Strength tensor

Most components of the strength tensors are defined in terms of the engineering strengths already discussed. For example, consider a uniaxial load on a specimen in the 1-direction. Under tensile load, the engineering strength is Xj, whereas under compressive load, it is (for example, Xg = -400 ksi (-2760 MPa) for boron-epoxy). Thus, under tensile load. 

The weak angles are defined trigonometrically by the terms g/(g2 + g 2) and g /(g2 + g2). This means that the field strength tensor satisfies... 

In electrodynamics, we define the strength tensor of electromagnetic field... 

Here G"v and F v are elements of the field strength tensors for the two SU(2) principal bundles. So far the theory is entirely parallel to the basic standard model of electroweak intereactions. In further work the Dirac and Yukawa Lagrangians that couple the Higgs field to the leptons and quarks will be included. It will then be pointed out how this will modify the B field. The < )4 field may be written according to a small displacement in the vacuum energy ... 

If these new fields are inserted in the Langrange density (30) and the field strength tensors are expanded, then one obtains the following Lagrange density [74]... 

A sufficient amount of oriented chiral molecules can be obtained in an induced cholesteric liquid crystal phase if the induced helical structure has been untwisted by an electric field. In the following description tensors are needed for the sake of simplicity (At least there are three tensors required the transition moment tensor (absorption tensor ,y), the rotational strength tensor (circular dichroism tensor A ,y), and the order tensor g,y33 (i,j= 1,2,3). If the molecules do not possess any symmetry, the principal axes of all of these tensors are differently oriented with respect to the molecular frame (the coordinate system in which only the three diagonal elements of a tensor are different from zero).) The only tensorial property, needed here explicitly, is the existence of three coordinates (components) of a tensor with respect to three specially chosen mutually perpendicular axes. This means that three information instead of one information about a molecule are needed instead of one CD value, namely Ae, three CD values, namely As, (i=l, 2, 3), have to be introduced. Ac is then one-third of a sum of the three so-called tensor coordinates of the CD tensor ... 

We shall explore how the electric and magnetic fields fit into the explicitly covariant framework developed so far. Since the fields E and B are physical observables and therefore gauge invariant quantities, we have to construct a gauge invariant quantity based on the gauge field A. The simplest choice is the so-called yield strength tensor F = (fl ) defined by... 

In terms of the dual field strength tensor the homogeneous Maxwell equations... 

We close this section on the field strength tensor by a discussion of two important invariants of the electromagnetic field, which are both Lorentz and gauge invariant. They can be evaluated by direct calculation and read. 

We are now in a position to discuss the transformation behavior of electric and magnetic fields under a change of the coordinate system, i.e., their transformation properties under Lorentz transformations. Since the electric and magnetic fields do not represent Lorentz 4-vectors, we have to analyze the transformation property of the field strength tensor Fi instead. 

In a more general case the right hand side of Eq. (3) is to be completed with one more item such as the magnetic strength tensor. However, in the present review the aspects of magnetohydrody-namical phenomena influence upon the dissipative structures, will not be considered. 

From the degree of anisotropy R in Figure 14 it follows that the ji and the a transitions are of A A and A B symmetry, respectively, for 9 < From the temperature dependence of the degree of anisotropy (Figure 16) with the help of a computational simulation the tensor coordinates of the absorption tensor and the dipole strength tensor D with respect to the principal axes of the order tensor can be obtained. Furthermore, information about the orientational order of the molecule in the used anisotropic system as stretched polymers or ordered liquid crystal phases can be obtained too. 

Upon reflection from the surface, the local field strength tensor L has to be... 

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