Paraboloid equations

A two-dimensional free energy reaction diagram may be described intuitively for a fundamental reaction (Figure 1). The curve at the saddle point and for most of the reaction coordinate can be modelled by the paraboloid Equation (6) but the equation fails to predict the minima in potential energy at the reactant and product positions. The value of will increase as increases and Equation (6) will have a as negative. The Px or p value for this change is defined as -3pNuy pAa . ... 

Symmetry 50. Intercepts 50. Asymptotes 50. Equations of Slope 51. Tangents 51. Equations of a Straight Line 52. Equations of a Circle 53. Equations of a Parabola 53. Equations of an Ellipse of Eccentricity e 54. Equations of a Hyperbola 55. Equations of Three-Dimensional Coordinate Systems 56. Equations of a Plane 56. Equations of a Line 57. Equations of Angles 57. Equation of a Sphere 57. Equation of an Ellipsoid 57. Equations of Hyperboloids and Paraboloids 58. Equation of an Elliptic Cone 59. Equation of an Elliptic Cylinder 59. 

For any constant pressure P, equation 2.78 is the equation of a parabola, and therefore all surfaces of constant pressure are paraboloids of revolution. The free surface of the liquid is everywhere at the pressure P0 of the surrounding atmosphere and therefore is itself a paraboloid of revolution. Putting P = Pa in equation 2.78 for the free surface... 

Equations 9.1 and 9.2 correspond with equations 2.80 and 2.79 in Volume 1, Chapter 2. Taking the base of the bowl as the origin for the measurement of zo, positive values of za correspond to conditions where the whole of the bottom of the bowl is covered by liquid. Negative values of za imply that the paraboloid of revolution describing the free surface would cut the axis of rotation below the bottom, and therefore the central portion of the bowl will be dry. 

Equation (E4.3.5) is a paraboloid, answering the first question of Exampfe 4.3. To answer the second question, we need to investigate some characteristics of equation (E4.3.5). 

Unlike the curves you may have seen in geometry books (such as bullet-shaped paraboloids and saddle surfaces) that are simple functions of x and y, certain surfaces occupying three dimensions can be expressed by parametric equations of the form x = f(u,v), y = g(u,v), z = h(u,v). This means that the position of a point in the third dimension is determined by three separate formulas. Because g, and h can be anything you like, the remarkable panoply of art forms made possible by plotting these surfaces is quite large. For simplicity, you can plot projections of these surfaces in the x-y plane simply by plotting (x,y) as you iterate u and in a... 

The Kelvin effect is not limited to spheres. For example, the neck of liquid between two supports is described by a paraboloid of revolution. In this case the radius of curvature of the concave surface is outside the liquid and therefore introduces a minus sign into Equation (40). In addition, the factor of 2 is not required to describe the Kelvin effect for this geometry. Example 6.2 illustrates a test of the Kelvin equation based on this kind of liquid neck. 

As early as 1819, J. L. Gay Lussac proposed to represent the solubility S of potassium chloride in water at a temp. 0 by the formula /S=29-23+0-27380 grms. per 100 grms. of water. Since that time it has been customary to represent solubility curves by empirical formula of the type /S=a+ 0+d02+..., where a, l, c,d,.. . are constants whose numerical values are calculated from the experimental data. Equations of the type S=a- -bO represent straight lines, equations with more terms represent curved lines the solubility equation /S=d+60+c02 represents a portion of a paraboloid curve. The greater the number of terms used in the formula the greater is supposed to be the accuracy of the result. 

Equation (3.340) yields a paraboloid-like change of dissipation with respect to forces A, andX2, as seen in Figure 3.4. The system tends to minimize the entropy and eventually reaches zero entropy production if there are no restrictions on the forces. On the other hand, if we externally fix the value of one of the forces, for example, A 2 = X20, then the system will tend toward the stationary state characterized by the minimum entropy production at X2 = X20. The system will move along the parabola of Figure 3.4 and stop at point [Pg.147]

The point of introducing (1.4.25) is that the equations f - fi, f - f2, f - f3,. . . (or in general, f - constant) have as their images a set of nested paraboloids of revolution, of which the one drawn in Fig. 1.4.1 is representative. This serves as an example of the more general statement that equations of the type f - f, are representable by surfaces in three dimensions or by hypersurfaces in a hyperspace (as is required when f involves more than three variables). 

The paraboloid representation can be extended to model the three-dimensional x.y-surface (Equation 7) (which is analogous to Equation 1). 

Equation 5.163 can be generalized for smooth surfaces of arbitrary shape (not necessarily spheres). Eor that purpose, the surfaces of the two particles are approximated with paraboloids in the vicinity of the point of closest approach Qi = h. Let the principal curvatures at this point be Cl and c[ for the hrst particle, and C2 and c for the second particle. Then the generalization of Equation 5.163 reads ... 

