Rotation of the vector field

This operator can now be shown to be identical with the operator for an infinitesimal rotation of the vector field multiplied by i, i.e. J = — M. The components of the angular momentum operator satisfy the commutation relations... 

In the previous section it was shown that the rotation of the vector field I—H on a sphere surrounding all the steady states is -hi. This rotation is equal to the sum of the indices of the steady states, therefore the number of the steady states is odd, since the indices take only values 1. These remarks prove the following theorem... 

Due to this feature, the family XfjL is called a rotation of the vector field X through a constant angle. This angle is positive if /x > 0 or negative if /x < 0, respectively. Hence, if at /x = 0, a separatrix of one saddle is connected to... 

Rough systems are also dense in the space of systems on two-dimensional orientable compact surfaces for which the necessary and sufficient conditions of roughness are analogous to those in the Andronov-Pontryagin theorem. The theory of such systems was developed by Peixoto [107]. The key element in this theory proves the absence of unclosed Poisson-stable trajectories in rough systems (they may be eliminated by a rotation of the vector field). 

In order to illustrate the mixed state, an example with five sample wavelets will be discussed in detail. Each wavelet is represented by its components ax and ay in the Cartesian basis (optical definition, see Section 9.2.2). If the polarization vector is described by a polarization ellipse with major and minor axes a = cos y and b = sin y, by a tilt angle X of this ellipse against a fixed coordinate frame (see Fig. 1.15), and by the direction of rotation of the electric field vector indicated by the sign of y, the components ax and cty follow from... 

This corresponds to two steps of rotation and inversion. However, since rotations and inversions commute, the result of changing the sign of two field component is equivalent to a rotation of the electric field vector. Such a rotation cannot affect the role of L and D and hence does not affect the enantiomeric selectivity. 

Properties of the vector field rotation may be applied to the investigation of the system of equations (Al). In this case the field F, in which the functions P, Q are the right-hand sides of the system (Al), is the velocity field tangent to its phase trajectories. The field is defined in the entire x, y plane and is continuous therefore, its only singular points are those at which F = 0, i.e. stationary points of the system which are generally isolated points). 

The first term represents the couple due to the electric field acting on an electron situated at the centre of gravity of the orbit the second term corresponds to Larmor s theorem, and signifies an additional rotation of the vector P about H with the angular velocity... 

In the following it will be reported on a straightforward method to provide information about the dipole matrix elements and phase shifts being essential for the theoretical description of the photoemission process in a relatively simple way and with a pronounced accuracy [2]. This can be achieved by means of photoelectron spectroscopy with linearly polarized light using the ability of a continuous rotation of the electric field vector. The method is exemplarily demonstrated at the system hydrogen on Gd(0001)/W(l 10) which possesses a pronounced adsorbate induced state. 

The dipole matrix element contains the radial parts Rp and Rf as well as phase shifts dp and df (see Eq. 4.5). In order to obtain information about these properties the photoelectron intensities were determined at a fixed detection angle d = 45° as a function of the rotation of the E-field vector. The spectra in Fig. 4.7 are shown for particular values of a which the maximum and minimum intensities are reached at for the peak 2 at 4.7 eV (4.0 eV in normal emission) with a = 170° and 80° as well as for the feature 1 at 1 eV and structure 3 at 6 eV with a = 140° and 50°, respectively. The intensity values are summarized in Fig. 4.8 (filled diamond Peak 1, open circle Peak 2, filled square Peak 3). The curves for peak 1 and 3 exhibit the same shape which may be caused by emission from orbitals with the... 

For a rubbed polymer surface we note that there are two principal, orthogonal planes perpendicular to the film surface. These are the planes oriented parallel and perpendicular to the rubbing direction (see Fig. 6.3A and B). The first one, which we will refer to as parallel plane, clearly possesses mirror symmetry, while the mirror symmetry of the plane perpendicular to the rubbing direction, referred to as perpendicular plane, may be broken by the directional nature of the rubbing process. Hence, the most general expression for the polarization dependence on rotation of the electric field vector within... 

Figure 4. Motion of spin 1/2 nuclei in the rotating frame and generation of transverse niagneti/.ation My by rotation of the vector of the macroscopic magnetization M,. The B, field caused hy the transmitter coil along the. r-axis deflects M, into the a -y -plane.

The vorticity is a vectorial quantity which informs about the local rotational character of the vector field v. 

The calculation of the rotation of a vector field is generally very difficult. In some cases, however, one can calculate the rotation by considering a simpler vector field. This is accomplished by using the very important concept of homotopy. 

Theorem A.4 [23]. Consider the Euclidean space The rotations 7, y" of the vector fields on the N-dimensional surface dV of... 

Circular polarization of a wave can occur in two directions right or left. For a right circularly polarized wave, the rotation of the electric field vector observed in the direction towards the source occurs in a clockwise sense the tip of the vector draws a right-handed helix while propagating in space. In the case of a left circularly polarized wave, the rotation of the electric field vector observed in the direction towards the source is counterclockwise the tip of the propagating vector draws a left-handed helix (Fig. 6.2). Therefore, a circularly polarized wave can be considered as a radiation possessing chirality. 

Note -. In the above definition of right and left circularly polarized light, we adopt the convention frequently used in optics, that is, from the point of view of an observer looking at the light head on. For R, the observer will see a clockwise rotation of the electric field vector, while for L, the observer will see a counterclockwise rotation of the electric field vector. This is also the convention used in Chapter 4 when we discussed circularly polarized fight in the context of the optical properties of cholesteric liquid crystals. 

Moreover, it follows from simple arguments based on the rotation of a vector field to be presented below that, if is a non-rough system, then given any > 0 there exists a rough system X which is -close to X. In other words, the rough systems form a dense set in Bq-... 

This representation is slightly inconvenient since Ey and 2 in equation (Al.6.56) are explicitly time-dependent. For a monocln-omatic light field of frequency oi, we can transfonn to a frame of reference rotating at the frequency of the light field so that the vector j s a constant. To completely remove the time dependence... 

Field variables identified by their magnitude and two associated directions are called second-order tensors (by analogy a scalar is said to be a zero-order tensor and a vector is a first-order tensor). An important example of a second-order tensor is the physical function stress which is a surface force identified by magnitude, direction and orientation of the surface upon which it is acting. Using a mathematical approach a second-order Cartesian tensor is defined as an entity having nine components T/j, i, j = 1, 2, 3, in the Cartesian coordinate system of ol23 which on rotation of the system to ol 2 3 become... 

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