Local coordinate systems

Figure 2.13 Local coordinate system in rectangular elements...

The one-to-one transformation between the global and local coordinate systems can be established using a variety of techniques (Zienkiewicz and Morgan,... 

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... 

Proof. By utilizing the local coordinate systems (1.135), the assertion of Lemma 1.13 reduces to the case... 

When we look in the local segment coordinate system, the symmetry of the equation seen in the global coordinate system is lost, and we will see azimuthal variations. We wish to express the equation for the segment surface in its local coordinate system... 

The Jk-th wave function of the electrons in a chain of hydrogen atoms results in a similar way. From every atom we obtain a contribution 2 cosnka, i.e. the Is function %n of the n-th atom of the chain takes the place of A0. All atoms have the same function x, referred to the local coordinate system of the atom, and the index n designates the position of the atom in the chain. The k-th wave function is composed of contributions of all atoms ... 

The description of fuzzy, local density fragments is facilitated by the use of local coordinate systems, however, some compatibility conditions of such local coordinate systems must be fulfilled, reflecting the mutual relations of the fragments within the complete molecule. Manifold theory, topological manifolds, and in particular, differentiable manifolds [153-158], are the branches of mathematics dealing with the general properties of compatible local coordinate systems. 

Local coordinate systems can be required to conform with certain mutual compatibility requirements which ensure that the local descriptions are compatible with a global description of the complete system. The branch of topology that deals with... 

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 differentiable manifolds the local coordinate systems must fulfill some compatibility conditions ensuring that in any overlapping region of two local coordinate systems any additional, differentiable functions expressed in either coordinate system are meaningful and differentiable in the other coordinate system as well. 

In many applications it is customary to define local coordinate systems indirectly by establishing their connection with the Cartesian coordinates in some underlying Euclidean space E if there is one. By labeling the points within each actual space (of local coordinate system) with the coordinate values in the underlying Euclidean space E there is a common reference for all local coordinate systems, and the compatibility conditions can be formulated within the Euclidean space E" of familiar and intuitively simple properties. 

The function cp, when restricted to the subset G n Y, is a local coordinate system for Y around point y. 

In order to fulfill compatibility condition (a), the local coordinate system of each parent molecule Mk can always be reoriented, resulting in a simple similarity transformation of the original fragment density matrix P (qS(Kk)) into a compatible fragment density matrix P (cp (K)),... 

In terms of the three-dimensional local coordinate transformations R leading to the local basis set transformations Tf k>, the entire macromolecular system is naturally covered with a family of local coordinate systems. These local coordinate systems are also pairwise compatible, since the actual transformation V< > between any two such local systems of some serial indices k and k can be given explicitly as... 

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. 

Many molecules contain chemically equivalent atoms, which, though in a different crystal environment, have, to a good approximation, the same electron distribution. Such atoms may be linked, provided equivalent local coordinate systems are used in defining the multipoles. In particular, for the weakly scattering hydrogen atoms, abundant in most organic molecules, this procedure can lead to more precisely determined population parameters. 

We demonstrate the use of local coordinate systems with the molecule of tetrasulfur tetranitride, S4N4, (Fig. 4.2) as an example. It occupies a general position in its crystal s space group, with one molecule in the asymmetric unit. Thus, there are eight crystallographically independent atoms. If multipoles up to and including the hexadecapoles are included, the number of population parameters... 

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. 

Response of the mean square dipole moment, < J2>, to excluded volume is evaluated for several chains via Monte-Carlo methods. The chains differ in the manner in which dipolar moment vectors are attached to the local coordinate systems for the skeletal bonds. In the unperturbed state, configurational statistics are those specified by the usual RIS model for linear PE chains. Excluded volume is introduced by requiring chain atoms participating in long-range interactions to behave as hard spheres. 

Note that q has the units of a velocity and is always smaller than the effective velocity in the pores, u, since < > < 1. It is assumed that the x-axis of the (local) coordinate system is always pointing in the direction of the mean flow. Since the mean flow usually does not follow a straight line, the x-axis does not necessarily point in a fixed direction either. 

Each boron atom has four valence shell orbitals, s, px, pv, and pz. We choose a local coordinate system for each boron atom so that its z axis is directed toward the center of the octahedron and the x and y axes are arranged as for the ligand atoms in Figure 8.12 where we were dealing with an octa-... 

The first two terms vanish because of the particular choice of local coordinate system. In the principal coordinate system, the Hessian (i.e, the matrix of second derivatives) is diagonal. 

The principal values of the calculated chemical shift tensors are compared to experimental values and IGLO ab initio calculations taken from literature in Table I. The experimental and most of the calculated values are given in the order 5, / 522 / 833. However in cases where the orientation of the principal axes was known from symmetry considerations or ab initio calculations, and this order was not adequately reproduced in our calculations the components were sorted according to their corresponding tensor orientation in the local coordinate system. In many of these small molecules the tensor axes are defined by the symmetry of the molecules. In other cases the axes are at least loosely defined by nearly symmetric local bond surroundings. For example the 533 component of the CH3 chemical shift tensor is close to the direction of the non CH bond or the 833 component of a sp2 carbon is perpendicular to the plane of the 71 system. 

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