Coordinate system, definition

The fimdamental disadvantage of the mean-field method is that it does not allow modes to respond in a correlated maimer to each other. This problem can be somewhat alleviated by a good definition of the relevant coordinate system [134. 136]. (An extension of mean-field methods that does allow for coupling [137. 138 and 139] will be discussed later.)... 

Furthermore, in a global syslena limits of definite integrals in the coefficient matrix will be different for each element. This difficulty is readily resolved using a local coordinate system (shown as x) to define the elemental shape functions as... 

Commonly Used Coordinate Systems, Basic Definitions and Operations... 

These definitions arise from the transformation properties of vectors and can be summarized as follows If in the transfonuation of the coordinate system to another system quantities Aj, A2, ., A trans-... 

Consider for definiteness the antisymmetric case. We choose the origin of the coordinate system in one of the wells, and the center of symmetry has the coordinates Qt,Q-) (fig- 31). Inside the well the classical trajectories are Lissajous figures bordering on the rectangle formed by the lines Q = Q , and Q = —Q , where Q are the turning-point coordinates. 

Such a matrix is not independent on the coordinate system, but the trace is. Cioslowski has proposed a definition of atomic charges as one-third of the trace over the APT, denoted Generalized Atomic Polar Tensor (GAPT) charges.The charge on atom A is defined as... 

We now formalize the definition of piston flow. Denote position in the reactor using a cylindrical coordinate system (r, 6, z) so that the concentration at a point is denoted as a(r, 9, z) For the reactor to be a piston flow reactor (also called plug flow reactor, slug flow reactor, or ideal tubular reactor), three conditions must be satisfied ... 

The underlying Euclidean space E" also simplifies the definition of individual coordinate systems considerably. 

Figure 1 Definition of the coordinate systems Oxyz and OXYZ and Euler angles 6, cp, and ij/.

I function which carries maximum information about that system. Definition of the -function itself, depends on a probability aggregate or quantum-mechanical ensemble. The mechanical state of the systems of this ensemble cannot be defined more precisely than by stating the -function. It follows that the same -function and hence the same mechanical state must be assumed for all systems of the quantum-mechanical ensemble. A second major difference between classical and quantum states is that the -function that describes the quantum-mechanical ensemble is not a probability density, but a probability amplitude. By comparison the probability density for coordinates q is... 

Given the thinness of a diffusion flamelet, it is possible to neglect as a first approximation curvature effects, and to establish a local coordinate system centered at the reaction interface. By definition, X is chosen to be normal to the reaction surface. Furthermore, because the reaction zone is thin compared with the Kolmogorov scale, gradients with respect to X2 and X3 will be much smaller than gradients in the x direction (i.e., the curvature is small).112 Thus, as shown in Fig. 5.18, the scalar fields will be locally onedimensional. 

We will follow the derivations presented in the work by Poshusta and Kinghorn [104] for the single-center Gaussians. The present work differs from theirs in that we use multicenter Gaussians to shift density away from the origin of the coordinate system. A definition used in the integral formulas is... 

Although we know that the total force field / is conservative, we lack such knowledge concerning the separate pieces fx or A- Therefore some specific path of integration must be chosen to complete the definition (210). Seemingly the most natural path is along the radius on which the point ro lies. As the center of the coordinate system a position of the nucleus is chosen in the case of single-ion system, the center of symmetry for a symmetrical molecule, and some inner point for molecule or cluster with lower symmetry. Then... 

Figure 1.19 Definition of a coordinate system for crystal structures.

If we examine a potential energy surface there are several features which play an important role in the interpretation of kinetic processes. These are minima (stable configurations of all the atoms), valleys (separate stable groups of atoms which we identify as reactants and products) and saddle points (transition states). However, before we give a more formal definition of these features we have to consider the coordinate system that is used. 

Figure 24.1 Coordinate system and geometric parameters used to characterize mixing processes in a river. See Table 24.1 for definitions of symbols.

Figure 24.1 and Table 24.1 give all the necessary definitions. The coordinate system at any location is chosen such that x always points in the direction of the main flow, y across the river, and z vertically upward. 

If the principal stresses had had shear components, which by definition they don t, then, in general, those shear components would have contributed to the stress vector on the rotated z plane. The a vector completely defines the stress state on the rotated z face. However, our objective is to determine the stress-state vector on the z face that aligns with the rotated coordinate system (z,r,G) x--, x-r, and x-e. The a vector itself has no particular value in its own right. Therefore one more transformation from cs to r is required ... 

Fig. 1. Definition of body-fixed and space-fixed coordinate systems. Refer to Table 1 and to the text for a detailed explanation...

The definition of angle may be broadened, however, by considering the location of a point (x,y) on the circumference of a circle, with centre at the origin of a Cartesian xy-coordinate system (sometimes referred to as a... 

Since H14 is a number that is obtained by evaluation of the definite integral in (2-54), it must not depeiid on the coordinate system that we choose to calculate this integral. 3C is the quantum mechanical expression for the energy it is in all cases independent of the coordinate system, since the energy of the system cannot depend on how we choose to describe the system. We have then, by using the inversion operator i,... 

Nevertheless, the definition of crystallinity must be somewhat modified. Landau [2] introduces the function p(x, y. z) as the probability of finding a particle (atom, electron, ion) in a particular position. In a crystal this function is periodic at any distance in a coordinate system x, y, z referred to some fixed group of particles in any disorderly state (gas, liquid), it becomes constant at sufficiently great distances. 

Fig. 8. C2vF(C2vT)2SRM (a) definition of the internal coordinates -n

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