Co-ordinates. Cartesian

Converting from polar to cartesian co-ordinates gives the closed-loop poles in the z-plane... 

The behavior of the different amines depends on at least four factors basicity, nucleophilicity, steric hindrance and solvation. In the literature (16), 126 aliphatic and aromatic amines have been classified by a statistical analysis of the data for the following parameters molar mass (mm), refractive index (nD), density (d), boiling point (bp), molar volume, and pKa. On such a premise, a Cartesian co-ordinate graph places the amines in four quadrants (16). In our preliminary tests, amines representative of each quadrant have been investigated, and chosen by consideration of their toxicity, commercial availability and price (Table 1). 

In principle, in the tests of the first type (Fig. 8), the force applied to the wedge need not be known. This force bends the two (partly separated) components of the solid, and the work of splitting can be calculated from the shape of the bent members. If x is the Cartesian co-ordinate along which the wedge moves forward, and y is the height of the bottom surface of the upper member above the middle of the slit, then97 98 ... 

In this section, quantities are expressed with respect to rectangular Cartesian co-ordinate axes, Ox, 0x2, Ox, except where otherwise stated. The components of a vector V are denoted Vi, V2 and F3 with respect to these axes. 

There are two types of molecular structure database two-dimensional (2D) and three-dimensional. The 2D type stores atoms (chemical elements) and connectivity information (i.e., which atoms are bonded to which in a molecule). The 3D type stores, in addition, the x, y, z Cartesian co-ordinates of each atom in a molecule. 

Let us denote the Cartesian co-ordinates of an arbitrary point of the body before deformation as x. The co-ordinates of the same point after deformation Xk are functions of the original co-ordinates... 

The transformation from cartesian co-ordinates to normal co-ordinates is determined in the preliminary harmonic force field calculation in the customary notation it is written... 

Boltzmann (3]. There it is proved for the special case of Cartesian co-ordinates. Cf. Boltzmann, Gastheorie, II, 124. 

X-ray diffraction data from good single crystals allows the determination of structure in the form of a so-called unit cell, a simple example of which is shown in Figure 8-44. The rest of the crystal structure is then obtained by packing identical cells along the crystallographic axes, in this case the x,y,z directions of a Cartesian co-ordinate system. 

You may need to draw a visualization of atomic orbitals, usually the s, p and d orbitals. This can be simplified by the use of Cartesian co-ordinates which allow a three-dimensional representation on paper. This is neither easy to replicate or often necessary. A simplified approach is, for example, to replace the spherical s orbital with a circle (Fig. 42.7). Similarly, the three p orbitals can be represented in two dimensions by the use of correct labelling of the axes (Fig. 42.7). Finally, the same approach can be replicated on the five d orbitals (Fig. 42.7). It is worthwhile remembering that the d , d,. and d, -orbitals do not reside on the axes (.x, y or z), but in the plane of their respective axes. In addition, the dy2, .2 orbital occupies the. v- and y-axes and the d-2 orbital occupies the z-axis. 

The actual configuration around disinclination lines was calculated by Oseen for the case kn — 33, 12 = 0, the latter assumption actually being exact. Then if L is parallel to a plane and is the azimuth of L in this plane, in which xi, xz are Cartesian co-ordinates, the free energy is minimized in the absence of body torques when... 

The capillary pressure 11.1.2) is a physical quemtlty and therefore independent of the actual choice of the two radii. At a particular location on a surface, the sum of the reciprocals of all pairs of radii is invariant. However, for practical purposes we shall only work with the principal radii, R and R. Often we shall draw in the plane of the paper and R normcd to it. Furthermore, in Cartesian co-ordinates the directions x and y define a horizontal plane and z is the vertical. 

As an introduction to a general description of spatial curvature, let us first discuss some simple geometries. First consider a sphere with radius a. Its equation in Cartesian co-ordinates is... 

Extending the analogy with bulk rheology, for linear shear deformation of an interface it is possible to define a surface (or interfacial) shear viscosity rj° and a surface (or interfacial) shear modulus G°. In a Cartesian co-ordinate system, with again the z-axis normal to the interface... 

