Vector fields, representations

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, vector fields uniquely defined on the cut plane. They cannot, however, be continued over the cut, but change their roles there instead. Thus, we have the situation of a crossing at which the eigenvector field is discontinuous and Assumption (A) of Thm. 3 is hurt. 

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. 

Besides its appearance in the FFMF equation in plasma physics, as well as associated with time-harmonic fields in chiral media, the chiral Beltrami vector field reveals itself in theoretical models for classical transverse electromagnetic (TEM) waves. Specifically, the existence of a general class of TEM waves has been advanced in which the electric and magnetic field vectors are parallel [59]. Interestingly, it was found that for one representation of this wave type, the magnetic vector potential (A) satisfies a Beltrami equation ... 

All intermolecular interactions can be adequately described, at least in principle, by multidimensional scalar and vector fields representing the energetics of a molecular system as functions of both intermolecular distances and orientations as well as intramolecular structure data. The visualization of these fields, however, has to be based on a three-dimensional picture or a two-dimensional projection because human pattern recognition ability is strongly related to the two- and three-dimensional world. Consequently, the multidimensional field has to be reduced to a two- or three-dimensional representation. In molecular science this can be done in many different ways. 

The molecular surface concept is not only useful for a representation of the bulkiness and the shape of molecules. These surfaces can also be used as screens for the visualization of many properties by means of color coding techniques. Color coding is a popular means of displaying scalar information on a surface. " " Every three-dimensional scalar or vector field that may be generated on the basis of the position of atomic or molecular fragments can be visualized by color coding on a given surface. 

This representation emphasizes the dependence of the fixed points on r. In the limit of a continuous stack of vector fields, we have a picture like Figure 3.1.3. The curve shown is r =, i.e.,... 

The starting point is a general representation theorem from vector calculus that states that any continuously differentiable vector field can be represented by three scalar functions , if, and x in the form10... 

Now specify a cartesian coordinate system with orthogonal unit vectors satisfying a 6y = in the laboratory reference frame. It is convenient to introduce a momentum representation for the fields through fourier transformation between x and k in physical applications the momentum vector k can often be identified with the wavevector of a photon. We choose the z-axis to be along the direction of k (i.e. k = k ej. A vector field U(k) may be decomposed into its longitudinal and transverse components... 

FIGURE 11.13 Schematic representations of vector fields with divergence (left) and curl (right). 

In Figure 6.3b, a GVF field representation of the MR image in Figure 6.3a is shown. As expected, the GVF vectors are pointing out into the boundary cavity, thus confirming the peculiarity of this operator to enter into boundary concavities. 

This is the generalization of (2.3.17) for the photon field, to the non-Abelian case. Note that for the group U 1), Cjki = 0 so only the first term of (2.3.35) appears in (2.3.17). Contrary to what one might have feared from (2.3.34), the transformation rule for the vector fields does not depend upon the representation matrices Lj. 

Besides, the mathematical properties of 7 and as dynamical systems constitute an object of investigation per se quite appealing from the purely mathematical point of view. The structure of these vector fields can be studied by the tools of differential topology. The phase portraits giving a geometric representation of the trajectories in the vicinity of points at which the modulus of the current density vanishes are particularly interesting. 

The Ramsey theory of indirect nuclear spin-spin coupling [12, 13] can be reformulated in terms of linear superposition of two current density fields, 7"" and 7"", induced in the electrons of a molecule by nuclear magnetic dipoles m/ and my [3, 51, 52, 61]. Graphical representations of the interference pattern within the total current density vector field, together with corresponding density maps, Eq. (7.57), are very useful to elucidate coupling pathways and to rationalize the exchange of spin information between coupled nuclei. 

Loads may be distinguished with respect to the location of their origin and nature of their action. External loads act from the outside of the mechanical system, while the internal loads appear within, and become visible when the system is cut. Applied or physical loads are regarded as given, whereas reactive or geometric loads are initially unknown, and result from restrictions on the motion and deformation. Loads may act upon a volume, a surface, or in ideal limit cases as line or point loads. They are representable by tensors of first order, thus taking the form of vectors. For a force F affecting a volume or an area, the vector fields of force density with a volume force f or an area force f describe the three- or two-dimensional distribution respectively ... 

The isosurface and volume rendering representation of 3D fields can only be applied when these fields are scalar. For 3D vector fields (e.g., the electric field surrounding a molecule in solution or a reaction field determining the most favorable... 

Under special conditions, discussed at the end of Chapter 13, it is possible for a mode to propagate with its electric field virtually parallel to either the x- or y-axis in the waveguide cross-section, assuming these are the principal axes. In such cases the displacement vector is parallel to the electric field, and, using the field representation of Eq. (30-4), the cartesian components of the transverse field, e, satisfy the transverse component of the vector wave equation of Eq. (30-31)... 

The above discussion relates to fibers of arbitrary profiles. If we restrict attention to weakly guiding fibers, the Green s function approach can be greatly simplified. To appreciate this simplification, we first set up the vector potential representation for the fields of arbitrary profile fibers and then specialize it to weakly guiding fibers. 

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