are defined in a pseudostandard phase system (see Introduction, Eq. (5)), for which... 

Equations of the kind (14.23) or (14.25) are inconsistent in the way that tensors in their left and right sides are defined in different (pseudostandard and standard) phase systems. This is done to underline that historically tensors composed of unit tensors were defined in pseudostandard phase systems whereas in this book main tensorial quantities obey standard phase systems. 

Unit tensors are especially important for group-theoretical methods of studying the lN configuration. We can express the infinitesimal operators of the groups [10, 24, 98], the parameters of irreducible representations of which are applied to achieve an additional classification of states of a shell of equivalent electrons, in terms of them. 

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