Synthetic Vector Analysis II

Abstract

Physicists prefer approximate calculations. This is natural, since physics is an empirical science. What is surprising judicious mathematicians is that many physicists use their favorite approximate reasoning to establish such theorems of pure mathematics as Gauss's divergence theorem and Stokes' theorem. What is more surprising is that their discussions impressively appeal to our geometric and physical intuitions, so that the discussions appear cryptically convincing, though mathematicians feel forced to contend offhand that such discussions are mathematically untenable and flimsy. In our previous paper (Nishimura, H. (2002). International Journal of Theoretical Physics 41, 1165–1190) we have shown that once we realize that their discussions in establishing Gauss's divergence theorem and Stokes' theorem are not approximate (with errors) but infinitesimal (without errors), the discussions are bona fide authentic. What we should do is only transfer between the standard universe of sets and mappings whose set of real numbers contains no infinitesimals but zero and an intuitionistic universe of sets and mappings whose set of real numbers contains nilpotent infinitesimals in abundance and in coherence. The principal objective in this paper is to show that the same finesse can establish the celebrated Gauss–Bonnet theorem relating the topology and the Gaussian curvature of a surface, opening the way to the geometric theory of characteristic classes.