Complex vector field

For more complex vector-field potentials depending on the relative orientation as well as the separation of the two particles, the corresponding vector expression is F = — V V, where V = (d/dx, d/dy, d/dz) is the gradient operator. Such vectorial aspects of intermolecular forces are obviously important for real molecules of nonspherical shape.]... 

The electromagnetic field is an example of a complex vector field, Q = B + (i/c)E. 

Knowledge of the spatial dimensions of a molecule is insufficient to imderstand the details of complex molecular interactions. In fact, molecular properties such as electrostatic potential, hydrophilic/lipophilic properties, and hydrogen bonding ability should be taken into account. These properties can be classified as scalar isosurfaces), vector field, and volumetric properties. 

A vector field defined over the field of real (complex, etc.) numbers, is called a real (complex, etc.) vector space. 

The equations satisfied by the fields and potentials may be reduced to a more compact form by the construction of a complex vector whose real and imaginary parts are formed from the vectors defining the magnetic and electric fields. 

We shall prove the integrability of / using the Newlander-Nirenberg theorem. (Replacing the argument with J and id, we find J and K are integrable.) If v and w are complex-valued vector field on X, we have... 

In field theory, electric charge [6] is a symmetry of action, because it is a conserved quantity. This requirement leads to the consideration of a complex scalar field . The simplest possibility [U(l)] is that have two components, but in general it may have more than two as in the internal space of 0(3) electrodynamics which consists of the complex basis ((1),(2),(3)). The first two indices denote complex conjugate pairs, and the third is real-valued. These indices superimposed on the 4-vector give a 12-vector. In U(l) theory, the indices (1) and (2) are superimposed on the 4-vector, 4M in free space, so, 4M in U(l) electrodynamics in free space is considered as transverse, that is, determined by (1) and (2) only. These considerations lead to the conclusion that charge is not a point localized on an electron rather, it is a symmetry of action dictated ultimately by the Noether theorem [6]. 

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... 

A three-way cross-check of the self-consistency of the 0(3) ansatz can be carried out starting from Eq. (459), in which A is complex because the electromagnetic field in 0(3) electrodynamics carries a topological charge k/,4i0 . The vector field A in the internal space of 0(3) symmetry must depend on by special relativity and can be written as... 

The divergenceless condition of the field A in the Coulomb gauge means that the complex vector a(k) is transverse, so that k a(k) = 0. Then, for every value of k, we can choose an orthonormal trihedron with by the real vectors k/co, ei(k) and e2(k), and we can represent the field a as... 

Now, if a Beltrami field is simultaneously complex lamellar, (1) combined with (3), then curl v is both perpendicular and parallel to v. This can happen only if curl v is zero (that is, the field v is curl-less, or lamellar). Hence a vector field that is simultaneously a complex lamellar and a Beltrami field is necessarily lamellar. If the divergence of (5) is taken, we obtain... 

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 utilizing a complex three-vector (self-dual tensor) rather than a real antisymmetric tensor to describe the electromagnetic field, Hillion and Quinnez discussed the equivalence between the 2-spinor field and the complex electromagnetic field [63]. Using a Hertz potential [64] instead of the standard 4-vector potential in this model, they derived an energy momentum tensor out of which Beltrami-type field relations emerged. This development proceeded from the Maxwell equations in free homogeneous isotropic space... 

In quantum mechanics we often encounter associative algebras of operators and matrices which are noncommutative. For example, the set of all n x n matrices over the real or complex number fields is an n2-dimensional vector space which is also an associative, noncommutative algebra whose multiplication is just the usual matrix multiplication. Also, the subset of all diagonal n x n matrices is a commutative algebra. 

APPENDIX 7 Vectors and vector fields. Introduction to tensors APPENDIX 8 Complex and imaginary quantities APPENDIX 9 Hamaker constants APPENDIX 10 Laplace and Fourier transformations APPENDIX 11 Time correlation functions... 

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