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**Principles of mathematical analysis. 3rd ed.**
*(English)*
Zbl 0346.26002

International Series in Pure and Applied Mathematics. Düsseldorf etc.: McGraw-Hill Book Company. X, 342 p. DM 47.80 (1976).

For a review of the first edition see Zbl 0052.05301 and for a review of the second edition see Zbl 0148.02903.

The main difference between the second and first edition was the treatment of functions of several variables. In the present edition this part of the book has been almost completely rewritten by the author. Many details are now supplied and illustrated by means of numerous examples. The inverse function theorem, the key result in Chapter 9, is proved by means of Banach’s contraction principle. This way the proof gains in clarity and is far easier to digest. Another novelty in this new edition is the more detailed discussion of differential forms contained in Chapter 10. Stoke’s theorem and the divergence theorem are proved and many examples are given to show the reader the wide range of applicability of these important theorems. The style of this chapter is modern. Other changes can be found in Chapter 1, where the construction of the real number system is moved to an appendix, and in Chapter 8 where the author has added a short do-it-yourself section on the gamma function. The treatment of the Riemann-Stieltjes integral has been trimmed a bit. But this seems to be a welcome trade-off against the important additions which we already indicated. Still within less than 350 pages the author has been able to squeeze a considerable portion of mathematical analysis in a very readable manner. In my optinion it is one of the best textbooks on mathematical analysis. The printing is beautiful and the author as well as the publisher are to be congratulated on the publication of this new edition.

The main difference between the second and first edition was the treatment of functions of several variables. In the present edition this part of the book has been almost completely rewritten by the author. Many details are now supplied and illustrated by means of numerous examples. The inverse function theorem, the key result in Chapter 9, is proved by means of Banach’s contraction principle. This way the proof gains in clarity and is far easier to digest. Another novelty in this new edition is the more detailed discussion of differential forms contained in Chapter 10. Stoke’s theorem and the divergence theorem are proved and many examples are given to show the reader the wide range of applicability of these important theorems. The style of this chapter is modern. Other changes can be found in Chapter 1, where the construction of the real number system is moved to an appendix, and in Chapter 8 where the author has added a short do-it-yourself section on the gamma function. The treatment of the Riemann-Stieltjes integral has been trimmed a bit. But this seems to be a welcome trade-off against the important additions which we already indicated. Still within less than 350 pages the author has been able to squeeze a considerable portion of mathematical analysis in a very readable manner. In my optinion it is one of the best textbooks on mathematical analysis. The printing is beautiful and the author as well as the publisher are to be congratulated on the publication of this new edition.

Reviewer: W.A.J.Luxemburg

### MSC:

26-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to real functions |