INDEX polar coordinates

The early paper [1] solves the Stokes problem in a wedge-like flow, with boundary conditions that I shall discuss later on (see below for the details). Let a liquid/vapor (1/v) flat interface cross an equally flat solid surface at a certain prescribed angle denoted T> and in between 0 and tt. The polar coordinates are such that r = 0 is at the intersection of the solid and 1/v interface. The fluid of index B is on the right side and corresponds to the values of the polar angle... 

It is more economical to use hyperspherical coordinate systems " for HLH systems. For collinear configurations, these coordinates are also plane polar coordinates, but the turning center is located at the origin. These coordinates have had a wide application to collinear re-actions,especially those of the HLH variety. The hyperspherical radius p is independent of the arrangement channel index... 

Consider the collision of two particles initially in internal states described by an index i. To simplify notation, it is convenient to use a single index to specify the states of both particles. The angle between the initial and final relative velocities v and 1/ is given by spherical polar coordinates and , where 0 is the deflection angle in the center of mass frame. We start with a well-defined beam of particles with a flux li (number of particles per unit area per unit time). After the collision, the flux Ij (number of particles per unit solid angle per unit time) is a function of the deflection angle 0 and is different for each possible set of final internal states j. We define the differential cross-section as... 

The source is located in a uniform medium of refractive index Mq is assumed to be sufficiently large that it fully illuminates at least the core cross-section of the fiber endface, as is normally the case in practice. A ray from the source is incident on the endface z = 0 at Q in Fig. 4-4, and makes angle 0q with the normal QN, or axial, direction. The polar coordinates of Q on the endface are (r, (f>) relative to the x-axis. We consider only rays incident over the core a ray incident over the cladding cannot become a bound ray in the core. 

The element of power dP radiated into solid angle dF by area dA of the source in the medium of refractive index is given by Eqs. (4-1) and (4-2). In Fig. 4-4, the angles (0o, 6 ) are spherical polar angles relative to the normal QN and (r, 4>) are polar coordinates relative to the fiber axis. Hence [4-6]... 

Fig. 18-4 Cross-section of the composite waveguide comprises two fibers of core radii p, and P2 with refractive-index profile n (x, y). The point P is defined by cylindrical polar coordinates r, and T2, 4>2 relative to the fiber axes, which are distance d apart.

Fig. 36-4 A fiber of refractive-index profile n (r) is bent into an arc of constant radius. Polar coordinates (r, (j)) describe the fiber cross-section relative to 0, where the COy-axis is parallel to the plane of the bend.

Figure 3.19 The Cartesian coordinates (a ,y) and polar coordinates (r, ) used to specify the orientation of the director n around a disclination which is perpendicular to the page and coincident with the 2 -axis placed at the origin. The director makes an angle with the a -axis and only depends upon the azimuthal angle 0. By equation (3.341), is of the form indicated where the integer n is the FVank index of the disclination.

Figures 1 and 2 show the corresponding conversTon curves in toluene and in methanol solutions respectively. In the latter case log-log coordinates are used to represent the data. The conversion curves are then linear and their slope B, which is the exponent of time in the relation per cent conversion = Kt, measures the extent of auto-acceleration. B is referred to as the "auto-accele-ration index". For pure acrylic acid B = 1.8 - 2.0 in non polar solvents 3 tends towards unity.

Reed and Allen, using their bond polarity index, have assigned values of 0.000, 0.027, and 0.050, respectively (compared to H —0.032 and F 0.189) [108]. Without attempting to be too quantitative, convenient values of the core energies of hybrid atomic orbitals, in units, recognizing that changes in coordination number also occur, are approximately... 

The electromagnetic fields of the right- and left-propagating polaritons, respectively, follow the wave equations with the speeds and damping rates of the different frequency components dispersed according to the frequency- and wavevector-dependent complex refractive index n = v/e(k, oj). A typical example of the dispersion of these modes is shown in Fig. 1 for the case of a real permittivity e. The term Ao(r,t) represents the envelope of the wavepacket on the phonon-polariton coordinate A. Note that this phonon-polariton coordinate is a linear combination of ionic and electromagnetic displacements, which both contribute to the polarization... 

There is only one way to cut a rutile crystal in (001) direction (Fig. 16.) Although this creates a non-polar, autocompensated surface, it does not represent a low-energy configuration. This becomes clear immediately when reviewing the coordination of the surface atoms. All the Ti atoms are 4-fold coordinated, and all the O atoms 2-fold coordinated. Hence the number of broken bonds on this surface is higher than on the other low-index rutile surfaces discussed so far. Consequently, the (001) surface has a high surface... 

As can be seen from the equations (21)-(22) and (23)-(24), there is an essential difference between the representations of plane and multipole waves of photons. In particular, a monochromatic plane wave of photons is specihed by only two different quantum numbers a = x, y, describing the linear polarization in Cartesian coordinates. In turn, the monochromatic multipole photons are described by much more quantum numbers. Even in the simplest case of the electric dipole radiation when X = E and j = 1, we have three different states of multipole photons in (23) with m = 0, 1. Besides that, the plane waves of photons have the same polarization a everywhere, while the states of multipole photons have given m. It is seen from (24) that, in this case, the polarization described by the spin index p can have different values at different distances from the singular point. In Section V we discuss the polarization properties of the multipole radiation in greater detail. 

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