Laplace field

In the last step, we have complemented the information gained in the first steps by an analysis of the Laplace field p(r) associated with p(r). It has been shown that the Laplaceian of p(r) indicates where electrons concentrate in the molecule which is useful for a description of electronic structure and chemical bonding [20]. 

In contrast to the natural bond orbital [159] picture, a detailed analysis of the electron density distribution and its associated Laplace field suggests that bonding in HeBeO, NeBeO and ArBeO is caused by strong charge induced... 

Due to the conservation law, the diffiision field 5 j/ relaxes in a time much shorter than tlie time taken by significant interface motion. If the domain size is R(x), the difhision field relaxes over a time scale R Flowever a typical interface velocity is shown below to be R. Thus in time Tq, interfaces move a distanc of about one, much smaller compared to R. This implies that the difhision field 6vj is essentially always in equilibrium with tlie interfaces and, thus, obeys Laplace s equation... 

Equation 9 is Laplace s equation which also occurs in several other fields of mathematical physics. Where the flow problem is two-dimensional, the velocities ate also detivable from a stream function, /. 

For many particles, the diffuse-charge layer can be characterized adequately by the value of the zeta potential. For a spherical particle of radius / o which is large compared with the thickness of the diffuse-charge layer, an electric field uniform at a distance from the particle will produce a tangential electric field which varies with position on the particle. Laplace s equation [Eq. (22-22)] governs the distribution... 

A variety of methodologies have been implemented for the reaction field. The basic equation for the dielectric continuum model is the Poisson-Laplace equation, by which the electrostatic field in a cavity with an arbitrary shape and size is calculated, although some methods do not satisfy the equation. Because the solute s electronic strucmre and the reaction field depend on each other, a nonlinear equation (modified Schrddinger equation) has to be solved in an iterative manner. In practice this is achieved by modifying the electronic Hamiltonian or Fock operator, which is defined through the shape and size of the cavity and the description of the solute s electronic distribution. If one takes a dipole moment approximation for the solute s electronic distribution and a spherical cavity (Onsager s reaction field), the interaction can be derived rather easily and an analytical expression of theFock operator is obtained. However, such an expression is not feasible for an arbitrary electronic distribution in an arbitrary cavity fitted to the molecular shape. In this case the Fock operator is very complicated and has to be prepared by a numerical procedure. 

In the simplest case of one-dimensional steady flow in the x direction, there is a parallel between Eourier s law for heat flowrate and Ohm s law for charge flowrate (i.e., electrical current). Eor three-dimensional steady-state, potential and temperature distributions are both governed by Laplace s equation. The right-hand terms in Poisson s equation are (.Qy/e) = (volumetric charge density/permittivity) and (Qp // ) = (volumetric heat generation rate/thermal conductivity). The respective units of these terms are (V m ) and (K m ). Representations of isopotential and isothermal surfaces are known respectively as potential or temperature fields. Lines of constant potential gradient ( electric field lines ) normal to isopotential surfaces are similar to lines of constant temperature gradient ( lines of flow ) normal to... 

Superposition of Flows Potential flow solutions are also useful to illustrate the effect of cross-drafts on the efficiency of local exhaust hoods. In this way, an idealized uniform velocity field is superpositioned on the flow field of the exhaust opening. This is possible because Laplace s equation is a linear homogeneous differential equation. If a flow field is known to be the sum of two separate flow fields, one can combine the harmonic functions for each to describe the combined flow field. Therefore, if d)) and are each solutions to Laplace s equation, A2, where A and B are constants, is also a solution. For a two-dimensional or axisymmetric three-dimensional flow, the flow field can also be expressed in terms of the stream function. 

Concentration averaged over the cross section of a tube cf (co) Laplace transform of the effective concentration field... 

Both Poisson s and Laplace s equations describe the behavior of the potential at regular points where the first derivatives of the field exist. To characterize the behavior of the potential at the boundary of media with different densities, let us make use of Equation (1.39) according to which a component of the field along some direction / is equal to the derivative of the potential in this direction ... 

Therefore, if the function U satisfies the Laplace s equation, then it possesses a remarkable interesting feature, namely, its average value calculated around some point p is exactly equal to the value of the function at this point. A certain class of functions has this feature only, and such functions are called harmonic. Correspondingly, we conclude that the potential of the attraction field is a harmonic function outside the masses. In accordance with Laplace s equation the sum of the second derivatives along coordinate lines, v, y, and z, equals zero, provided that U(p) is a harmonic function. At the same time we know that in the one-dimensional case there is a class of functions for which the second derivative is equal to zero, that is. 

