This two-volume set is devoted to a quite well known and many times presented topic – differential and integral calculus for functions of several real variables. There are a considerable number of textbooks explaining this subject at various levels of mathematical rigour. Nevertheless, these books bring a new and fresh point of view to an old theme. The first important feature is that these books do not follow the usual strict division into analysis, geometry and algebra; they combine ideas from these different branches (together with those coming from mathematical physics, Lie groups theory, number theory, probability, and special functions) whenever it is suitable. Another unusual feature is that half of the content is constructed of exercises (altogether more than 550) ranging from standard ones to those indicating relations to other fields of mathematics and to challenging ones (useful for accompanying seminars). A logical dependence inside the set of exercises is carefully indicated. It is assumed that the reader has a good understanding of differential and integral calculus with one real variable, otherwise everything is fully proved. The book offers a nice solution to the usual problem of how to present vector calculus and the Stokes type theorem in a basic course of analysis.

The first book is devoted to differential calculus in several variables. The first chapter studies topology of Rn and continuous maps between Euclidean spaces (distance, open, connected and compact sets, and continuous maps). The topics treated in the second chapter are differentiable maps between Euclidean spaces (approximation by linear maps, partial derivatives, critical points, the Taylor formula, and commutativity of a limit process with differentiation and integration). The inverse function theorem and the implicit function theorem are the main tools for a description of local behavior of differentiable maps, which are studied in chapter 3. All that having been covered, smooth submanifolds in Rn are defined in chapter 4 (the Morse lemma and a discussion of critical points of functions is included here) and their tangent spaces are studied in chapter 5 (including a study of curves in R3, Gaussian curvature of a surface and Lie algebras of linear Lie groups).

The second book is devoted to integral calculus in Rn. It starts with Riemann integration in Rn (Fubini’s theorem and change of variables, with several proofs) and continues with applications involving the Fourier transform and the basic limit theorems in integral calculus. Chapter 7 contains the theory of integration with respect to a density over a submanifold in Rn, the main point being the Gauss divergence theorem. The last chapter is devoted to differential forms and a general form of the Stokes theorem. As a whole, the monograph is an excellent addition to the existing literature. Its most valuable feature is that it preserves (and presents) natural relations to many other fields of mathematics, which are cut off in standard treatments. The carefully worked out and comprehensive set of exercises will be very useful for teachers in their lectures and seminars. These books can be very much recommended to any teacher of real analysis as well as to a general mathematical audience.

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