Spherical polar coordinates properties

To describe the velocity profile in laminar flow, let us consider a hemisphere of radius a, which is mounted on a cylindrical support as shown in Fig. 2 and is rotating in an otherwise undisturbed fluid about its symmetric axis. The fluid domain around the hemisphere may be specified by a set of spherical polar coordinates, r, 8, , where r is the radial distance from the center of the hemisphere, 0 is the meridional angle measured from the axis of rotation, and (j> is the azimuthal angle. The velocity components along the r, 8, and (j> directions, are designated by Vr, V9, and V. It is assumed that the fluid is incompressible with constant properties and the Reynolds number is sufficiently high to permit the application of boundary layer approximation [54], Under these conditions, the laminar boundary layer equations describing the steady-state axisymmetric fluid motion near the spherical surface may be written as ... 

The usual spherical polar coordinate system (r, 6, right-handed Cartesian coordinate system whose coordinates occur in the equations. The Z-axis is the unique axis around which r-functions (A = 0) have rotational symmetry. Functions with 0 < A g / are characterized 19) through A by their standard transformation properties when they undergo rotations around the Z-axis ... 

P10.15 The general rule to use in deciding commutation properties is that operators having no variable in common will commute with each other. We first consider the commutation of 1 with the Hamiltonian. This is most easily solved in spherical polar coordinates. 

These cylindrical and spherical polar coordinates are also displayed in Fig. 4.12. A typical equipotential surface is displayed schematically in Fig. 4.13a, at an energy below the dissociation energy. This mapping suffers from two crucial defects. First, it does not have the basic one-to-one correspondence property between configurations and points Q. Indeed, for collinear configurations in which A coincides with the center of mass of, 7a is arbitrary. Therefore, all points on the circle C on the X Y plane, whose radius is the scaled distance, are associated with that single configuration. Second,... 

The shape of such a surface may be described in a mathematical way which has several important advantages. The shape is analysed into harmonic components functions much as one might analyse a wave into a Fourier series. The spherical harmonics, as these components are called, have the necessary property of orthogonality however, their form is more complicated than the cos(njc) type of component of a Fourier series for a plane wave. The spherical harmonics are functions of the polar coordinate angle, referred to the director axis. The first four components, abbreviated Pq, P2(cosa), P4(cos Of) and PeCcos a) or simply Pq, P2, P4, Pe, are defined below and drawn in Fig. 4. 

In the general investigation of symmetry characters it is possible to introduce spatial polar coordinates instead of i 2 3- For small values of the coordinates the eigenfunction becomes a product of a function of the radius with a spherical function. One then has to investigate the symmetry properties of the spherical function with respect to the regular tetraeder, a task which we will perhaps undertake later. 

The multipole expansion may be carried out in several coordinate systems, which may be chosen depending on the symmetry properties of the problem under investigation. Spherical polar and Cartesian coordinates are used most commonly when calculating two-electron integrals. We outline here the derivation for the spherical series. The interested reader may find more detailed discussions, for example, in the books of Eyring, Walter and KimbalP or Morse and Feshbach. A discussion of the multipole expansion in the framework of atomic and molecular interactions and potentials may be found in the article of Williams in this series of reviews (Ref. 34) or the book by Hirschfelder, Curtiss and Bird. ... 

Transition metals (TMs) possess nine orbitals in the valence shell, which are depicted in Scheme 9.R. 1. Letters that describe the orbitals shapes also coincidentally correspond to their spatial properties s, spherical p, polar (namely, pointed in a given direction) and d, dipolar (pointed in two different directions). The directed orbitals have also labels describing their directions in space for example, p refers to a p orbital oriented along the x direction of the x, y, z coordinate system placed around... 

Equations (9.2.4), (9.2.5), (9.2.7), and (9.2.9) are sufficient for describing the motions of a planetary atmosphere. The vector notation used in these expressions is convenient for the study of their general properties, but for most applications it is necessary to write the equations in a specific coordinate system. Eor some calculations, a local rectangular system may suffice. More generally, spherical coordinates are employed with the origin at the center of the planet and the polar axis coincident with fi. hi this coordinate system, the equations take the form... 

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