A generalization of the fundamental theorem of calculus (Q2510994)

The purpose of this paper is to present a new way of rewriting the fundamental theorem of calculus for functions with values in a \(2\)-dimensional real associative algebra which has zero divisors (in the paper called dual real number algebra). For this purpose, firstlythe author generalizes in two ways the classical order relation from \(\mathbb{R}\) to this new algebra, so that the new order relations are compatible with the multiplication. Then, introducing the concepts of \(\varphi\)-derivative and \(\varphi\)-integral based on a function \(\varphi : \mathbb{R}\to \mathbb{R}\), several properties (including a fundamental theorem of calculus) are obtained for them. The details are too complicated to be reproduced here. Reviewer's remark. In several places in the paper (on pages 242, 245, 248--250, 252--253), the strange symbol \(\clubsuit\) appears instead of some indexes, a fact which may confuse the reader.