Zero divergence

If there exist no free charges within the domain of interest, the time-harmonic electromagnetic held must also satisfy zero divergence conditions... 

Let M V X tij/) and N = (V x M)/< with p given by Eq. (53), where r is the radius vector of the spherical coordinate system. Then both M and N satisfy the vector wave equation and both have zero divergence. In addition, M and N satisfy... 

Let us consider a second field, B, also with zero divergence and the same periodicity (a magnetic field). The advection of the magnetic field B by the velocity field v is governed by the frozenness condition an infinitesimally small vector proportional to B is advected by the fluid flow as if it were drawn onto the particles (and stretched together with them). 

Since the expression enclosed in parentheses has a zero divergence, we arrive at the following equation for the solenoidal field... 

The waves of the second type, corresponding to the solenoidal component of the elastic displacement U, should have zero divergence, i.e.. 

A large class of molecular properties arise from the interaction of molecules with electromagnetic fields. As emphasized previously, the external fields are treated as perturbations and so one considers only the effect of the fields on the molecule and not the effect of the molecule on the field. The electromagnetic fields introduced into the electronic wave equation is accordingly those of free space. From (79) one observes that in the absence of sources the electric field has zero divergence, and so both the electric and magnetic fields are purely transversal. It follows that the scalar potential is a constant and can be set to zero. In Coulomb gauge the vector potential is found from the equation... 

If in addition is required to have zero divergence (Capria, 2005), then... 

The right-hand side of (6.4) is implied by the fact that the Ricci tensor has zero divergence for any metric, which requires that the stress-energy tensor should likewise have V T > = 0. 

Helmholtz s theorem states that A general vector field, that vanishes at infinity, can be completely represented as the sum of two independent vector fields, one that is irrotational (zero curl) and another that is solenoidal (zero divergence) [Morse and Feshbach, 1953 Plonsey and CoUin, 1961]. The impressed current density P is a vector field that vanishes at infinity and, according to the theorem, maybe expressed as the sum of two components ... 

Table 4.1 Connections between the existence of a Hamiltonian, the existence of a first integral, and zero divergence.

Thus, in the case of a two-dimensional symplectic manifold, the locally Hamiltonian vector fields are exactly the flows of incompressible liquid, that is, the vector fields with zero divergence. In other words, the condition for the local Hamiltonian properties of the field v in the two-dimensional case is equivalent to the condition div(v) = 0. 

In the second step the convective terms are dropped and a Poisson equation for the pressure emerging from the zero divergence of the velocity field, must be solved. In this step, the tangential stress balance evaluates F explicitly. 

Figure Ala illustrates a vector function of x and y that has a non-zero curl but a divergence of zero. A vector with a non-zero divergence but a curl of zero is shown in Fig. Alb.

Fig. A1 (a) The vector function —yx + yx has non-zero curl but zero divergence, (b) The function xx + yy has non-zero divergence but zero curl...

The superscript identifies the conformation at the beginnin of the time-step. For small timesteps At this should be reasonable to do. Fj in the above equation is the force exerted on particle j. The so-called spurious drift , i.e., the third term in the r.h.s. of eq. (3.20) usually vanishes, since most diffusion tensors which have been used in the literature have zero divergence (this is directly related to the assumption of incompressible flow). p] At) is the random displacement by the coupling to the heat bath. The crucial difficulty comes from the connection of the displacement by the heat bath and the hydrodynamic interaction tensor Dy via the fluctuation dissipation theorem. This fixes the first two moments to be... 

Substitution of the last equation for the total flux into the zero divergence condition provides an elliptic equation for the total pressure, p. 

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