A Course in Advanced Calculus (Dover Books on Mathematics) by Robert S. Borden (PDF)

Description

This remarkable undergraduate-level text offers a study in calculus that simultaneously unifies the concepts of integration in Euclidean space while at the same time giving students an overview of other areas intimately related to mathematical analysis. The author achieves this ambitious undertaking by shifting easily from one related subject to another. Thus, discussions of topology, linear algebra, and inequalities yield to examinations of innerproduct spaces, Fourier series, and the secret of Pythagoras. Beginning with a look at sets and structures, the text advances to such topics as limit and continuity in En, measure and integration, differentiable mappings, sequences and series, applications of improper integrals, and more.Carefully chosen problems appear at the end of each chapter, and this new edition features an additional appendix of tips and solutions for selected problems.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Professor Borden writes with a very personable style. I find the concepts explained in a friendly, conversation like manner, with subject matter cadenced in a way that really doesn’t leave the reader feeling lost. Borden provides very understandable examples to this subject, that I have previously found lacking in many other texts on the material in this book.I have a Math & Comp Sci degree from a local Tech University, from some years ago. In that program during the 80’s they usually didn’t teach an Advanced Calculus class, however the topics contained in a more traditional A.C. class (such as the topics contained in David V. Widder’s Advanced Calculus) were covered in other classes anyway (minus the Lebegue/Stieltjes Integrals). I can agree that for an undergraduate A.C. book with that scope, this would probably not be the right book.This could be a Real Analysis book with the wrong title; if so, I think it’s pretty darn good, with the way the subject matter is presented. Prior to reading Borden’s book, I read Angus E. Taylor’s General Theory of Functions and Integration. While the reviews just glow over this one, the level of Taylor’s book was a bit over my head, certainly being geared for beginning graduate students probably the reason. Now by reading Bordens book and seeing the same material a second time, the subject matter is making much more sense, and explained in a more conversational manner certainly helps.To some extent, Math books in the 80’s were starting to be written in this style, at least for undergrads, by relaxing some of the stringency in the prose while retaining the rigor (or even lacking prose to begin with; apparently Rudin’s book is supposed to be famous for that). For the material presented in Borden’s book, this is probably the most accessible it can get, at least what I have found so far.I rated this only four stars because I don’t do the problems, so I can’t judge that aspect of the text. I did hate it when the problems were out of sync with the text, so if that is true here, and this truly should not be an Advanced Calculus book, 4 stars it is.

⭐I had Professor Borden as a visiting professor at Bradley University in 1989 using a hardcover version of the book and had my eyes opened to math when I thought I wouldn’t understand the beauty of it. I later got my degree in mathematics from Aurora University in Illinois and would like to hear from Robert if he’s still alive. My name is Bill Bouris and I live in Aurora, IL. I remember that his wife proofread his material and he always came to class in a suit with chalk in hand… what an inspriration. I now teach two algebra courses and know that his book was meant as a text book.

⭐

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