This is the title of Chapter 3 in Ref. [9], Advances in Quantum Chemistry, Vol. 57, dedicated to confined quantum systems. The conoidal boundaries involve spheres, circular cones, dihedral angles, confocal paraboloids, con-focal prolate spheroids, and confocal hyperboloids as natural boundaries of confinement for the hydrogen atom. In fact, such boundaries are associated with the respective coordinates in which the Schrodinger equation is separable and the boundary conditions for confinement are easily implemented. While spheres and spheroids model the confinement in finite volumes, the other surfaces correspond to the confinement in semi-infite spaces. 

The thermal boundary-layer equation, (9-257), also apphes for axisymmetric bodies. One example that we have already considered is a sphere. However, we can consider the thermal boundary layer on any body of revolution. A number of orthogonal coordinate systems have been developed that have the surface of a body of revolution as a coordinate surface. Among these are prolate spheroidal (for a prolate ellipsoid of revolution), oblate spheroidal (for an oblate ellipsoid of revolution), bipolar, toroidal, paraboloidal, and others.22 These are all characterized by having h2 = h2(qx, q2), and either h2/hx = 1 or h2/hx = 1 + 0(Pe 1/3) (assuming that the surface of the body corresponds to q2 = 1). Hence the thermal boundary-layer equation takes the form... 

The computational domain is the unit square in u and v, and this was divided into a 15 x 15 mesh i.e., 225 elements, and 16 x 16 = 256 nodes, so 256 basis functions and 256 residual equations. The Jacobian matrix was banded with a total bandwidth of 35. The first solution computed was the minimal surface, for which the initial estimate was an hyperbolic paraboloid. The nonlinear system of residual equations was solved by Newton iteration on a Cyber 124, each iteration using about 1 second cpu time. For nearly all the surfaces calculated, the mesh was an even mesh over the entire unit square. However, for the surfaces just near the close-packed spheres (CPS) limit, the nodes were evenly spaced in the u-direction but placed as follows in the i -direction i = 0,1/60,1/30,0.05,0.075,0.1,0.15,0.2,0.3,0.4,0.5,0.6,0.7,... 

The ionization limit of the Schrodinger equation and its eigenfunctions for the free hydrogen atom, at a vanishing energy value, corresponds to Bessel functions in the radial coordinate as known in the literature and illustrated in 2.1. The counterparts for paraboloidal [21], hyperboloidal [9], and polar angle [22] coordinates have also been shown to involve Bessel functions. These limits and their counterparts for the other coordinates are reviewed successively in this section. 

The ionization limit for either paraboloidal degree of freedom follows from the corresponding Equation (64) or (65) with E = 0, which become... 

We consider the confinement of the hydrogen atom by a paraboloid defined by a fixed value off = fo- Its wave functions have the form of Equation (74), and the condition that it vanishes at the position of the confining boundary is expressed as... 

Indeed, the removal of the degeneracy of the free hydrogen atom energy levels due to the presence of a paraboloidal boundary far away from the nucleus follows from the application of Equation (12) in Equation (105), and here we limit ourselves to writing the final result ... 

The positions of the paraboloidal boundaries where the successive energy levels become degenerate at zero energy follow from Equation (75) with Aj = 2e2, with the explicit result,... 

The free electron confined by a paraboloidal boundary is described by Weber parabolic functions [16] Chapter 9, which may be obtained from Equations (64) and (65), with the following changes in the separation constants ... 

To avoid these aberrations a surface would have to be ground to fixed curvature with ellipsoidal or paraboloidal shape (Heald 1982). This is extremely difficult to achieve with any accuracy with the very large values needed for i v at X-ray reflection grazing incident angles (equation (5.24)). This approach is more feasible for longer wavelengths where 6C (equation (5.20)) and <3V (equation (5.25)) are larger and Rv (equation (5.24)) is smaller. 

We now turn to the inner-sphere redox reactions in polar solvents in which the coupling of the electron with both the inner and outher solvation shells is to be taken into account. For this purpose a two-frequency oscillator model may the simplest to use, provided the frequency shift resulting from the change of the ion charges is neglected. The "adiabatic electronic surfaces of the solvent before and after the electron transfer are then represented by two similar elliptic paraboloids described by equations (199.11), where x and y denote the coordinates of the solvent vibrations in the outer and inner spheres, respectively. The corresponding vibration frequencies and... 

As shown in [384], Balandin s hypothesis is a particular case of the Buzhinsky hypothesis, and it is interpreted by a paraboloid of rotation in the stress space that, in accordance with Equation (7.4), is observed for the condition... 

Almost two decades previous to the Doering papers a reasonable model for substituent rate effects was proposed that was based on a geometric model for the MOE-J energy surface for the 3,3-shift. Thus, a hyperbolic paraboloid surface equation could be differentiated to obtain coordinates and the activation free energy for the saddle point (the transition state) cast in terms of the relative free energies for formation of the diyl and the two allyl radicals, the same independent variables of Eqs. (7.1) and (7.2). Equation 7.3, which relates the independent variables by the harmonic mean is based on the simplest hyperbolic paraboloid surface, that is, one with linear edge potentials. Slightly more realistic models were also explored. 

The values of rf/ of the cylinder, truncated cone, paraboloid, and hemisphere are given in the following, based on the equations given above and in ref. [17] ... 

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