A generalization of Gauss’ divergence theorem

This paper is devoted to the proof Gauss’ divegence theorem in the framework of “ultrafunctions". They are a new kind of generalized functions, which have been introduced recently by Benci in 2013 and developed by the authors. 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.