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Book review

Author(s) Bröcker, Theodor
Title Analysis
Publisher Wissenschaftsverlag
Year of publication 1992
Reviewed by Cristina Blaga

These two little volumes, written by one of the most respected German expert in differential geometry and topology, constitute, in fact, an introductcry course to differential and integral calculus of one and several variables for undergraduate students.

The first volume is concerned with analysis on R. Topics touched include: number sets, sequences, continuous functions, differential and integral calculus, power series, Taylor development and approximation theorems. The last chapter of this volume introduces the metric spaces and the topological spaces.

The second volume starts with a chapter devoted to the several variables differential calculus. The following chapter is dealing with some problems related to the differential geometry of R^n. It is proved the inverse function theorem, then there are introduced the notions of submanifold and tangent space to a submanifold. The chapters three and four contain basic results related to measure theory and Lebesgue integral. The last one is devoted to some more special topics, including the rang theorem, the Morse lemma and the Sard theorem.

Both volumes contain, at their ends, alot of exercises, a set for each chapter. There are, also, short literature lists and indices.

Everyone who attended a course of analysis knows how vast the matter is, so it is easy to understand that a book of this size cannot include all the results and all the proofs. On the other hand, it is not hard to understand what amount of pedagogical care is needed to choose only the essential. I can say that, from this point of view, this book is an incontestable success. It can be highly recommended to all undergraduate students in mathematics or physics. It would be, also, of a great help to anyone who wish to have a first contact with analysis, especially if he (or she) wants to use the knowledge aquired to learn modern differential geometry.

I ought to say a few words more. First, that the author has written several books of a great success (e.g. Einführung in die Differentialtopologie, Springer 1990, with K. Jänich). Second, the author intend to publish a third volume, devoted to global analysis. Finaly, I have to say that an English version of this book would be, also, very useful for those who cannot read in German.