Mathematical Analysis: An Introduction (Undergraduate Texts in Mathematics) [Hardcover]

Andrew Browder (Author)

Book Description

Publication Date: December 15, 1995 | ISBN-10: 0387946144 | ISBN-13: 978-0387946146 | Edition: Corrected

Among the traditional purposes of such an introductory course is the training of a student in the conventions of pure mathematics: acquiring a feeling for what is considered a proof, and supplying literate written arguments to support mathematical propositions. To this extent, more than one proof is included for a theorem - where this is considered beneficial - so as to stimulate the students' reasoning for alternate approaches and ideas. The second half of this book, and consequently the second semester, covers differentiation and integration, as well as the connection between these concepts, as displayed in the general theorem of Stokes. Also included are some beautiful applications of this theory, such as Brouwer's fixed point theorem, and the Dirichlet principle for harmonic functions. Throughout, reference is made to earlier sections, so as to reinforce the main ideas by repetition. Unique in its applications to some topics not usually covered at this level.

Editorial Reviews

Review

This is a very good textbook presenting a modern course in analysis both at the advanced undergraduate and at the beginning graduate level. It contains 14 chapters, a bibliography, and an index. At the end of each chapter interesting exercises and historical notes are enclosed.\par From the cover: ``The book begins with a brief discussion of sets and mappings, describes the real number field, and proceeds to a treatment of real-valued functions of a real variable. Separate chapters are devoted to the ideas of convergent sequences and series, continuous functions, differentiation, and the Riemann integral (of a real-valued function defined on a compact interval). The middle chapters cover general topology and a miscellany of applications: the Weierstrass and Stone-Weierstrass approximation theorems, the existence of geodesics in compact metric spaces, elements of Fourier analysis, and the Weyl equidistribution theorem. Next comes a discussion of differentiation of vector-valued functions of several real variables, followed by a brief treatment of measure and integration (in a general setting, but with emphasis on Lebesgue theory in Euclidean spaces). The final part of the book deals with manifolds, differential forms, and Stokes' theorem [in the spirit of M. Spivak's: ``Calculus on manifolds'' (1965; Zbl 141.05403)] which is applied to prove Brouwer's fixed point theorem and to derive the basic properties of harmonic functions, such as the Dirichlet principle''. ZENTRALBLATT MATH A. Browder Mathematical Analysis An Introduction "Everything needed is clearly defined and formulated, and there is a reasonable number of examples…. Anyone teaching a year course at this level to should seriously consider this carefully written book. In the reviewer's opinion, it would be a real pleasure to use this text with such a class."—MATHEMATICAL REVIEWS

Product Details

• Hardcover: 370 pages
• Publisher: Springer; Corrected edition (December 15, 1995)
• Language: English
• ISBN-10: 0387946144
• ISBN-13: 978-0387946146
• Product Dimensions: 9.3 x 6.7 x 1 inches
• Shipping Weight: 1.4 pounds (View shipping rates and policies)
• Average Customer Review: 3.8 out of 5 stars  See all reviews (10 customer reviews)
• Amazon Best Sellers Rank: #852,996 in Books (See Top 100 in Books)

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Most Helpful Customer Reviews

10 of 10 people found the following review helpful:

2.0 out of 5 stars okay book, November 27, 2007

By 

mathematician - See all my reviews

This review is from: Mathematical Analysis: An Introduction (Undergraduate Texts in Mathematics) (Hardcover)

I taught a year analysis with this book. The book contains two semesters at least of undergraduate abstract analysis. The first half or so covers a first semester course with the usual things, metric spaces, continuity, sequences and series, and so forth. The second half or so has Lebesgue integration and multivariable analysis. I found it difficult to find another book having both semesters content in one volume, except for Rudin, which I chose not to use due to its price. So it fits a certain need. But I probably will not use this textbook again in my courses. I found the text to be way to concise, which might have been a result of the effort to fit so much into one volume. This is especially hard for the beginning half of the book, since the transition from calculus to abstract analysis tends to be conceptually challenging for students. Rudin is similar in some respect, since it is also concise. However personally I found it very natural to "fill in the blanks" in sketched proofs in Rudin, whereas in Browder I was more likely to develop a headache doing the same. Also I don't like the problems in the text. There's not really that many, and they tend to be multipart, too difficult and time consuming. The book could be improved by increasing the number of shorter exercises.

10 of 11 people found the following review helpful:

5.0 out of 5 stars Concise and extremely well written book, February 10, 1999

By A Customer

This review is from: Mathematical Analysis: An Introduction (Undergraduate Texts in Mathematics) (Hardcover)

This is a very well written book. It is concise, rigorous and contains all the usual material of an undergraduate analysis course. I prefer the treatment of manifolds and differential forms in this book to that given in Rudins' classic book.

6 of 6 people found the following review helpful:

4.0 out of 5 stars content OK, but problems with the typo density..., June 9, 2004

By A Customer

This review is from: Mathematical Analysis: An Introduction (Undergraduate Texts in Mathematics) (Hardcover)

(...) First let me remark that talking about content, the book is very good. It contains elegance, rigor and the explanations seem OK. The problem however with the book is the typo density of most of the proofs. Some long proofs are presented as one continuous block without even no linefeed in between the facts. This makes some parts of the book rather uncomfortable to study from. I can 't stop starting to read these beautifull theory, but mostly, after one hour I give up, because "my eyes start dancing" ....
Tip for the author and editor : take exactly the same text, but spread it over 600 pages instead of 350 pages. This will make the book more expensive, but I will definitely buy it..