Invariance from Coordinate System

Foremost, we have to remark that a steepest descent path is in general not identical with a valley floor at all, even if it is coming from a saddle point (see Sect. 3.3). This is due to the fact that a valley floor has to be described by a gradient extremal curve which can have a direction other than the gradient itself (see Sect. 3.2, Example 3, and Examples 4-6 with valleys which flatten out somewhere at the slope). In Fig.18 we give a view of such a drastic case. 

We discuss the influence of a coordinate transformation on the path definition using the case of steepest descent, because the definition pf a gradient path is a mathematically very simple, and we may here better understand the problems of coordinate dependence than in a more complicated path formulation. 

We obtain E(p,q)= p / 2 + q, where circles, as the original equi-potential lines, become ellipses. In Fig.19, we show the relations of corresponding gradient paths. 

The mathematical conception of an independent definition of geometric subjects (as reaction paths) in the configuration space starts with the idea of an analogous transformation of the coordinates as well as the angle relations in the new system. The distortion of equipotential lines in the new system should be compensated by an inverse distortion of the scalar product defining the angles  

If we treat two coordinate systems, x and q, we should have the transformation equations 

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The practical way of calculating 2 is different from that used in the derivation of (4.18). Since 2 is invariant with respect to canonical transformations, it is preferable to seek it in the initial coordinate system. Writing the linearized equation for deviations from the instanton solution 6Q,... 

To look ahead a little, there are properties that depend on the choice of coordinate system the electric dipole moment of a charged species is origin-dependent in a well-understood way. But not the charge density or the electronic energy Quantities that have the same value in any coordinate system are sometimes referred to as invariants, a term borrowed from the theory of relativity. 

From Invariance, in an arbitrary coordinate system, we have... 

Rate of change of observables, 477 Ray in Hilbert space, 427 Rayleigh quotient, 69 Reduction from functional to algebraic form, 97 Regula fold method, 80 Reifien, B., 212 Relative motion of particles, 4 Relative velocity coordinate system and gas coordinate system, 10 Relativistic invariance of quantum electrodynamics, 669 Relativistic particle relation between energy and momentum, 496 Relativistic quantum mechanics, 484 Relaxation interval, 385 method of, 62 oscillations, 383 asymptotic theory, 388 discontinuous theory, 385 Reliability, 284... 

The value of the dot product is a measure of the coalignment of two vectors and is independent of the coordinate system. The dot product therefore is a true scalar, the simplest invariant which can be formed from the two vectors. It provides a useful form for expressing many physical properties the work done in moving a body equals the dot product of the force and the displacement the electrical energy density in space is proportional to the dot product of electrical intensity and electrical displacement quantum mechanical observations are dot products of an operator and a state vector the invariants of special relativity are the dot products of four-vectors. The invariants involving a set of quantities may be used to establish if these quantities are the components of a vector. For instance, if AiBi forms an invariant and Bi are the components of a vector, then Az must be the components of another vector. 

The elements with such an electronic configuration known as luminescent centers are Cr, Mn and They are capable of substituting in a wide variety of metal oxide host systems. They are invariably oxygen-coordinated with six nearest neighbors, and may be in a pure octahedral or a distorted octahedral symmetry site. These luminescent centers exists in a d configuration (Fig. 5.23), and the electronic repulsion, which results from placing three electrons in the same set of d-orbitals yields several states identified as free... 

The sum of the diagonal elements is an invariant of the stress tensor. That is, regardless of the particular orientation of the coordinate system, or the coordinate system itself (e.g., cartesian versus cylindrical), the sum of the diagonal elements of the stress tensor is unchanged. From Eq. 2.180 it is easily seen that... 

The differential cross-section d/i/dfl is not invariant when we change the description from one coordinate system to another. Clearly, due to the relation in Eq. (4.55) a change in y will not lead to the same change in 0 and the space angle d 2c.m. = si n ydycif/) is not identical to the space angle dfl = sin GdGd in the laboratory frame. Thus,... 

The differential cross-section is not invariant when we change our description from one coordinate system to another, since the space angle dQ, is not. To find a relation between the differential cross-section in the laboratory coordinate system, (da/dQ.)iab, and in the center-of-mass coordinate system, (da/d l)c.m., we write... 

The gauge-invariant metric tensor of internal space and classical equations of internal motion in terms of the PAHC are given explicitly as follows, from which the general properties of a metric force associated with this coordinate system will be deduced. By applying Eq. (20) to Eq. (6), we obtain for the moment of inertia tensor referred to the body frame as... 

The object s inherent structure, being fixed, remains the same no matter how the spatial coordinate system is chosen, or where its origin is taken to be. Because the structure is invariant, even if its constituent points are transformed from one coordinate system to another, the relative positions of the points within the object remain the same. The structure is not dependent on the coordinate system we choose. Thus, if the structure of a molecule is defined by specifying the coordinates in space xj, yj, Zj of each atom j in the molecule, atom 5 (or 7 or 18, or whatever) maintains the same relationship in space to atom 14 (or 3, or whatever) no mater what the coordinate system. 

Selection Rules for all Laue Classes. The selection rules for the harmonic coefficients are derived from the invariance of the pole distribution to the operations of the crystal and sample Laue groups. The invariance conditions are applied to every function t/ (h, y) from Equation (36), as the terms of different / in this equation are independent. If we compare Equations (38) and (39) with (37) we observe that they have an identical structure. On the other side the sample and the crystal coordinate systems were similarly defined. As a consequence the selection rules for the coefficients of", flf", and respectively y) ", resulting from the sample symmetry must be identical with the selection rules for the coefficients A P and resulting from the crystal symmetry, if the sample and the crystal Laue groups are the same. The exception is the case of cylindrical sample symmetry that has no correspondence with the crystal symmetry. In this case, only the coefficients af and y l are different from zero, if they are not forbidden by the crystal symmetry. 

Translational invariance of sheared systems takes a special form for two-time correlation functions, because a shift of the point in coordinate space from F to F gives... 

The velocity field is statistically homogeneous if all statistics are invariants under a shift in position. If the field is also statistically invariant under rotations and reflections of the coordinate system, then it is statistically isotropic. In chemical reaction engineering these mathematical definitions are usually somewhat relaxed, since turbulence is said to be isotropic if the individual velocity fluctuations are equal in all the three space dimensions. Otherwise it is said to be an-isotropic. Similarly, a flow field where turbulence levels do not change from one point to another is called homogeneous. 

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