Vectors fields

According to the Helmholtz theorem the Hilbert space of 2-D vector fields p x, y) with the inner product... 

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 ... 

The integrals are connected with the transversal measurements of the 2-D tensor field (Tij and the 2-D vector field respectively. 

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... 

Braun, H., Hauck, A. Tomographic reconstruction of vector fields. IEEE Trans, on Signal Processing, 1991, 39(2) fOf-fll. 

Prince, J. Tomographic reconstruction of S-D vector fields. Proc IEEE Int. Gonf. ASSP 93, 1993, V48S-386. 

The Champ-Sons model is based upon this approximation. It results into a modified Rayleigh integral where specific terms appear. The resulting formula for the refracted field (e.g., displacement vector-field), is given by... 

Above we described tire nature of Maxwell s equations in free space in a medium, two more vector fields need to be... 

Now the Lagrangean associated with the nuclear motion is not invariant under a local gauge transformation. Eor this to be the case, the Lagrangean needs to include also an interaction field. This field can be represented either as a vector field (actually a four-vector, familiar from electromagnetism), or as a tensorial, YM type field. Whatever the form of the field, there are always two parts to it. First, the field induced by the nuclear motion itself and second, an externally induced field, actually produced by some other particles E, R, which are not part of the original formalism. (At our convenience, we could include these and then these would be part of the extended coordinates r, R. The procedure would then result in the appearance of a potential interaction, but not having the field. ) At a first glance, the field (whether induced internally... 

The occurrence of the argument pj2 shows that these eigenvectors are defined up to a sign only. For a unique representation we have to cut the plane along a half-axis. By this, [Pg.389]

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. 

The representation of molecular properties on molecular surfaces is only possible with values based on scalar fields. If vector fields, such as the electric fields of molecules, or potential directions of hydrogen bridge bonding, need to be visualized, other methods of representation must be applied. Generally, directed properties are displayed by spatially oriented cones or by field lines. 

Vector field gradient of the electrostatic potential, i.e., force... 

This theorem provides a convenient means for obtaining rate of change of a vector field function over a volume V(t) as... 

For massless particles, only a postprocessing of the result field is needed, where so-called streaklines that follow the given vector field are calculated. If the particles have a certain mass, additional equations have to be solved. 

In order to provide a more general description of ternary mixtures of oil, water, and surfactant, we introduce an extended model in which the degrees of freedom of the amphiphiles, contrary to the basic model, are explicitly taken into account. Because of the amphiphilic nature of the surfactant particles, in addition to the translational degrees of freedom, leading to the scalar OP, also the orientational degrees of freedom are important. These orientational degrees of freedom lead to another OP which has the form of the vector field. 

The first term in (44) is related to the deviation of the surfactant concentration at point r from the average concentraction, p. The second part is related to the orientational ordering and defines the vector field u. 

The vector field u(r) describing the local orientational ordering of amphi-philes is split into two fields s and t... 

Recently an alternative approach for the description of the structure in systems with self-assembling molecules has been proposed in Ref. 68. In this approach no particular assumption about the nature of the internal interfaces or their bicontinuity is necessary. Therefore, within the same formahsm, localized, well-defined thin films and diffuse interfaces can be described both in the ordered phases and in the microemulsion. This method is based on the vector field describing the orientational ordering of surfactant, u, or rather on its curlless part s defined in Eq. (55). 

Suppose that the vector field u(f) is a continuous function of the scalar variable t. As t varies, so does u and if u denotes the position vector of a point P, then P moves along a continuous curve in space as t varies. For most of this book we will identify the variable t as time and so we will be interested in the trajectory of particles along curves in space. 

The divergence of a vector field u, written div u, is given in Cartesian coordinates by... 

Figure 0.4 shows a vector field u on which I have drawn a surface S. The surface could correspond to a real physical boundary (such as a metallic surface, or the boundary between air and water), or it could just be an abstract entity. Lines of u cross the surface, and we speak colloquially of the flux of u through the surface. Again speaking colloquially, the more lines through S, the greater the flux. I will have cause to mention flux in this volume, so we need to investigate the concept in mure detail. 

Figure 0.4 Flux of vector field through surface...

The flux of u through 5Si is defined as u n i55i, that is the projection of the vector field along the unit normal for 5Si multiplied by the area of 58 j. It is usual to define a surface element 5S = n S. 

In the case of a uniform magnetic induction along the z-axis, B = you will find that the vector field... 

In vector calculus, the flux 4> of an arbitrary vector field A through a surface S is given by the surface integral... 

The previous results become somewhat more transparent when consideration is given to the manner in which matrix elements transform under Lorentz transformations. The matrix elements are c numbers and express the results of measurements. Since relativistic invariance is a statement concerning the observable consequences of the theory, it is perhaps more natural to state the requirements of invariance as a requirement that matrix elements transform properly. If Au(x) is a vector field, call... 

Guckenheimer, J. Holmes, P. Nonlinear Oscillations. Dynamical Systems and Bifurcation of Vector Fields Springer, 1983. 

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