This book presents the main ideas and results on functions of several variables. Both differential and integral calculus are treated here. The reader is motivated with many examples and exercises. In chapter 1, basic notions of differential calculus of functions of several variables are discussed. In section 1.5, extensions of the material to Banach spaces are given. The Lebesgue integral in several variables is presented in chapter 2. The most relevant results are explained, including the Fubini theorem, the area and co-area theorems and the Gauss-Green formulae. In chapter 3, the authors deal with potentials and integration of differential 1-forms focusing on solenoidal and irrotational fields. Chapter 4 is dedicated to a wide introduction to the theory of holomorphic functions of one complex variable. In chapter 5, the notions of immersed and embedded surfaces in Rn are discussed. The implicit function theorem is given, accompanied by the most important applications. Chapter 6 is devoted to the stability theory of non-linear systems and the Poincaré-Bendixon theorem. This material is discussed in order to show that dynamical systems with one degree of freedom do not present chaos, in contrast to one-dimensional discrete dynamics and higher-dimensional continuous dynamics. The book may be used for advanced undergraduate and graduate students or as a self-study guide. In my opinion, it is one of the best books in the field. This is due, among others reasons, to the considerable pedagogical experience of both authors.

Reviewer:

oj