Coordinate systems described

To illustrate, again consider the H2O molecule in the coordinate system described above. The 3N = 9 mass weighted Cartesian displacement coordinates (Xl, Yl, Zl, Xq, Yq, Zq, Xr, Yr, Zr) can be symmetry adapted by applying the following four projection operators ... 

The polar coordinate system describes the location of a point (denoted as [r,0]) in a plane by specifying a distance r and an angle 0 from the origin of the system. There are several relationships between polar and rectangular coordinates, diagrammed in Figure 1-30. From the Pythagorean Theorem... 

Sets of local coordinate systems describing certain local features of complicated objects are often advantageous when compared to a single, global coordinate system. Within a topological framework, the general theory of sets of local coordinate systems is called manifold theory. Often, the local coordinate systems are interrelated, and these relations can be expressed by continuous, and in the case of differentiable manifolds, by differentiable mappings, called homeomorphisms (see Equation (15)), and diffeomorphisms, respectively. 

In this case, we must therefore begin with the full Navier Stokes and continuity equations for a 2D flow, (2 91) and (2 20). In terms of the Cartesian coordinate system described in Fig. 4-8, these are... 

A number representing the difference between two colors can be calculated in any of the coordinate systems described above. The equation is based on the square root of the sum of the squares of the differences in each axis. For example ... 

Throughout this book it has been necessary to refer to several cartesian coordinate systems described as follows ... 

Figure 16.17 Guiding of light in a planar waveguide by total reflection, together with a designation of directions in the coordinate system, described in the text...

In particular, a change of coordinate system described by an origin shift d, i.e.,... 

Liquid Storage Tanks Seismic Analysis, Fig. 1 Cylindrical coordinate system describing tank geometry... 

In any of the coordinate systems described above, it is useful to define a very small or differential volume element... 

Use now this equation to describe liquid film flow in conical capillary. Let us pass to spherical coordinate system with the origin coinciding with conical channel s top (fig. 3). It means that instead of longitudinal coordinate z we shall use radial one r. Using (6) we can derive the total flow rate Q, multiplying specific flow rate by the length of cross section ... 

Here the ijk coordinate system represents the laboratory reference frame the primed coordinate system i j k corresponds to coordinates in the molecular system. The quantities Tj, are the matrices describing the coordinate transfomiation between the molecular and laboratory systems. In this relationship, we have neglected local-field effects and expressed the in a fomi equivalent to simnning the molecular response over all the molecules in a unit surface area (with surface density N. (For simplicity, we have omitted any contribution to not attributable to the dipolar response of the molecules. In many cases, however, it is important to measure and account for the background nonlinear response not arising from the dipolar contributions from the molecules of interest.) In equation B 1.5.44, we allow for a distribution of molecular orientations and have denoted by () the corresponding ensemble average ... 

The effect of an MW pulse on the macroscopic magnetization can be described most easily using a coordinate system (x, y, z) which rotates with the frequency about tlie z-axis defined by the applied field B. 

In a crossed-beam experiment the angular and velocity distributions are measured in the laboratory coordinate system, while scattering events are most conveniently described in a reference frame moving with the velocity of the centre-of-mass of the system. It is thus necessary to transfonn the measured velocity flux contour maps into the center-of-mass coordmate (CM) system [13]. Figure B2.3.2 illustrates the reagent and product velocities in the laboratory and CM coordinate systems. The CM coordinate system is travelling at the velocity c of the centre of mass... 

Basically, two different methods arc commonly used for representing a chemical struchiive in 3D space. Both methods utilize different coordinate systems to describe the spatial arrangement of the atoms of a molecule under con.sidcration. The most common way is to choose a Cartesian coordinate system, i.e., to code the X-, y-, and z-coordinates of each atom, usually as floating point numbers, For each atom the Cartesian coordinates can be listed in a single row. giving consecutively the X-, )> , and z-valnc.s. Figure 2-90 illustrates this method for methane. 

It is always possible to convert internal to Cartesian coordinates and vice versa. However, one coordinate system is usually preferred for a given application. Internal coordinates can usefully describe the relationship between the atoms in a single molecule, but Cartesian coordinates may be more appropriate when describing a collection of discrete molecules. Internal coordinates are commonly used as input to quantum mechanics programs, whereas calculations using molecular mechanics are usually done in Cartesian coordinates. The total number of coordinates that must be specified in the internal coordinate system is six fewer... 

