Homogeneous field equation

The homogeneous field equation of 0(3) electrodynamics is inferred from the Jacobi identity of variant derivatives... 

The development just given illustrates the fact that the topology of the vacuum determines the nature of the gauge transformation, field tensor, and field equations, as inferred in Section (I). The covariant derivative plays a central role in each case for example, the homogeneous field equation of 0(3) electrodynamics is a Jacobi identity made up of covariant derivatives in an internal 0(3) symmetry gauge group. The equivalent of the Jacobi identity in general relativity is the Bianchi identity. 

As a result of this Jacobi identity, the homogeneous field equation... 

The Jacobi identity (40) means that the homogeneous field equation of electrodynamics for any gauge group is... 

The homogeneous field equation (515) of 0(3) electrodynamics therefore reduces 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. 

Thus the kinetic equation may be derived for operator (7.21), though it does not exist for an average dipole moment. Formally, the equation is quite identical to the homogeneous differential equation of the impact theory with the collisional operator (7.27). It is of importance that this equation holds for collisions of arbitrary strength, i.e. at any angle of the field reorientation. From Eq. (7.10) and Eq. (7.20) it is clear that the shape of the IR spectrum... 

In the previous section we found components of the attraction field due to masses of a homogeneous spheroid. Equations (2.326 and 2.327). Correspondingly, the components of the gravitational field which include the influence of the centrifugal force are... 

The remaining part of the mode seeking procedure can formally proceed exactly in the same as for the straight waveguide starting from both innermost and outermost slices, two values of the immittance matrix in some suitably chosen radial position p are found and are then used to construct the set of homogeneous linear equations for the mode field amplitudes ... 

The return to equilibrium of a polarized region is quite different in the Debye and Lorentz models. Suppose that a material composed of Lorentz oscillators is electrically polarized and the static electric field is suddenly removed. The charges equilibrate by executing damped harmonic motion about their equilibrium positions. This can be seen by setting the right side of (9.3) equal to zero and solving the homogeneous differential equation with the initial conditions x = x0 and x = 0 at t = 0 the result is the damped harmonic oscillator equation ... 

Note that the right-hand side of this equation is independent of k, showing the homogeneous-field characteristic that... 

From the last line of Eq. (2.17) it is obvious that, as we are dealing with an open system, the source field breaks the symmetry of 7 " . As an immediate consequence 7 " does not satisfy a homogeneous continuity equation but rather the external potential acts as a source of momentum,... 

The corresponding field p(r, cu), obtained as a result of a Fourier transform from the time to the frequency domain, satisfies the homogeneous Helmholtz equation ... 

In the case of a non-homogeneous, non-isotropic FGM, the gauges A(x) are considered to be uniquely determined by the structure of the material. Proceeding from the single domain to the bulk component, the non-steady field T could be evaluated from the invariance of L (T, V jT, A) over the entire component by integration of the particular local field equations, taking into account coupling equations deduced above (6-7). 

In the previous section, we have seen that it cannot suffice to consider the order parameter alone. A crucial role is played by order parameter fluctuations that are intimately connected to the various singularities sketched in fig. 11. We first consider critical fluctuations in the framework of Landau s theory itself, and return to the simplest case of a scalar order parameter (j ) with no third-order term, and u > 0 [eq. (14)], but add a weak wavevector dependent field <5 H(x) = SHqexp(iq x) to the homogeneous field H. Then the problem of minimizing the free energy functional is equivalent to the task of solving the Ginzburg-Landau differential equation... 

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