Rectilinear coordinates

Indicating the appropriate graphical coordinates (rectilinear or semilogarithmic), sketch the profiles of rate of absorption against the dose administered and rate of elimination against dose administered. What will be the relationship between the rate of absorption and the rate of elimination at peak time ... 

The behaviour of te,2 (tj) is qualitatively different. In the dense media this dependence also satisfies the Hubbard relation (6.64), and in logarithmic coordinates of Fig. 6.6 it is rectilinear. As t increases, it passes through the minimum and becomes linear again when results (6.25) and (6.34) hold, correspondingly, for weak and strong collisions ... 

For axial capillary flow in the z direction the Reynolds number, Re = vzmaxI/v = inertial force/viscous force , characterizes the flow in terms of the kinematic viscosity v the average axial velocity, vzmax, and capillary cross sectional length scale l by indicating the magnitude of the inertial terms on the left-hand side of Eq. (5.1.5). In capillary systems for Re < 2000, flow is laminar, only the axial component of the velocity vector is present and the velocity is rectilinear, i.e., depends only on the cross sectional coordinates not the axial position, v= [0,0, vz(x,y). In turbulent flow with Re > 2000 or flows which exhibit hydrodynamic instabilities, the non-linear inertial term generates complexity in the flow such that in a steady state v= [vx(x,y,z), vy(x,y,z), vz(x,y,z). ... 

When N > 4 there appears to be too many Zn, since N(N — l)/2 > 3N — 6. However, the Zn are not globally redundant. All Zn are needed for a global description of molecular shape, and no subset of ZN — 6 Zn will be adequate everywhere.49 The space of molecular coordinates which defines the shape of a molecule is not a rectilinear or Euclidean space, it is a curved manifold. It is well known in the mathematical literature that you cannot find a single global set of coordinates for such curved spaces. 

By the introduction of the (x, y) coordinate system, one has reduced the problem to the motion of a particle of mass (i in a two-dimensional rectilinear space (x, y). Thus, the problem of the collision between an atom and a diatomic molecule in a collinear geometry has been converted into a problem of a single particle on the potential energy surface expressed in terms of the coordinates x and y rather than the coordinates rAB and rBc The coordinates x and y which transform the kinetic energy to diagonal form in such way that the kinetic energy contains only one (effective) mass are referred to as mass scaled Jacobi coordinates. 

To obtain the anharmonic terms in the potential, on the other hand, the choice of coordinates is important 130,131). The reason is that the anharmonic terms can only be obtained from a perturbation expansion on the harmonic results, and the convergence of this expansion differs considerably from one set of coordinates to another. In addition it is usually necessary to assume that some of the anharmonic interaction terms are zero and this is true only for certain classes of internal coordinates. For example, one can define an angle bend in HjO either by a rectilinear displacement of the hydrogen atoms or by a curvilinear displacement. At the harmonic level there is no difference between the two, but one can see that a rectilinear displacement introduces some stretching of the OH bonds whereas the curvilinear displacement does not. The curvilinear coordinate follows more closely the bottom of the potential well (Fig. 12) than the linear displacement and this manifests itself in rather small cubic stretch-bend interaction constants whereas these constants are larger for rectilinear coordinates. A final and important point about the choice of curvilinear coordinates is that they are geometrically defined (i.e. independent of nuclear masses) so that the resulting force constants do not depend on isotopic species. At the anharmonic level this is not true for rectilinear coordinates as it has been shown that the imposition of the Eckart conditions, that the internal coordinates shall introduce no overall translation or rotation of the body, forces them to have a small isotopic dependence 132). 

We then consider a set of rectilinear coordinates representing the energy, entropy, and volume we take the vertical axis to represent the energy. Experience has shown that the thermodynamic functions are single-valued. 

Equation (11) is written in the form of Newton s second law and states that the mass times acceleration of a fluid particle is equal to the sum of the forces causing that acceleration. In flow problems that are accelerationless (Dx/Dt = 0) it is sometimes possible to solve Eq. (11) for the stress distribution independently of any knowledge of the velocity field in the system. One special case where this useful feature of these equations occurs is the case of rectilinear pipe flow. In this special case the solution of complex fluid flow problems is greatly simplified because the stress distribution can be discovered before the constitutive relation must be introduced. This means that only a first-order differential equation must be solved rather than a second-order (and often nonlinear) one. The following are the components of Eq. (11) in rectangular Cartesian, cylindrical polar, and spherical polar coordinates ... 

