Tensors definition

In arriving at this conclusion, we have used antisymmetric tensor definitions such as... 

Now recall the definition of an elementary tensor (Definition 2.15). It tons out that elementary tensors always correspond to unentangled states, and vice versa. 

The St. Venant compatibility equations [13,14,15] follow immediately from the strain tensor definition of Equation (2), and are... 

The Janak s theorem (eq.17) and the hardness tensor definition (eq. 20) allows the calculations of Tiij as the first derivative of the Kohn-Sham orbital eigenvalues with respect to the orbital occupation numbers [17] ... 

It is important to use the exact strain tensor definition, Eq. (6), to achieve rotational invariance with respect to lattice rotation the conventional linear strain tensor only provides differential rotational invariance of u in Eq. (7).hierarchy of approximations may be used for the elastic tensor 7. The most rigorous approach is to transform the bulk elastic tensor c according to... 

Introducing the expressions for the first order approximation of the distribution function (2.246) and thereafter the partial solution for (2.247) into the pressure tensor definition, (2.64), we get [39] ... 

Other approaches have been suggested to more formally define the form of damage that occurs during fatigue (34-37). These methods model damage as an array of microcracks (34,35) that inevitably require a tensoral definition of... 

In the OKE response only the off-diagonal elements, i j, of the dielectric tensor are involved, so the isotropic polarizability, a, does not contribute. Furthermore, the experimental q wave-vector value is zero, hence the translational part of (2.27) is not relevant. Concerning the OKE response the susceptibility tensor definition can be simplified as... 

D. L. Bryce, R. E. Wasylishen and M. Gee. Characterization of tricoordinate boron chemical shift tensors Definitive high-field solid-state NMR evidence for anisotropic boron shielding. J. Phys. Chem. A 105, 2001, 3633. 

Note that strains transform with the same transformation equation as stresses provided that the tensor definition of shear strain is used (see Table 8.3). 

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