The practical and computational complications encountered in obtaining solutions for the described differential or integral viscoelastic equations sometimes justifies using a heuristic approach based on an equation proposed by Criminale, Ericksen and Filbey (1958) to model polymer flows. Similar to the generalized Newtonian approach, under steady-state viscometric flow conditions components of the extra stress in the (CEF) model are given a.s explicit relationships in terms of the components of the rate of deformation tensor. However, in the (CEF) model stress components are corrected to take into account the influence of normal stresses in non-Newtonian flow behaviour. For example, in a two-dimensional planar coordinate system the components of extra stress in the (CEF) model are written as... 

There are 3M-6 vibrations of a non-linear molecule containing M atoms a linear molecule has 3M-5 vibrations. The linear molecule requires two angular coordinates to describe its orientation with respect to a laboratory-fixed axis system a non-linear molecule requires three angles. 

The Hamiltonian in this problem contains only the kinetic energy of rotation no potential energy is present because the molecule is undergoing unhindered "free rotation". The angles 0 and (j) describe the orientation of the diatomic molecule s axis relative to a laboratory-fixed coordinate system, and p is the reduced mass of the diatomic molecule p=mim2/(mi+m2). 

For symmetric top species, Pave hes along the symmetry axis of the molecule, so the orientation of Pave can again be described in terms of 0 and (j), the angles used to locate the orientation of the molecule s symmetry axis relative to the lab-fixed coordinate system. As a result, the El integral again can be decomposed into three pieces ... 

The simpler model can be derived to describe a shallow shell which is characterized by the closeness of the mid-surface to the plane. In other words, it is assumed that a = b = 1 and the coordinate system a, (5) coincides with the Descartes system X, X2- Then differentiating the fourth and the fifth equilibrium equations with respect to Xi and X2, respectively, and combining with the third equilibrium equation give... 

Let a punch shape be described by the equation z = ip(x), and xi,X2,z be the Descartes coordinate system, x = xi,X2). We assume that the mid-surface of a plate occupies the domain fl of the plane = 0 in its non-deformable state. Then the nonpenetration condition for the plate vertical displacements w is expressed by the inequalities... 

Now we intend to derive nonpenetration conditions for plates and shells with cracks. Let a domain Q, d B with the smooth boundary T coincide with a mid-surface of a shallow shell. Let L, be an unclosed curve in fl perhaps intersecting L (see Fig.1.2). We assume that F, is described by a smooth function X2 = i ixi). Denoting = fl T we obtain the description of the shell (or the plate) with the crack. This means that the crack surface is a cylindrical surface in R, i.e. it can be described as X2 = i ixi), —h < z < h, where xi,X2,z) is the orthogonal coordinate system, and 2h is the thickness of the shell. Let us choose the unit normal vector V = 1, 2) at F,, ... 

The starting point for obtaining quantitative descriptions of flow phenomena is Newton s second law, which states that the vector sum of forces acting on a body equals the rate of change of momentum of the body. This force balance can be made in many different ways. It may be appHed over a body of finite size or over each infinitesimal portion of the body. It may be utilized in a coordinate system moving with the body (the so-called Lagrangian viewpoint) or in a fixed coordinate system (the Eulerian viewpoint). Described herein is derivation of the equations of motion from the Eulerian viewpoint using the Cartesian coordinate system. The equations in other coordinate systems are described in standard references (1,2). 

Blt-M ppedImages. A bit map is a grid pattern composed of tiny cells or picture elements called pixels. Each pixel has two attributes a location and a value or set of values. Location is defined as the address of the cell in a Cartesian, ie, x andjy coordinate, system. Value is defined as the color of the pixel in a specified color system. Geometric quaUties of images are a function of the location attribute, ie, the finer the grid pattern, the more precisely can the geometric quaUties be controlled. Color quaUties are a function of the value attribute, ie, the more bytes of computer memory assigned to describe each pixel, the more precisely can the color quaUties be controlled. 

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