Treatise of differential calculus and integral calculus

Résumé : This treatise in 1 volumes is originated from the course of «Mathematical Analysis» delivered by Constantin Meghea along 1 years at the University POLITEHNICA of Bucharest and also from his scientific seminar «Functional Analysis and Applied Mathematics» held for teachers and engineers at the same university.

Based on the Zermelo - Fraenkel axiom system while the natural numbers are defined by Peano axioms, this treatise represents a modern, rigorous and unitary exposure of differential and integral calculus in Rⁿ and in C, with flights to general topology and functional analysis, massively implemented, and to integration on s-algebras as well. The treatise is also self-contained: except for the chapter O and for some isolated cases, every concept used is defined, every proposition used is proven. A special and continuous attention is paid to the concepts, propositions and methods which are at the basis of applied mathematics in natural sciences and engineering.

The Lebesgue measure and integral are, probably, the most important creation in the mathematics of the XX^th century. Together with their axiomatization - the integration with respect to a measure on a s-algebra, they have directly, and indirectly through integration on differential manifolds from Rⁿ, a huge impact on natural sciences. Thus, presentation of these theories takes into account for this impact. Moreover, we note that «integration on differential manifolds» has been a ticklish problem. To get together rigor, intuition and at all simple possible proofs, and to balance them within the above mentioned subject is a difficult task.

Finally, the abstract character of the treatise is strongly tempered on one hand by a torrent of solved exercises and also by figures, these clarifying the concepts and being their intuitive support, and, on the other hand, by numerous applications. To some of them the funtional analysis itself offers simple and powerful techniques.