It is nice to have a distinctive notation for the curvilinear co-ordinates, which emphasizes their difference from and yet their one-to-one correlation with the Rt co-ordinates. Most authors reporting anharmonic calculations do not in fact make any distinction they denote the curvilinear co-ordinates by the same symbols customarily used to denote the corresponding rectilinear coordinates in harmonic calculations. For many purposes this is satisfactory, particularly since the harmonic force constants are not altered by the change from rectilinear to curvilinear co-ordinates. However, in a general discussion it is important to distinguish the two sets, and so for the remainder of this section we shall follow Hoy et al.12 and write the curvilinear co-ordinates with the symbol Hi. 

Symmetry, and the Number of Independent Force Constants.—As in harmonic calculations, the rather general discussion of the preceding section can be simplified in particular cases by making use of symmetry, as discussed by Hoy et a/.12 Thus we may choose the curvilinear co-ordinates Jfin linear combinations that span the irreducible representations of the point group we denote such symmetrized curvilinear co-ordinates by the symbol S, and we define them by means of a U matrix exactly analogous to that used for rectilinear coordinates ... 

The continuity equation for the boundary layer in two-dimensional rectilinear coordinates is... 

This constrained relationship between the two rectilinear coordinates 2xi and 2x3 defines implicifly fhe locus offhe seam in fheplane (2xi, 2xs) (fhe complementary equation being /2 (2x2) = f Qx2 = 0, i.e., 2x2 = 0, ia the fiill three-dimensional space). The graph of the seam is a parabola given by the explicit equation... 

Although equal in value to the rectilinear coordinate Qx, the parameter /s can be treated as a curvilinear coordinate that follows the infinitesimal displacement of a point on the seam along the local tangent vector to the curve, t(/3). This moving frame is completed by the normal vector, n(/3). At the expansion point (origin of the frame fj, = 0), the normal and tangent vectors to the seam are parallel to xi and X3 (unit vectors), respectively. However, away from that point, these vectors are different and combine xi and X3 because the seam is curved (Fig. 5). 

We now introduce a pair of mass-weighted nuclear coordinates, gxi and 2 2 > (see Section 1) that describe mass-weighted rectilinear displacements along xi (Qo) and X2 (Qo), respectively. This basis set is the most convenient one for the present analysis because it simplifies the degeneracy-lifting terms of (17). The expansions for 8f and 8/i in this basis are ... 

Curvilinear internal bond coordinates versus rectilinear normal coordinates [9,10] is, of course, not the only choice to be made. There is a larger selection of coordinates to choose from Radau, Jacobi, hyperspherical, and so on, coordinates see, for example, Refs. 11-14, and the review by Bacic and Light [15] (and references therein). In addition to the rovibrational states of semirigid molecules, these can be used for different types of problems for example, for systems where molecular bonds are broken and formed, chemical reactions occur, and so on. It is clear that both kinetic energy operators and... 

Rapid convergence of the potential energy expansion depends on the type of internal coordinates involved. Of particular importance here is whether we use curvilinear coordinates that are close to the true geometrical variables (e.g. the valence coordinates comprising bond lengths and angles, etc.) or whether we use rectilinear coordinates. We shall here define rectilinear coordinates as a subset, qi, q2,.. . q n < 3 N—6, of internal coordinates which enter only linearly in the expression for the rag components... 

A priori we expect, and experience has confirmed 6 53 that the most rapid convergence is obtained if the potential energy is expanded using curvilinear coordinates. However, this advantage is opposed by complications in deriving the kinetic energy. In this respect the rectilinear coordinates are superior. Hoy, Mills and Strey6 ... 

In this section we shall see how the principles outlined above are applied to evaluate the Wilson-Howard Hamiltonian1,2 However, most of the derivation may be worked out without explicitly assuming that rectilinear internal coordinates are used. We shall take advantage of this in that we will also examine the general consequences of the Eckart conditions as opposed to the special properties connected with the introduction of linearized coordinates. As an intermediate result we will therefore obtain a Hamiltonian which is exactly equivalent to the one which Quade derived for the case of geometrically defined curvilinear coordinates7 ... 

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