Magnetfeldbeschreibung mit verallgemeinerten poloidalen und toroidalen Skalaren

1972 ◽  
Vol 27 (8-9) ◽  
pp. 1167-1172 ◽  
Author(s):  
Gerhard Gerlich
Keyword(s):  
Vector Field ◽  
Magnetic Fields ◽  
Vector Fields ◽  
Solenoidal Vector ◽  
Integrability Conditions ◽  
Solenoidal Vector Field ◽  
Complicated Geometry

Abstract Representation of Magnetic Fields by Generalized poloidal and Toroidal Scalars Every solenoidal vector field can be represented by unique poloidal and toroidal scalars. This description is especially appropriate to the geometry of a sphere. A generalization which can be applied to a more or less complicated geometry could be elaborated by means of transforming integrability conditions of space into integrability conditions of surfaces. This formalism enables us to give simple proofs of other important representations of vector fields by two scalars (magnetic coordinates, complex-lamellar fields).

scholarly journals A Reconstruction Approach for Imaging in 3D Cone Beam Vector Field Tomography

2008 ◽  
Vol 2008 ◽  
pp. 1-17 ◽  
Author(s):  
T. Schuster ◽  
D. Theis ◽  
A. K. Louis
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Three Dimensional ◽  
Simulated Data ◽  
Point Of View ◽  
Cone Beam ◽  
Solenoidal Vector ◽  
Approximate Inverse ◽  
Solution Approach ◽  
Moving Fluid

3D cone beam vector field tomography (VFT) aims for reconstructing and visualizing the velocity field of a moving fluid by measuring line integrals of projections of the vector field. The data are obtained by ultrasound measurements along a scanning curve which surrounds the object. From a mathematical point of view, we have to deal with the inversion of the vectorial cone beam transform. Since the vectorial cone beam transform of any gradient vector field with compact support is identically equal to zero, we can only hope to reconstruct the solenoidal part of an arbitrary vector field. In this paper we will at first summarize important properties of the cone beam transform for three-dimensional solenoidal vector fields and then propose a solution approach based on the method of approximate inverse. In this context, we intensively make use of results from scalar 3D computerized tomography. The findings presented in the paper will continuously be illustrated by pictures from first numerical experiments done with exact, simulated data.

Global regularity for a model Navier–Stokes equations on ℝ3

Analysis ◽  
2015 ◽  
Vol 35 (3) ◽  
Author(s):  
Dongho Chae
Keyword(s):  
Vector Field ◽  
Global Regularity ◽  
Stokes Equations ◽  
Time Dependent ◽  
Navier Stokes ◽  
Nonlinear Parabolic System ◽  
Solenoidal Vector ◽  
Navier Stokes Equations ◽  
Nonlinear Parabolic ◽  
Solenoidal Vector Field

AbstractWe study a nonlinear parabolic system for a time dependent solenoidal vector field on ℝ

scholarly journals Pairings between bounded divergence-measure vector fields and BV functions

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Graziano Crasta ◽  
Virginia De Cicco ◽  
Annalisa Malusa
Keyword(s):  
Vector Field ◽  
Conservation Laws ◽  
Bounded Variation ◽  
Vector Fields ◽  
Hyperbolic Conservation Laws ◽  
Divergence Measure ◽  
Compensated Compactness ◽  
Hyperbolic Conservation ◽  
Bv Functions ◽  
Bounded Functions

AbstractWe introduce a family of pairings between a bounded divergence-measure vector field and a function u of bounded variation, depending on the choice of the pointwise representative of u. We prove that these pairings inherit from the standard one, introduced in [G. Anzellotti, Pairings between measures and bounded functions and compensated compactness, Ann. Mat. Pura Appl. (4) 135 1983, 293–318], [G.-Q. Chen and H. Frid, Divergence-measure fields and hyperbolic conservation laws, Arch. Ration. Mech. Anal. 147 1999, 2, 89–118], all the main properties and features (e.g. coarea, Leibniz, and Gauss–Green formulas). We also characterize the pairings making the corresponding functionals semicontinuous with respect to the strict convergence in \mathrm{BV}. We remark that the standard pairing in general does not share this property.

scholarly journals The geometry of null-like disformal transformations

2019 ◽  
Vol 16 (11) ◽  
pp. 1950180 ◽  
Author(s):  
I. P. Lobo ◽  
G. G. Carvalho
Keyword(s):  
Vector Field ◽  
Conformal Transformation ◽  
Vector Fields ◽  
Killing Vector ◽  
Killing Vector Fields ◽  
Spinor Structure ◽  
Schwarzschild Metric ◽  
Killing Vectors ◽  
Symmetry Properties ◽  
Geometric Objects

Motivated by the hindrance of defining metric tensors compatible with the underlying spinor structure, other than the ones obtained via a conformal transformation, we study how some geometric objects are affected by the action of a disformal transformation in the closest scenario possible: the disformal transformation in the direction of a null-like vector field. Subsequently, we analyze symmetry properties such as mutual geodesics and mutual Killing vectors, generalized Weyl transformations that leave the disformal relation invariant, and introduce the concept of disformal Killing vector fields. In most cases, we use the Schwarzschild metric, in the Kerr–Schild formulation, to verify our calculations and results. We also revisit the disformal operator using a Newman–Penrose basis to show that, in the null-like case, this operator is not diagonalizable.

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