Laczkovich M., Sós V.T. Real Analysis: Functions of Several Variables, and Applications

5th Edition. — Springer, New York, 2015. — 485 p. — (Undergraduate Texts in Mathematics) — ISBN: 978-1-4939-2765-4.Analysis forms an essential basis of mathematics as a whole, as well as of the natural sciences, and more and more of the social sciences too. The theory of analysis (differentiation and integration) was created—after Galileo’s insight—for the purposes of describing the universe in the language of mathematics. Working out the precise theory took almost 300 years, with a large portion of this time devoted to definitions that encapsulate the essence of limits and continuity. Mastering these concepts can be a difficult process; this is one of the reasons why analysis is only barely present in most high-school curricula.A Brief Historical Introduction
Basic Concepts
Real Numbers
Infinite Sequences I
Infinite Sequences II
Infinite Sequences III
Rudiments of Infinite Series
Countable Sets 9 Real-Valued Functions of One Real Variable
Continuity and Limits of Functions
Various Important Classes of Functions (Elementary Functions)
Differentiation
Applications of Differentiation
The Definite Integral
Integration
Applications of Integration
Functions of Bounded Variation
The Stieltjes Integral
The Improper Integral