Eigenfunctions of the curl operator in spherical coordinates
01 Jan 1994-Journal of Mathematical Physics (American Institute of Physics)-Vol. 35, Iss: 1, pp 499-507
TL;DR: In this article, it was shown that the eigenfunctions of the curl operator with vanishing divergence can be written in terms of a single scalar potential that satisfies the Helmholtz equation.
Abstract: The eigenfunctions of the curl operator are obtained by separation of variables in spherical coordinates, making use of the spin‐weighted spherical harmonics. It is shown that the eigenfunctions of the curl operator with vanishing divergence can be written in terms of a single scalar potential that satisfies the Helmholtz equation. It is also shown that these eigenfunctions give a complete basis for the divergenceless vector fields.
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TL;DR: In this article, a method for the study of vortex knots is developed for a special class of ideal fluid flows, the axisymmetric ones satisfying the Beltrami equation curl V (x ) = λ V ( x ).
30 citations
01 Jan 1997
TL;DR: In this article, the authors propose a method to solve the problem of "uniformity" and "uncertainty" in the context of health care, and propose a solution.
Abstract: 1
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TL;DR: In this paper, the Moses curl eigenfunctions are used to describe linear force-free magnetic fields, enabling a proof that such fields are defined entirely by the value of their curl transform on the unit hemisphere in transform space.
Abstract: The Moses curl eigenfunctions are used to describe linear force‐free magnetic fields, enabling a proof that such fields are defined entirely by the value of their curl transform on the unit hemisphere in transform space. This change of viewpoint suggests an orderly approach to the exploration and classification of the properties of such fields, which is briefly sketched. The simplest force‐free fields defined on zero‐ , one‐ , and two‐dimensional sets on the transform sphere are exhibited, and the possibility of fields with fractal support sets suggested. Connections with other mathematical descriptions are pointed out, as well as several promising directions for further exploration. All results apply equally well to the description of the Trkalian subset of Beltrami fields in fluid dynamics.
19 citations
01 Jan 1989
16 citations
TL;DR: In this paper, the Dirac equation for a particle subject to a Coulomb potential, a 1/r scalar potential, and the potential of a magnetic monopole is solved by separation of variables using the spin-weighted spherical harmonics and the bound states are obtained.
Abstract: The Dirac equation for a particle subject to a Coulomb potential, a 1/r scalar potential, and the potential of a magnetic monopole is solved by separation of variables using the spin-weighted spherical harmonics and the bound states are obtained. It is shown that the separation constants are the eigenvalues of the z-component and the square of the total angular momentum, which includes that of the electromagnetic field and the spin of the particle. We find that, under certain conditions, there exist solutions where the spin is in the outward or inward radial direction.
16 citations
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Book•
01 Jan 1984
TL;DR: The calculus of 2-spinors was introduced and systematically developed in this article, which leads not only to a deeper understanding of the structure of space-time, but also provides shortcuts to some very tedious calculations.
Abstract: This volume introduces and systematically develops the calculus of 2-spinors. This is the first detailed exposition of this technique which leads not only to a deeper understanding of the structure of space-time, but also provides shortcuts to some very tedious calculations. Many results are given here for the first time.
2,260 citations
TL;DR: In this article, the relationship of the sTlm (θ, φ) to the spherical harmonics of R 4 is also indicated, and the behavior of sYlm under the conformal group of the sphere is shown to realize a representation of the Lorentz group.
Abstract: Recent work on the Bondi‐Metzner‐Sachs group introduced a class of functions sYlm (θ, φ) defined on the sphere and a related differential operator ð. In this paper the sYlm are related to the representation matrices of the rotation group R 3 and the properties of ð are derived from its relationship to an angular‐momentum raising operator. The relationship of the sTlm (θ, φ) to the spherical harmonics of R 4 is also indicated. Finally using the relationship of the Lorentz group to the conformal group of the sphere, the behavior of the sTlm under this latter group is shown to realize a representation of the Lorentz group.
733 citations
TL;DR: In this paper, it was shown that, in space-times which are asymptotically flat, there are reasonable physical restrictionsthat allow one to impose coordinate conditions (in addition to the usual Bondi-type conditions) which restrict the allowed coordinate group to a subgroup of the Bondi Metzner-Sachsgroup.
Abstract: It is shown that, in space-times which are asymptotically flat, there are reasonable physical restrictionsthat allow one to impose coordinate conditions (in addition to the usual Bondi-type conditions)which restrict the allowed coordinate group to a subgroup of the Bondi-Metzner-Sachsgroup. This subgroup is isomorphic to the improper orthochronous inhomogeneous Lorentz group.
717 citations