A generalization of Gauss' divergence theorem

TL;DR

This paper extends Gauss' divergence theorem within the framework of ultrafunctions, a novel class of generalized functions based on non-Archimedean fields, enabling solutions to equations lacking classical or distributional solutions.

Contribution

It introduces a generalized version of Gauss' divergence theorem using ultrafunctions, broadening the scope of solvable equations beyond traditional distributions.

Findings

Ultrafunctions are based on non-Archimedean fields with infinite and infinitesimal numbers.

The generalized divergence theorem applies to equations without classical solutions.

Ultrafunctions provide new tools for solving generalized equations.

Abstract

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 [2] and developed in [4], [5] and [6]. 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.