Comment on divergence, Stokes' theorem, the delta function, and source-free Maxwell's equations (original) (raw)
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A vector field is a function with direction, and because of this directional property, many new kinds of differentiation and integration can be performed on it. For instance, a vector field can be made to pierce a surface or an element thereof, and as it pierces that surface its variation from point to point can be monitored. This leads to one kind of differentiation and integration which we discuss next. The integration leads to the concept of the flux of a vector field, and the associated differentiation to the notion of divergence.
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This paper is devoted to the proof Gauss' divergence theorem in the framework of "ultrafunctions". They are a new kind of generalized functions, which have been introduced recently in [2] and developed in [4], and . Their peculiarity is that they are based on a non-Archimedean field, namely on a field which contains infinite and infinitesimal numbers. Ultrafunctions have been introduced to provide generalized solutions to equations which do not have any solutions, not even among the distributions.
1997
Let (P, Q) be a C 1 vector field defined in a open subset U ⊂ R 2 . We call a null divergence factor a C 1 solution V (x, y) of the equation P ∂V ∂x + Q ∂V ∂y = ∂P ∂x + ∂Q ∂y V . In previous works it has been shown that this function plays a fundamental role in the problem of the center and in the determination of the limit cycles. In this paper we show how to construct systems with a given null divergence factor. The method presented in this paper is a generalization of the classical Darboux method to generate integrable systems. * Research partially supported by a University of Lleida Project 93-3.
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An analytical expression for displacement current density has been derived for both a line current and a current element or a point charge particle in motion. We have shown that the divergence of the inverse square field is zero, in contrast to the accepted notion that it is a delta function. As a result, Maxwell's equations are source-free and the displacement current density is free of divergence; furthermore, both the displacement current and magnetic field are produced simultaneously by a current. Résumé : Une expression analytique pour le courant densité de déplacement a ´ eté tirée pour un courant de ligne et un courantélément ou une particule point-chargée en marche. Nous avons demontré que le divergence de l'inverse square field est zéro, en contraste au notion accepteé que c'est une fonction delta. Conséquemment, leséquations de Maxwell sont source-free et le courant densité de déplacement n'a aucun divergence ; de plus les deux, le courant de déplacement et le champ magnétique sont produit simultanément par un courant.
Characterizations of the existence and removable singularities of divergence-measure vector fields
Indiana University Mathematics Journal, 2008
We study the solvability and removable singularities of the equation div F = µ, with measure data µ, in the class of continuous or L p vector fields F , where 1 ≤ p ≤ ∞. In particular, we show that, for a signed measure µ, the equation (∂U) for any open set U with smooth boundary. For non-negative measures µ, we obtain explicit characterizations of the solvability of div F = µ in terms of potential energies of µ for p ≠ ∞, and in terms of densities of µ for continuous vector fields. These existence results allow us to characterize the removable singularities of the corresponding equation div F = µ with signed measures µ.