Rotation of coordinates

Fig. 5. Rotation of coordinate axis where y = XR and R is a rotation matrix. Coordinates refer to sample / in the original reference system to the new...

Since S, S, E, E, and by assumption K and K are all invariant under a rotation of coordinate frame, it is easily verified that C, , , , and si are similarly invariant, and that all the equations of this section are objective. 

The factors in Q on each side cancel, so that the stress rate relation is invariant to a rotation of coordinate frame, and is objective. Note that it is the special dependence of c and b on which makes the stress rate relation objective. If... 

Vectors are commonly used for description of many physical quantities such as force, displacement, velocity, etc. However, vectors alone are not sufficient to represent all physical quantities of interest. For example, stress, strain, and the stress-strain iaws cannot be represented by vectors, but can be represented with tensors. Tensors are an especially useful generalization of vectors. The key feature of tensors is that they transform, on rotation of coordinates, in special manners. Tsai [A-1] gives a complete treatment of the tensor theory useful in composite materials analysis. What follows are the essential fundamentals. 

Note Another way, probably less ambiguous, to compute the azimuth is to make a rotation of coordinates around Oz to bring Ox in the vertical plane and Oy in the horizontal plane. Then, make another rotation around Oy to bring Oz vertical and Ox In the horizontal plane. The azimuth is the clockwise angle between the new OH and Ox. 

Eqs. (1-76) show that these values of the coefficients produce the Navier-Stokes approximations to pzz and qz [see Eq. (1-63)] the other components of p and q may be found from coefficient equations similar to Eq. (1-86) and (1-87) (or, by a rotation of coordinate axes). The first approximation to the distribution function (for this case of Maxwell molecules) is ... 

Our convention is that a symmetry operation R changes the locations of points in space, while the coordinate axes remain fixed. In contrast in Section 1.2 we considered a change (proper or improper rotation) of coordinate axes, while, the points in space remained fixed Let x y z be a set of axes derived from the xyz axes by a proper or improper rotation. Consider a point fixed in space. We found that its coordinates in the x y z system are related to its coordinates in the xyz system by (1.120) or (2.29) ... 

Direction cosines and rotation of coordinates. Suppose we have a set of Cartesian coordinates x,yyz. Let line L pass through the origin and make angles a, / , and y with the positive x, y, and z axes, respectively. The direction cosines /, m, and n of line L are defined as... 

The equations (1.120) for a proper or improper rotation of coordinate axes can be written as... 

Conditions (2.76) and (2.77) define independence of the design from rotation of coordinates. When selecting the null/centerpoints points (points in experimental center) take into consideration a check of lack of fit of the model, an estimate of experimental error and conditions of uniformity [37]. Centerpoints are created by setting all factors at their midpoints. In coded form, centerpoints fall at the all-zero level. The centerpoints act as a barometer of the variability in the system. All the necessary data for constructing the rotatable design matrix for k<7 are in Table 2.137. This kind of designing is called central, because all experimental points are symmetrical with reference to the experimental center. This is shown graphically for k=2 and k=3 in Fig. 2.40. 

As pointed out above in regard to the hyperpolarizability effect, the lability of a biopattern is closely related to its stability. Consider, for example, the lability of the Fucus egg polarity in its early development. In the absence of an applied gradient, the equations describing the dynamics of the egg are invarient under a rotation of coordinates about the center o the egg. This implies that if we have a solution of tt e form Y(r,t) then fo any ota ion operation R taking a vector r to a new vector r = Rr, r = Jr, then Y(r, t) is also a solution (state) of the Fucus system. This implies that the state Y(r,t) is only marginally stable as follows. Assume Y obeys the equation... 

Overlap matrix elements in the x,y-plane involve the additional complication of the rotations of coordinates to allow the alignment of the primitive function with the locations of the sources in the unit cell n. Thus, one finds for the diagonal elements and ... 

In practice, the translational invariance is very much easier to use than the rotational property, because a rotation of coordinates sends any non-s-type basis function into a linear combination of other basis functions within the same shell and this leads to book-keeping problems which are not easy to handle. Translational invariance simply generates one free integral from a given group of related integrals and so is quite straightforward to implement. 

There is a close connection between symmetry and the constants of the motion (these are properties whose operators commute with the Hamiltonian H). For a system whose Hamiltonian is invariant (that is, doesn t change) under any translation of spatial coordinates, the linear-momentum operator p will commute with H, and p can be assigned a definite value in a stationary state. An example is the free particle. For a system with H invariant under any rotation of coordinates, the operators for the angular-momentum components commute with H, and the toted angular momentum and one of its components are specifiable. An example is an atom. A linear molecule has axial symmetry, rather than the spherical synunetry of an atom here only the axial component of angular momentum can be specified (Chapter 13). 

The carbon atom is at the center of the molecule, and the carbon Ij and Is AOs are each sent into themselves by every symmetry operation. These AOs transform according to the totally symmetric species A]. The carbon 2p 2py, and 2p AOs are given by x, y, or z times a radial function. Their symmetry behavior is the same as that of the functions x, y, and z, respectively. From the formulas for rotation of coordinates [Eq. (15.52)], we see that any proper rotation sends each of the functions x, y, and z into some linear combination of x, y, and z. Any improper rotation is the product of some proper rotation and an inversion (Problem 12.15) the inversion simply converts each coordinate to its negative. Hence the three carbon 2p orbitals are sent into linear combinations of one another by each symmetry operation. They must therefore transform according to one of the triply degenerate symmetry species. Further investigation (which is omitted) shows the symmetry species of the 2p AOs to be T2. 

Figure 2D.1. Rotation of coordinate axis used in derivation of equation 2C-41.

Rivlin s choice of a strain-energy function U that involved the squares of the extension ratios arose because he assumed that negative values of the extension ratios A were a mathematical possibility, whereas it was necessary for U always to be greater than zero. We have seen, however, that by choosing suitable rotations of coordinate axes the most general deformation can be described in terms of pure strain, i.e. three principal extension ratios Ai, A2, A3 which are all positive (although some are necessarily less than unity, because A1A2A3 = 1). 

The structures and dynamic behaviors of amido, ketimido, oximato, hy-drazido, and hydrazonato complexes of Group 5 and 6 transition metals have been compared. The factors that determine the barrier to rotation of coordinated ligands have been discussed. 

Softness of neutral acid (from Drago-Wayland parameters by rotation of coordinates)... 

We have mentioned that a matrix may be used to represent the rotation of coordinates by some angle 6. Such a rotation is a geometric operation, so we have. 

To satisfy the continuity of the field across the surface of sphere i we have to transpose all modes to lattice site rj. This implies that we have to rotate the coordinates at lattice site tj to the connecting line r-, — tj to site , and transpose all modes from site j to site i by means of addition theorem (5.36), and rotate the coordinates back to the standard direction n. The rotation of coordinates couples multipoles of the same type but different orientation. Using the notation shown in Fig. 26 we have... 

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