V. A. Zorich Libri

This second edition of a very popular two-volume work presents a thorough first course in analysis, leading from real numbers to such advanced topics as differential forms on manifolds; asymptotic methods; Fourier, Laplace, and Legendre transforms; elliptic functions; and distributions. Especially notable in this course are the clearly expressed orientation toward the natural sciences and the informal exploration of the essence and the roots of the basic concepts and theorems of calculus. Clarity of exposition is matched by a wealth of instructive exercises, problems, and fresh applications to areas seldom touched on in textbooks on real analysis. The main difference between the second and first editions is the addition of a series of appendices to each volume. There are six of them in the first volume and five in the second. The subjects of these appendices are diverse. They are meant to be useful to both students (in mathematics and physics) and teachers, who may be motivated by different goals. Some of the appendices are surveys, both prospective and retrospective. The final survey establishes important conceptual connections between analysis and other parts of mathematics. The first volume constitutes a complete course in one-variable calculus along with the multivariable differential calculus elucidated in an up-to-date, clear manner, with a pleasant geometric and natural sciences flavor.

This second English edition of a popular two-volume work offers a comprehensive introduction to analysis, covering topics from real numbers to advanced concepts like differential forms on manifolds, asymptotic methods, and various transforms. It emphasizes a clear connection to the natural sciences and explores the essence of calculus concepts and theorems. The clarity of the exposition is complemented by a wealth of exercises, problems, and applications often overlooked in typical real analysis textbooks. The second edition includes new appendices—six in the first volume and five in the second—designed to benefit both students and teachers with diverse goals. These appendices include surveys that highlight important connections between analysis and other mathematical fields. The second volume presents classical analysis as part of a unified mathematical framework, illustrating its interactions with modern areas such as algebra, differential geometry, differential equations, complex analysis, and functional analysis. This work lays a strong foundation for advanced studies in any of these mathematical directions.