Coordinate system right handed

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... 

Consider for example the theory under discussion, namely, quantum electrodynamics under the circumstance that the frame of reference with respect to which phenomena are described is changed from a right-handed to a left-handed coordinate system, i.e., from one in which the space-time coordinates are labeled by to one in which they are labeled by sc, with... 

Particle-Antipartide Conjugation.—If quantum electrodynamics is invariant under space inversion, then it does not matter whether we employ a right- or left-handed coordinate system in the description of ptnely electrodynamio phenomena. To speak of right and left is, an arbitrary convention in a worlcl ip which only electrodynamics operates. 

Because of small flattening, we let R — a at the right hand side of Equation (2.235). Thus, we again arrive at the known equation of the spheroid in the spherical system of coordinates... 

It should be noted that the positive sign of this result depends on the choice of a right-handed coordinate system in which the angle is acute. The relation developed here for the volume of a parallelepiped is often employed in crystallography to calculate the volume of a unit cell, as shown in the following section. 

The term inversion is used here to refer to a feasible physical phenomenon, whioh in this case is similar to an umbrella that turns inside-out in the wind. The word has already been used twice in different senses (i) the inversion of coordinates, e.g. the passage from a right-handed system to a left-handed one and (ii) a traditional symmetry operation applied to a molecule with a center of invmioa. Accordingly, this term must be used with care ... 

When treating CF parameters in any of the two formalisms, non-specialists often overlook that the coefficients of the expansion of the CF potential (i.e. the values of CF parameters) depend on the choice of the coordinate system, so that conventions for assigning the correct reference framework are required. The conventional choice in which parameters are expressed requires the z-direction to be the principal symmetry axis, while the y-axis is chosen to coincide with a twofold symmetry axis (if present). Finally, the x-axis is perpendicular to both y- and z-axes, in such a way that the three axes form a right-handed coordinate system [31]. For symmetry in which no binary axis perpendicular to principal symmetry axis exists (e.g. C3h, Ctt), y is usually chosen so as to set one of the B kq (in Wybourne s approach) or Aq with q < 0 (in Stevens approach) to zero, thereby reducing the number of terms providing a non-zero imaginary contribution to the matrix elements of the ligand field Hamiltonian. Finally, for even lower symmetry (orthorhombic or monoclinic), the correct choice is such that the ratio of the Stevens parameter is restrained to X = /A (0, 1) and equivalently k =... 

Any three-dimensional orthogonal coordinate system may be specified in terms of the three coordinates q, q2 and q3. Because of the orthogonality of the coordinate surfaces, it is possible to set up, at any point, an orthogonal set of three unit vectors ex, e2, e3, in the directions of increasing qx, q2, q3, respectively. It is important to select the qt such that the unit vectors define a right-handed system of axes. The set of three unit vectors defines a Cartesian coordinate system that coincides with the curvilinear system in... 

Ligand positions about the iron are designated X, Y, Z, —X, -Y, and -Z, where X, Y, and Z, define the directions of an idealized right-handed orthogonal coordinate system centered about the iron site. 

Fig. 2.6 Translation and deformation of a fluid element in the r-z plane. For the right-hand coordinate system (z, r, 9), note that the positive 6 direction is into the page. The displacements in the figure are grossly exaggerated to facilitate annotation. In the limit dr - 0, on which the analysis is based, the two elements approach being colocated.

The shear-stress convention is a bit more complicated to explain. In a differential control volume, the shear stresses act as a couple that produces a torque on the volume. The sign of the torques defines the positive directions of the shear stresses. Assume a right-handed coordinate system, here defined by (z, r, 9). The shear-stress sign convention is related to ordering of the coordinate indexes as follows a positive shear xzr produces a torque in the direction, a positive xrg produces a torque in the z direction, and a positive x z produces a torque in the r direction. Note also, for example, that a positive xrz produces a torque in the negative 6 direction. 

Assuming a right-handed (z, r, 6) coordinate system, explain why the shear stresses zrz are shown in their nominal positive directions. Assuming a flow in the positive z direction, resulting in velocity profiles as illustrated qualitatively, determine the directions of the forces that results from the xrz at the inner and outer radii. 

A coordinate system that is natural for the conical channel can be established as illustrated in right-hand panel of Fig. 5.20. The origin of the new coordinate system begins on the tube wall at the entrance of the conical section. The x coordinate aligns with the surface of the tube wall and the y coordinate measures the distance across the channel and is normal to the tube wall. The 4> coordinate measures the circumferential angle around the conical... 

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