Pseudo-vector

In arriving at Eq. (11-249) we have made use of Eq. (11-241), of the (pseudo)vector character of the surface element dau(x) and of the invariance of the vacuum state expressed by Eq. (11-239). We now insert into the right-hand side of Eq. (11-249) the expansion of iftin(x) in terms of operators, and find... 

H. M. McConnell, A pseudo-vector nuclear hyperfine interaction. Proc. Natl. Acad. Sci. USA, 1958, 44, 766-767. 

Note that A with two subscripts denotes the thermal conductivity which is a second rank tensor and A with one subscript denotes the director constraint torque which is a pseudo vector.) Heat conduction is a dissipative process. The entropy production per unit time and unit volume, a, caused by the heat flow... 

In the case of degenerate E states we have [E2] = A + E and E2 = A2, and therefore here the pure rotational Raman spectra of quadrupole type are possible. In the case of T terms also, pseudovector scattering is allowed, since in this case T2 contains T. Finally, analogously one can make sure that in the states E n and E5/2 pure rotational scattering of pseudovector type only is possible, while in the states G3 2 both quadrupole and pseudo vector scattering should take place. 

Viewing the LC as an anisotropic continuum, a pseudo-vector n of unit length, called the director, is used to indicate the orientation field. Thus, orienfafional... 

Two equivalent forms of Maxwell s field equations in terms of the standard vector formalism are Eq. (5), or (6) with the Lorentz gauge 9 4 = 0. The former is in terms of the antisymmetric second-rank tensor solution F, which is a combination of the electric and magnetic field variables. The latter is in terms of the vector potential, A, shown in Eq. (6) [as well as Eq. (7) in terms of the pseudo vector potential Bassuming that the parameter E, is nonzero]. (Experimental results to this point in time indicate that indeed this parameter is zero to within experimental accuracy [15]—even though the symmetry of relativity theory has no reason to exclude it. Henceforth, we will assume that this parameter is zero.)... 

Problem 2-5. Pseudo-Vectors. Demonstrate that the pseudo-vector equation... 

A and B must have the same parity (that is, if A is a pseudo-vector, then B is a pseudovector if A is a true vector, then B is a true vector and so on). 

Because det L can have only two values for an orthogonal transformation, +1 or —1, it follows that the difference between pseudo-vectors or pseudo-tensors and normal vectors and... 

Common examples of pseudo-vectors that will be relevant later include the angular velocity vector f2, the torque T, the vorticity vector co (or the curl of any true vector), and the cross product of two vectors. The inner scaler product of a vector and a pseudo-tensor or a pseudo-vector and a regular tensor will both produce a pseudo-vector. It will also be useful to extend the notion of a pseudo-vector to scalers that are formed as the product of a vector and a pseudo-vector. The third-order, alternating tensor e is a pseudo-tensor of third order as may be verified by reviewing its definition... 

The subspaces of this algebra correspond to the subspaces of Af 4, scalar, vector, bivector, pseudo-vector, and pseudo-scalar spaces of R4. 

In general, polarized radiation can be described by four Stokes parameters (I,Q,U,V) the linear polarization is a pseudo-vector P = + U )/I ... 

It follows that is a vector, whereas 7 75 is a pseudo-vector,... 

The pseudo vector order parameter w vanishes in the SmA phase. If we neglect the biaxiality of the SmC phase, the order parameter Qap is expressed in terms of the director n and the vector order parameter (Eq. 16) is simplified ... 

Remark. Another representation of a Gaussian beam is the integral representation over plane waves [48,116]. This can be obtained by using Fourier analysis and by replacing the pseudo-vector potential of a nth Davis beam by an equivalent vector potential (satisfying the wave equation), so that both vector fields have the same values in a plane 2 = const. 

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