Figure 4.25. Pictorial representation of a non-equilibrated surface layer, indicating the transport processes that may occur in it. Cartesian co-ordinates zl surface, counting downwards from z = 0 in the surface x parallel to the surface on the right y parallel to the surface, normal to the plane of drawing. Desorption is ignored. Further discussion in the text.

Initially the calcium ions were placed on the faces of rings as well as at random locations in the cavity. Once many of the distinct minima had been found for specific aluminium configurations, calciums were placed in these positions to verify that no minima had been missed and optimised for all of the aluminium configurations The calcium ion positions were obtained in Cartesian co-ordinates relative to the centre of region 1, rather than fractional coordinates since ions in a defect model are not replicated periodically. In order to compare with X-ray diffraction data, approximate fractional co-ordinates for the calcium ion were found by assuming that the fractional co-ordinates of the defect centre did not change when the geometry of the defect was relaxed. Positions could then be related to the unit cell of the pure material. 

For calculation of the spontaneous strain components, a Cartesian co-ordinate system must be defined. For convenience we choose the reference system of Guimares (1979) with i parallel to the trigonal c-axis,y parallel to the trigonal 6-axis, and k parallel to the [210] direction in the trigonal unit-cell (Fig. A2). From Equations (43)-(48) of Carpenter et al. (1988), we can now write down the expressions for the spontaneous strain components in terms of the observed monoclinic unit-cell parameters (here without subscript) and the extrapolated trigonal unit-cell parameters transformed into the monoclinic setting (here subscripted with o to indicate their use as the reference state) ... 

The gradient of 0 at the cell face center can be expressed either in terms of the derivatives with respect to global Cartesian co-ordinates or local orthogonal co-ordinates. If the local orthogonal system attached to the cell face center is used, then only the derivative in the normal (n) direction contributes to the diffusive flux ... 

In equations 30 and 31 (7 denotes the Laplacian operator in Cartesian co-ordinates,... 

Figure 23. A static magnetic field applied to the nucleus causes it to process at a rate dependent on the field strength and the magnetogyric ratio of the spin. The field is conventionally applied along the z-axis of a Cartesian co-ordinate frame and the motion of the nucleus represented as a vector moving on the surface of a cone.

Consider a set of Cartesian co-ordinate axes, as in Fig. 1, which are fixed in space these are often called the laboratory frame (Bloch, 1946). The z direction is taken to be that of a large polarizing field H0 about which the angular momentum vector and associated magnetic... 

The idea that physical quantities may7 be represented by operators was suggested by Bom and Wiener [17]. Equation 6.4 is satisfied by the choice of the multiplication operator qr to represent the dynamical variable qr, and the differential operator —i%djdqr to represent the dynamical variable pri conjugate to qr. A particular representation5 in quantum mechanics is made if, for the qr> we take Cartesian co-ordinates. The Hamiltonian function of classical mechanics H(pr> qr), the energy7 of the system, then becomes... 

The expansions of the cartesian co-ordinates as functions of the angle variables (to be calculated from (26), 22) must now be introduced, to provide a starting-point for the calculation of the perturbations. In this connection, however, there is one point to be borne in mind. In the unperturbed Kepler motion (without taking account of the variation in mass) only Jx is fixed by the quantum theory, whilst J2, i.e. the eccentricity, remains arbitrary in the relativistic Kepler motion, J2 is also to be quantised and, for a one-quantum orbit, J2=J1=A. We shall not take account quantitatively of the relativistic variation of mass, but we shall assume that the initial orbit of each electron is circular with limiting degeneration J1=A,... 

The Cartesian co-ordinate system shown in Figure 22 is used to orientate and locate the measuring devices. The origin is located at the intersection of the tunnel axis and the contact plane between the concrete plug and the bentonite buffer. Positive X axis is directed along the tunnel axis towards the other end of the test section. The Z-axis is vertical, pointing upwards and the Y-axis is perpendicular to the (XZ) plane in the position indicated in Figure 22. 

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