Global coordinate system

In the figure operation (M) represents a one-to-one transformation between the local and global coordinate systems. This in general can be shown as... 

Isoparametric transformation functions between a global coordinate system and local coordinates are, in general, written as... 

Although the elemental stiffness Equation (2.55) has a common form for all of the elements in the mesh, its utilization based on the shape functions defined in the global coordinate system is not convenient. Tliis is readily ascertained considering that shape functions defined in the global system have different coefficients in each element. For example... 

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... 

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. 

Coordinates of molecules may be represented in a global or in an internal coordinate system. In a global coordinate system each atom is defined with a triplet of numbers. These might be the three distances x,, y,-, z, in a crystal coordinate system defined by the three vectors a, b, c and the three angles a, / , y or by a, b, c, a, P, y with dimensions of 1,1,1,90°, 90°, 90° in a cartesian, i. e. an orthonormalized coordinate system. Other common global coordinate systems are cylindrical coordinates (Fig. 3.1) with the coordinate triples r, 6, z and spherical coordinates (Fig. 3.2) with the triples p, 9, . 

Distances and angles. Structures can be presented in an internal coordinate system (symmetry adapted coordinates used in spectroscopy or Z-matrices, that is interatomic distances, three center angles and four center angles) instead of a global coordinate system (coordinate triples, for example cartesian, crystal, cylindrical or spherical coordinates). 

If a Gaussian primitive is expressed in terms of a global coordinate system, the components of the position vector of the centre on which it is based appear in an obvious way ... 

The only new feature here is the crucial one the appearance of the radial distance squared in the exponential factor which, of course, means that, unlike the STO exponential factor, the GTF exponential factor splits into a product of separate functions of the (local) Cartesian coordinates. Further, since the relationship between any two (parallel) Cartesian coordinate systems is simply addition of a constant to each coordinate, this separation process is possible in the global coordinate system. 

Once the inclusion assembly has been constmcted, the homogenised stiffness matrix of the composite is calculated as follows. First, calculate Eshelby tensors S,- for the inclusions [97,98] in local coordinates CS,. Transform the result in the global coordinate system GCS. Then calculate the strain concentration tensors for all the inclusions ... 

The approach (5) may be seen as a simultaneous similarity transformation (Ackermann, 1970). The substituted observables u and t are the local coordinates, which are transformed into the global coordinate system using the additional parameters a, b. The extended system of error equations (5), (8) is then a similarity transformation including scale observations. 

Let us take a reference electron at position r in a global coordinate system (Fig. 11.8a) and try to approach it with a probe electron of the same spin coordinate, shown by the radius vector r - -Tp (i.e., the probe electron would have the radius vector r, when seen from the reference electron shown by r). We will consider only such Tp that ensure that the probe electron be enclosed around the reference one in a sphere of radius Rp, i.e., < Rp. The key function... 

The vector R indicates the origin of the external magnetic field H vector potential from the global coordinate system (cf. Appendix G available at booksite.elsevier.comy978-0-444-59436-5 and the commentary there related to the choice of origin). 

The configuration of the nuclei of the total system can be defined by a set of coordinates given by vector R. We divide the whole system into the interacting subsystems ( molecules ) A, B,C,... with their intemal geometries (configurations of the nuclei) defined by Ra, Rfi, Rc, and the fixed numbers of electrons Na, Nb,, Nc, , respectively. The rest of the coordinates ( external ) that determine the intermolecular distances and the orientations of the molecules in a global coordinate system will be denoted as R j ... 

We have the dipole-dipole term in the form R (p axt bx + l ayl by — f azi bz) = —2R fiazpibz- because the x- and j-components of our dipole moments equal zero. Since A and B are neutral, then it is irrelevant which coordinate system is chosen to compute the dipole moment components. Therefore, let us use the global coordinate system, in which the positions of the charges have been specified. Thus, fiaz = (+1) 0 -f (—1) 1 = —1 and Mfc = (+l)10-f (-1)-11 = -1. 

The total milling forces Fxi and Fx2 (Eq. 9) are resulted by transforming the cutting forces (Eq. 8) into the global coordinate system x, X2 and summing them up over all teeth ... 

With kinematic tolerancing, each FE involved in a tolerance chain is associated with a local 4x4 transformation matrix. Each transformation is done with respect to a global coordinate system attached to the geometric entity where the functional requirement (FR) is defined. The effects that small displacements have on functionality are obtained by first-order derivation of the coordinate frames position vectors in the tolerance chain. 

The lamina stress-strain relation in the global coordinate system is (Herakovich 1998, GangaRao et aL 2007) ... 

For different values of the slot eccentricity, the trajectory of the workpiece s center, expressed in the global coordinates system, is altered gradually from a circle at fg = 0 to a three-cusp concave curve with accentuated curvature for higher eccentricity values, as it can be observed in Figure 12.8. As it can be observed in Figure 12.9, the trajectories of four points, located at the intersection of the workpiece diameter with irmer and... 

Inversely, if the functions / and h satisfy the conditions specified above, then after the indicated gluings of the infinite cross we obtain a smooth Riemannian metric on the sphere. The transition to the global coordinate system is realized by the Weierstrass P-function. Conditions a-d are henceforth regarded as fulfilled. 

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