Vector solenoidal

The function (p is called the potential of an irrotational vector field and the function is called the potential of a solenoidal vector field. 

Braun and Hauck [3] discovered that the irrotational and solenoidal components of a 2-D vector field can be imaged separately using the transverse and longitudinal measurements, respectively. This result has a clear analogy in a 2-D tensor field. We can distinguish three types of measurements which determine potentials of the symmetric tensor field separately ... 

According to the Helmholtz theorem, the two-dimensional vector field can be represented as a sum of an irrotational field and of a solenoidal one... 

Since the condition V B= 0 is generally valid the vector B is solenoidal (source free) and can therefore be expressed as the curl of another vector A0... 

In this final section, it is shown that the three magnetic field components of electromagnetic radiation in 0(3) electrodynamics are Beltrami vector fields, illustrating the fact that conventional Maxwell-Heaviside electrodynamics are incomplete. Therefore Beltrami electrodynamics can be regarded as foundational, structuring the vacuum fields of nature, and extending the point of view of Heaviside, who reduced the original Maxwell equations to their presently accepted textbook form. In this section, transverse plane waves are shown to be solenoidal, complex lamellar, and Beltrami, and to obey the Beltrami equation, of which B is an identically nonzero solution. In the Beltrami electrodynamics, therefore, the existence of the transverse 1 = implies that of , as in 0(3) electrodynamics. 

As argued by Reed [4], the Beltrami vector field originated in hydrodynamics and is force-free. It is one of the three basic types of field solenoidal, complex lamellar, and Beltrami. These vector fields originated in hydrodynamics and describe the properties of the velocity field, flux or streamline, v, and the vorticity V x v. The Beltrami field is also a Magnus force free fluid flow and is expressed in hydrodynamics as... 

This set of conditions implies that the held is intrinsically irrotational (without vortex) and derives from a scalar potential , while the held is a solenoidal held (without divergence) dehned on the basis of the rotational of a vector potential A ... 

The vector potential A is dehned only up to the gradient of a scalar function this additional degree of freedom can be used to turn A into a solenoidal potential... 

If d is the diameter of the solenoid infinite in length, the A(r, t) vector potential is characterized by a single component only... 

In search for an explanation, Aharonov and Bohm worked out quantum mechanics equations based on the measurable physical effect of the vector potential, which is nonnull in a region outside the solenoid. Like many other paradoxes in physics, including the twin paradox, the interpretation of this experiment proposed in 1959 was the subject of an intense controversy among researchers. This controversy is well summarized in a review article [55] and in other references of interest [56-67]. 

Many physicists opposed both the interpretation of this effect attributed to the vector potential, and the experimental conditions of Chambers [57] and Mollenstedt [58] experiments. Criticisms insisted the solenoid was not infinite in length. As a consequence, they claimed, the magnetic field leaks out in a region too close to the area crossed by the electrons, to have no effect. This leak was even employed to quantify the F flux. In addition, electron beams can interfere, as we shall soon demonstrate below, with the magnetic field created inside the solenoid. 

As a consequence, the vector potential A inside the solenoid must be written as... 

We must insist that internal and external vector potentials reconnect with each other in a continuous manner on the solenoid boundary for r = r0. 

We observe that the gauge transform is unique and cannot allow us to eliminate the vector potential outside the solenoid. In addition, the vector potential A derives from a multiform scalar potential F. This result contradicts the solenoidal characteristic of the A = — VA r In (r)Bo/2] vector potential. Henceforth, the gauge transform given above represents nonobservable stationary waves in vacuum since the Lorentz gauge ... 

If the velocity U of an electron within the beam is constant outside the solenoid, the variation of the vector potential A as a function of time in the medium, and thus also in the solenoid, will induce a modification of the phase, as indicated by the equations written above. This will produce a modification of the boundary conditions on the boundary of the solenoid for the quantities a and b. We must also stress that the modification of the vector potential outside the solenoid is generated by either an external or an internal source feeding the solenoid. This can explain the existence of the Aharonov-Bohm effect for toroidal, permanent magnets. The interpretation of the Aharonov-Bohm effect is therefore classic, but the observation of this effect requires the principle of interference of quantum mechanics, which enables a phase effect to be measured. 

Finally, recall that subtraction of (5.2.1b) from (5.2.1a) yields, taking into account (5.2.1c) and (5.2.5a), ix = 0 (this corresponds to solenoidality of the total current density vector in more than one dimension). 

