Cartesian polar systems

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. 

Working equations of the streamline upwind (SU) scheme for the steady-state energy equation in Cartesian, polar and axisymmetric coordinate systems... 

Figure 6.4 shows the magnetic field vector B in the molecular Cartesian axes system xyz whose orientation is defined by the polar angles 0 (between B and z) and < ) (between the projection of B on the x-y plane and x). The third dimension of the... 

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. 

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. 

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... 

Systems of spherical symmetry are more amenable to analysis in a spherical polar, rather than cartesian, coordinate system. In this case the Lapla-cian operator is a function of (r,9,(p) rather than (x,y,z). Separation of... 

In the general case, when s-polarized light is converted into p-polarized light and/or vice versa, the standard SE approach is not adequate, because the off-diagonal elements of the reflection matrix r in the Jones matrix formalism are nonzero [114]. Generalized SE must be applied, for instance, to wurtzite-structure ZnO thin films, for which the c-axis is not parallel to the sample normal, i.e., (1120) ZnO thin films on (1102) sapphire [43,71]. Choosing a Cartesian coordinate system relative to the incident (Aj) and reflected plane waves ( > ), as shown in Fig. 3.4, the change of polarization upon reflection can be described by [117,120]... 

The hrst suffix of each tensor component gives the row and the second the column in which the component appears. The xis term, for example, measures the component of the polarization parallel to X2 (usually the y direction in a Cartesian coordinate system) when a field is applied parallel to X3 (the z direction). The susceptibility tensor must conform to any restrictions imposed by crystal symmetry, see Eqs. 6.5-6.9. 

The induced polarization in a piezoelectric, Pj, is a first-rank tensor (vector), and mechanical stress, is a second-rank tensor (nine components), which is represented in a Cartesian coordinate system with axes x, y, and z, as ... 

Figure 11 To the medium under investigation is rigidly attached the laboratory Cartesian reference system x, y, z, whereas to the molecule under consideration is attached the Cartesian system of axes x, Xj, Xj. The static electric field E, which polarizes the medium non-linearly, acts along the z-axis. The light wave which serves for analysing the induced non-linearity propagates along y...

In his analysis of the dynamic behavior of M, Brown made use of the spherical polar coordinate system which relates to the Cartesian coordinate system as follows ... 

To relate Dff to the anisotropic motion of a molecule in a liquid crystalline solvent, we employ the function P(d, < ), defined as the probability per unit solid angle of a molecular orientation specified by the angles 6 and <3>, the polar coordinates of the applied magnetic field direction relative to a molecule-fixed Cartesian coordinate system. We expand P(0, ) in real spherical harmonics ... 

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 ... 

Figure A.5 1 Points in the complex plane, (a) Cartesian system, (b) Polar system.

Use a spherical polar system on centre C to re-express the Cartesians on C in a form adapted to the ciction of the angular projectors ... 

These definitions enable the various quantities related to the Cartesian and polar systems on each nucleus to be written ... 

Figure 8.12 Polar and cartesian coordinate systems used to describe stress field around a crack.

Some coordinate transformations are non-linear, such as converting from a Cartesian to a spherical polar system. Here the r, 6, (p coordinates are related to the x, y, z coordinates by square root and trigonometric functions, as shown in Figure 16.1. Other coordinate transformations are linear, with the new coordinates given as linear combinations of the old ones. A linear transformation can be described as a rotation of the coordinate system. 

A vector in 3-D space may be defined in a Cartesian coordinate system by giving the x, y, and z components, but also in the spherical coordinate system by giving R,6,(j) polar coordinates. Now, let us write the dipole moment / .oo(R) in spherical coordinates / .oo(R) =... 

Let us consider a molecule having a fixed position in a Cartesian coordinate system. To excite the molecule, the IR light (because the separation of the vibrational levels corresponds to the infrared region) is used that is polarized along the x-axis. The electromagnetic theory says that what decides the intensity of the absorption is the square of the transition integral ... 

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