Cartesian coordinates polar

The parameters of this matrix are the image / and the vector d written by [dx, dy] in cartesian coordinates or [ r, 0] in polar coordinates. The number of co-occurrence on the image / of pairs of pixels separated by vector d. The latter pairs have i and j intensities respectively, i.e. 

In what follows, the 2D space is assumed to be a plane, and therefore we apply either the polar coordinates q, 0) or the Cartesian coordinates (x,y). 

To shift it to some arbirtrary point ( yo,0jo) we first express Eq. (161b) in terms of Cartesian coordinates, and then shift the solution to the point of interest, namely, to (xjo, o)[= ( T/Oi 0yo)]- Once completed, the solution is transformed back to polar coordinates (for details see Appendix F). Following... 

Pio. 2.1 The relationship between spherical polar and Cartesian coordinates. di pends on 6 and 0 ... 

In this section the governing Stokes flow equations in Cartesian, polar and axisymmetric coordinate systems are presented. The equations given in two-dimensional Cartesian coordinate systems are used to outline the derivation of the elemental stiffness equations (i.e. the working equations) of various finite element schemes. 

Gaussian type orbitals can be written in terms of polar or cartesian coordinates... 

Some coordinate transformations are non-linear, like transforming Cartesian to polar coordinates, where the polar coordinates are given in terms of square root and trigonometric functions of the Cartesian coordinates. This for example allows the Schrodinger equation for the hydrogen atom to be solved. Other transformations are linear, i.e. the new coordinate axes are linear combinations of the old coordinates. Such transfonnations can be used for reducing a matrix representation of an operator to a diagonal form. In the new coordinate system, the many-dimensional operator can be written as a sum of one-dimensional operators. 

As the argument is the same in both cases, the difference being in the system of coordinates used—cartesian coordinates in the van dm Pol case and polar ones in the K.B. case, we shall follow the K.B. exposition, which is somewhat more convenient, as it deals directly with the amplitude a and the phase tp.16 We comply with the K.B. notations by writing Eq. (6-87)... 

Our next objective is to find the analytical forms for these simultaneous eigenfunctions. For that purpose, it is more convenient to express the operators Lx, Ly, Zz, and P in spherical polar coordinates r, 6, q> rather than in cartesian coordinates x, y, z. The relationships between r, 6, q> and x, y, z are shown in Figure 5.1. The transformation equations are... 

Equation (6.12) cannot be solved analytically when expressed in the cartesian coordinates x, y, z, but can be solved when expressed in spherical polar coordinates r, 6, cp, by means of the transformation equations (5.29). The laplacian operator in spherical polar coordinates is given by equation (A.61) and may be obtained by substituting equations (5.30) into (6.9b) to yield... 

Since the density is invariant to the velocity coordinates (Cartesian or polar) used, one has, expressing the density in terms of differential cross-section,... 

In this form the equation is rather cumbersome and not easily solved, so it is customary to express it in spherical polar coordinates r, 6, and, (p, where r is the distance from the nucleus and 6 and (p are angular coordinates, rather than in the Cartesian coordinates x, y, and z. The relationship of the polar coordinates to the Cartesian coordinates is shown in Figure 3.5. In this form V = e2/r, and the equation is easier to solve particularly because it can be expressed as the product R(r)Q(9)(dimensional functions R, the radial function, and 0 and , the angular functions. Corresponding to these three functions there are three quantum numbers, designated n, /, and m. 

Figure 3.5 The relationship between polar coordinates and Cartesian coordinates.

It has two components, (x, y) in cartesian coordinates, and these can be transformed into plane polar coordinates to give... 

The parameters of 0(3) can be chosen in various ways. In a cartesian coordinate system (x, y, z) two parameters are required to fix the direction of the rotation axis u, e.g. the angular polar coordinates 6,(j>, and another, ijj to specify the angle of rotation. 

Transformation from polar to Cartesian coordinates clarifies the 2px designation, noting that the new wave function becomes... 

Fig. 13.4 Profiles of the intensities l J2, Ey 2, and EZ 2 of the Cartesian coordinate components of the electric field in a fundamental mode with quasi linear polarization. The insets show the inner parts of the profiles, which correspond to the field inside the fiber. The parameters used a 200 nm, X 1,300 nm, 1.4469, n2 1. Reprinted from Ref. 61 with permission. 2008 Elsevier...

The classical potential energy term is just a sum of the Coulomb interaction terms (Equation 2.1) that depend on the various inter-particle distances. The potential energy term in the quantum mechanical operator is exactly the same as in classical mechanics. The operator Hop has now been obtained in terms of second derivatives with respect to Cartesian coordinates and inter-particle distances. If one desires to use other coordinates (e.g., spherical polar coordinates, elliptical coordinates, etc.), a transformation presents no difficulties in principle. The solution of a differential equation, known as the Schrodinger equation, gives the energy levels Emoi of the molecular system... 

While it is possible to solve the vibrational problem using any one of a number of coordinate systems (Cartesian coordinates, spherical polar coordinates, valence coordinates, etc.), it is often most convenient to employ some type of valence... 

Figure 9e. A projection of the Cremer-Pople sphere see Fig. 9A) onto a plane perpendicular to the polar axis. 8 and values for experiment and models were converted to cartesian coordinates, with the model points connected by solid lines. The central (0,0) point corresponds to a perfect chair, and the dashed line follows the 60 -240 meridian.

For compactness, the subscript M for the electronic density parameters has been omitted in Eq. (8.49). The polar coordinate system has the z axis of the local Cartesian coordinate system as the polar axis, and the vector RMP is referred to this local coordinate system. 

Cartesian coordinates is a product of N — 1 Jacobians for the local transformations from polar to Cartesian components for each bond vector Q for j < — 1, times the Jacobian det[A ]p = 1 for the transformation of Q... 

Basis sets for use in practical Hartree-Fock, density functional, Moller-Plesset and configuration interaction calculations make use of Gaussian-type functions. Gaussian functions are closely related to exponential functions, which are of the form of exact solutions to the one-electron hydrogen atom, and comprise a polynomial in the Cartesian coordinates (x, y, z) followed by an exponential in r. Several series of Gaussian basis sets now have received widespread use and are thoroughly documented. A summary of all electron basis sets available in Spartan is provided in Table 3-1. Except for STO-3G and 3 -21G, any of these basis sets can be supplemented with additional polarization functions and/or with diffuse functions. It should be noted that minimal (STO-3G) and split-valence (3-2IG) basis sets, which lack polarization functions, are unsuitable for use with correlated models, in particular density functional, configuration interaction and Moller-Plesset models. Discussion is provided in Section II. 

Here, ma is the mass of the nucleus a, Zae2 is its charge, and Va2 is the Laplacian with respect to the three cartesian coordinates of this nucleus (this operator Va2 is given in spherical polar coordinates in Appendix A) rj a is the distance between the jth electron and the a1 1 nucleus, rj k is the distance between the j and k electrons, me is the electron s mass, and Ra>b is the distance from nucleus a to nucleus b. 

This equation was solved numerically with a fourth-order Runge-Kutta scheme. It was usually more convenient to recast (12.25) as a differential equation in the rectangular Cartesian coordinates sometimes, however, the advantage was tipped in favor of the polar coordinates. The results shown in Fig. 12.4 were obtained with a mixture of the two approaches. 

The incident plane wave has only field components perpendicular to the direction of propagation. In contrast, the evanescent field has components along all directions X, y, and z of a Cartesian coordinate system attached to the IRE, as shown in Fig. 2. The direction of the incident field vector can be selected by use of a polarizer. The symbols II and denote electric field vectors parallel and perpendicular to the... 

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