Local Cartesian coordinate system

Assuming that all vectors on atom M are referred to the same local Cartesian coordinate system of that atom, the results for the peripheral contributions to the potential are2... 

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. 

Figure 11. Unit orthogonal base vectors a , a2, and a3 for local Cartesian coordinate system associated with chain interior atom / . Atoms / — 1,/ ,/ + 1 lie in ah a2 plane and a2 bisects bond angle 8. Also shown are bond vectors W1 and r 1.

Consider a set A and a (possibly approximate) symmetry element R, where the associated symmetry operator R leaves at least one point of the convex hull C of set A invariant. We assume that a reference point c g C, a fixed point of R, and a local Cartesian coordinate system of origin c are specified, where the coordinate axes are oriented according to the usual conventions with respect to the symmetry operator R. For example, if R is a Cy rotation axis, then the z axis of the local Cartesian system is chosen to coincide with this Cj axis, whereas if R is a reflection plane, then the z axis may be chosen perpendicular to this plane. 

It is important to note that (5—23)—(5—25) are identical in form to the equations that would be obtained if the problem had been formulated from the beginning in terms of a local Cartesian coordinate system, withy normal to the surface of the inner cylinder, and... 

One interesting feature is that the operator V20, which is expressed on the left-hand side of (9-222) in terms of rescaled spherical coordinate variables, takes a form in the limiting approximation (9-225) that appears to be just the normal derivative term in V20 for a Cartesian coordinate system. In fact, we shall see that boundary-layer equations always can be expressed in terms of a local Cartesian coordinate system, with one variable normal to the body surface at each point (Y in this case) and the others tangent to it. This reduction of the equations in the boundary layer to a local, Cartesian form is due to the fact that the dimension of the boundary layer is so small relative to that of the body that surface curvature effects play no role. 

Because this inner boundary-layer region is infinitesimal in thickness relative to a, all curvature terms that appear when the equations of motion are expressed in curvilinear coordinates will drop out to first order in Re thus leaving boundary-layer equations that are effectively expressed in terms of a local Cartesian coordinate system. 

For every element there is a local cartesian coordinate system element-coordinate-system U = (u,v,w) parallel to the element axis, in which all essential constructional parts and survey points are coordinated. The built-in adjustment plates were measured and the rigging data ascertained, in this system. 

We fix a local Cartesian coordinate system (e, e, e ) to every contour point of an elastic wire of finite thickness. In so doing, the C axis is taken in the direction of the unit tangent vector u of the chain centroid (or contour), and the and axes in the directions of the principal axes of inertia for the cross-section of the wire at the same point. When the wire is deformed (bent and twisted), the local coordinate system at contour point 5 4- is related to that at contour point s by an infinitesimal rotation dQ. The deformed state of the wire is then represented by a vector 0 (0, 0, 0 ) as a function of s, where 0 is defined by dil/ds. 

Stage 1. The global Cartesian coordinate system is chosen. In this system, we draw the equilibrium configuration of the molecule, with the atoms numbered. On each atom, a local Cartesian coordinate system is located with the axes parallel to those of the global one. For each atom we draw the arrows of its displacements along x, y and z oriented toward the positive values (3Af displacements aU together), assuming that the displacements of equivalent atoms have to be the same. When symmetry operations are applied, these displacements transform into themselves and therefore form a basis set of a (reducible) representation T of the symmetry... 

The local Cartesian coordinate system depicted in Fig. 3.3 is affixed to each bond in the chain. Bond i runs fmm chain atom / — 1 to chain atom i. The x-axis for this bond is parallel with the bond, and oriented from chain atom 1 to chain atom i. The y-axis is in the plane of bonds i and / 1, and oriented with a positive projection on bond / — 1. 

We define a local Cartesian coordinate system for each of the bonds. We assume that the axis x, is directed along the bond i, and the y, axis lies in the plane formed by bonds i and /—I, while the z, axis is directed to make the coordinate system right-handed. The components of the (i-l-l)th bond l,+i can be expressed in the coordinate system of the preceding bond i... 

The form adopted in equation (17) assumes that r and all of the /j are expressed in the same coordinate system. Specification of the elements in any one of the li requires knowledge of the length of the bond and its orientation in the coordinate system common to all It. The rotational isomeric state model uses a different approach. Each li is expressed in a local Cartesian coordinate system with axis xt along bond i, axis yt in the plane of bonds i — 1 and i, with a positive projection on bond i -1, and zt completing a right-handed Cartesian coordinate system. The x and y axes for the local coordinate systems of the first two bonds are depicted in Figure 3. In its own coordinate system, each li is easily written in terms of the length of this bond. 

Description of the Stereochemical Sequence. The stereochemical sequence is described using dl pseudoasymmetric centers, as defined on page 175 of Mattice and Suter [4]. The C-C bonds in the chain are indexed sequentially from 1 to . A local Cartesian coordinate system is associated with each bond. Axis x,- lies along bond i, with a positive projection on that bond. Axis yi is in the plane of bonds i — 1 and i, with a positive projection on bond i—l. Axis Zi completes a right-handed Cartesian coordinate system. The chain atom at the junction of bonds i — 1 and / is a if pseudoasymmetric center if it bears a methyl group with a positive coordinate along z, it is an I pseudoasymmetric center if this coordinate is negative. 

A primitive Gaussian-type function can be written in a local Cartesian coordinate system in the form... 

To each bond k of a molecule a local Cartesian coordinate system with z-axis directed along the bond direction is assigned. The other two bond coordinate axes are perpendicular to the z-axis and their particular orientation depends on the local symmetry of the bond. A bond polarizability tensor is presented in the form... 

Geological systems are manifestly non-Euclidean. Layers are twisted, pinched-out, or faulted. At best there is a local Cartesian coordinate system. 

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