In other words, in a solenoidal Beltrami field the vector lines are situated in the surfaces c = constant. This theorem was originally derived by Ballabh [4] for a Beltrami flow proper of an incompressible medium. For the sake of completeness, we mention that the combination of the three conditions (1), (2), and (3) only leads to a Laplacian field, that is better defined by a vector field that is both solenoidal (divergence-less) and lamellar (curlless). 

There are two known standard methods for decomposition of any smooth (differentiable) vector field. One is that attributed to Helmholtz, which splits any vector field into a lamellar (curl-free) component, and a solenoidal (divergenceless) component. The second, which divides a general vector field into lamellar and complex lamellar parts, is that popularized by Monge. However, the relatively recent discovery by Moses [7] shows that any smooth vector field— general or with restraints to be determined—may also be separable into circularly polarized vectors. Furthermore, this third method simplifies the otherwise difficult analysis of three-dimensional classical flow fields. The Beltrami flow field, which has a natural chiral structure, is particularly amenable to this type analysis. 

In this application we consider EM fields in free space consequently both E and H are solenoidal and satisfy Trkalian field relations. Thus, taking the curl of (71), both vector fields satisfy Helmholtz vector wave equations ... 

We consider now the Aharonov-Bohm effect as an example of a phenomenon understandable only from topological considerations. Beginning in 1959 Aharonov and Bohm [30] challenged the view that the classical vector potential produces no observable physical effects by proposing two experiments. The one that is most discussed is shown in Fig. 10. A beam of monoenergetic electrons exists from a source at X and is diffracted into two beams by the slits in a wall at Y1 and Y2. The two beams produce an interference pattern at III that is measured. Behind the wall is a solenoid, the B field of which points out of the paper. The absence of a free local magnetic monopole postulate in conventional... 

If i is the current in the windings of the solenoid and l the length of the solenoid per turn, the vector B, called the magnetic induction, is defined by... 

In order to obtain Green s identities for the flow field (u,p), a vector z is defined as the dot product of the stress tensor a(u, p) and a second solenoidal vector field v (divergence-free). The divergence or Gauss Theorem (10.1.1) is applied to the vector z... 

Let us analyze the space and time structure of the elastic displacement field in detail. We will demonstrate that equation (13.26) describes the propagation of two types of body waves in an elastic medium, i.e., compressional and shear waves travelling at different velocities and featuring different physical properties. To this end, let us recall the well-known Helmholtz theorem according to which an arbitrary vector field, in particular an elastic displacement field U(r), may be represented as a sum of a potential, Up(r), and a solenoidal, Us(r), field (Zhdanov, 1988) ... 

In other cases the fluid velocity vector may be considered solenoidal ° even though the mixture density is not constant. 

A vector field v satisfying V v = 0 is called solenoidal. A volume-preserving motion is called isochoric, i.e., a motion for which the density in the neighborhood of any particle remains constant as the particle moves. The flow of an incompressible fluid is necessarily isochoric, but there may also be isochoric flows of compressible fluids [104] (p. 212). 

The transverse part of the induced adiabatic vector potential, (2.16), which appears in the kinetic energy of the adiabatic nuclear Hamiltonian, (2.18), governs electronic geometric phase development [25,28]. The two dimensional curl of Aj yields a solenoidal magnetic field of the form 7rftS(, )S( 2) general case of zt O, it yields a monopolar field). 

Vector fields whose divergence vanishes are sometimes referred to as solenoidal. A more comprehensive discussion of the conditions for approximating the velocity field as solenoidal has been given by Batchelor.8 These imply that, in cases in which the fluid is subjected to an oscillating pressure, the characteristic velocity in the Mach number condition should be interpreted as the product of the frequency times the linear dimension of the fluid domain, and that the difference in static pressures over the length scale of the domain must be small compared with the absolute pressure. Because our subject matter will frequently deal with incompressible, isothermal fluids, we shall often make use of (2-20) in lieu of the... 

In effect, (7-29) represents a decomposition of the general vector field a into an irrotational part, associated with V, and a solenoidal (or divergence-free) part, represented by V A 0AVX). It should be noted that general proofs exist that show not only that (7-29) can represent any arbitrary vector field a but also that an arbitrary, irrotational vector field can be represented in terms of the gradient of a single scalar function

solenoidal vector field can be represented in the form of the second term of (7-29). Because... 

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