In essence, Equation (1.162) is an example, when the function is presented as a sum of partial solutions of Laplace s equation. Further, we will use Equation (1.170) to describe the gravitational field of the earth. 

Certainly, the expression for the potential is much simpler than that for the field, and this is a very important reason why we pay special attention to the behavior of this function U(p). As follows from the behavior of the gravitational field, the potential U has a maximum at the earth s center and with an increase of the distance from this point it becomes smaller, since the first derivative in the radial direction, that is, the component of the gravitational field, is negative. At very large distances from the earth the function U has a minimum and then it starts to increase, but this range is beyond our interest. In the first chapter we demonstrated that the potential of the attraction field obeys Poisson s and Laplace s equations inside and outside the earth, respectively ... 

In Section 2.4 we have studied the behavior of the gravitational field of the spheroid outside of masses. Now let us focus our attention on the field of attraction inside masses. It may be proper to notice that the determination of the field caused by masses in the spheroid and, in general, by an ellipsoid, was a subject of classical works performed by Maclaurin, Lagrange, Laplace, Poisson, and others. As is well known, the equation of the ellipsoid, when the major axes are directed along coordinate lines is... 

It may be useful to point out a few topics that go beyond a first course in control. With certain processes, we cannot take data continuously, but rather in certain selected slow intervals (c.f. titration in freshmen chemistry). These are called sampled-data systems. With computers, the analysis evolves into a new area of its own—discrete-time or digital control systems. Here, differential equations and Laplace transform do not work anymore. The mathematical techniques to handle discrete-time systems are difference equations and z-transform. Furthermore, there are multivariable and state space control, which we will encounter a brief introduction. Beyond the introductory level are optimal control, nonlinear control, adaptive control, stochastic control, and fuzzy logic control. Do not lose the perspective that control is an immense field. Classical control appears insignificant, but we have to start some where and onward we crawl. 

Laplace s equation, V2V = 0, in any number of dimensions, describes a system of balanced forces in a potential field. The equation is satisfied by a variety of functions, such as... 

The field between the planes with the given potentials will be electrostatic and plane-parallel and satisfies the two-dimensional Laplace s equation... 

For potential flow, ie incompressible, irrotational flow, the velocity field can be found by solving Laplace s equation for the velocity potential then differentiating the potential to find the velocity components. Use of Bernoulli s equation then allows the pressure distribution to be determined. It should be noted that the no-slip boundary condition cannot be imposed for potential flow. 

The conversion came at a time when the Newtonian program of explanation had lost ground in several fields of laboratory studies, including physical optics, electricity, and heat. Intellectually, this loss of influence was epitomized by the publication in 1826 of Augustin Fresnel s 1819 prize memoir on the diffraction of light, in which he abandoned the Newtonian corpuscular theory. Institutionally, the decline was registered by the 1822 election of Fourier to the office of permanent secretary of the Academy of Sciences, despite the opposition of Laplace, who along with Berthollet had earlier personified the Newtonian tradition in France.37... 

If the excitation electric field is an s-polanzed evanescent field instead of the above p-polarized example, then wH 11 [ = wHJI(z)] does not depend upon p. Therefore, an approximate C(z) can be calculated from the observed fluorescence (P) (obtained experimentally by varying 0) by ignoring the z dependence in the bracketed term in Eq. (7.45) and by inverse Laplace transforming Eq. (7.44) after the ,(0, /J) 2 term has been factored out.,37 39)... 

In order to stably levitate an object, the net force on it must be zero, and the forces on the body, if it is perturbed, must act to return it to its original position. The object must be at a local potential minimum that is, the second derivatives with respect to all spatial coordinates of the potential must be positive. This may seem, at first sight, to be trivial to arrange. However, any system whose potential is a solution to Laplace s equation is automatically unstable A statement in words of Laplace s equation is that the sum of the second partial derivatives of the potential is zero, and so not all can be simultaneously positive. This has long been known for electrostatic potentials, having been stated by Earnshaw(n) Millikan s scheme for suspending charged particles is thus only neutrally stable, since the fields within a Millikan capacitor provide no lateral constraint. 

At a nuclear site in a solid an electric field gradient, due to charges around the nucleus, will generally exist if the atomic symmetry around the site is lower than cubic. For cubic symmetry the gradient of the electric field VE vanishes, since in cubic symmetry the x, y, and z directions are equivalent and in general v E = 0 (Laplace equation) so that VE = 0. The electric field itself at a nuclear site in a solid is of course zero, since a nucleus is bound to the site. Deviations from cubic symmetry may be caused by the inherent noncubic symmetry of the solid or the presence of defects in an otherwise cubic solid. 

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