You are here

Advanced Calculus

Avner Friedman
Dover Publications
Publication Date: 
Number of Pages: 
[Reviewed by
Blair F. Madore
, on

This is a Dover Edition reprint, published in 2007, of a book first published in 1971 by Holt, Rinehart and Winston. Two things struck me immediately concerning this textbook: How affordable it is and how little things have changed since 1971.

I was born in 1969, probably around the time this book was planned and written. I learned about analysis about 20 years later. Now, almost 20 more years later, the same material is covered and even the same notation is commonly used. This is not a bad thing. The lack of change in nearly 40 years is not an indication of stagnation; rather it is an indicator of a mature subject that has been nearly completely investigated and is well thought out pedagogically. Many people say that the study and teaching of analysis was so influenced by Rudin's Principles of Mathematical Analysis (1953) that we are still teaching the same topics in the same way today. I take Friedman as an example of this school — a good example.

I am a mathematics professor at a small Liberal Arts college and regularly teach "Advanced Calculus" using textbooks similar in content to Part One of Friedman's book (limits, continuity, differentiation, integration, sequences and series). Unfortunately most of the textbooks we use are not nearly as affordable. Though some texts we use, those by Lay and Gaughan, are seen to be more readable by the students. I prefer texts by Abbott and Ross  which are more reasonably priced, seem more rigorous to me and, like Friedman, are more traditional.

Both Ross and Friedman recognize the pedagogical superiority of the sequential definition of continuity of a function at a point. Ross relies almost completely on the sequential definition for developing most needed results with clear and simple proofs, though this does leave the student wondering how this connects to what he learned back in Calculus class. Friedman uses the traditional definition of continuity at a point then develops the equivalence to the sequential definition. He uses both concepts — in some cases providing two different proofs for one theorem. In most cases he chooses the concept that does best in providing an understandable proof.

Where Friedman differs significantly from the many of the newer introductory analysis texts is in Part Two, which covers the rigorous fundamentals of multivariate calculus from limits and continuity to Stokes' theorem. Our multivariate calculus course, while trying to emphasize understanding and rigor, does not leave room for the complete development of the subject found in Friedman's Part Two. In fact no course in our program, and I suspect in most programs, would have students study the fundamentals of multivariate calculus in such detail. So, from one point of view, Part Two of Friedman's text would seem irrelevant. From my point of view, Part Two is excellent! Even if we only used Part One of the text in class, by putting the whole book in the students' hands we ensure that they are not completely ignorant of the careful and precise development of the multivariate calculus, along the lines of the careful development of the foundations of single variable calculus that they have studied. The keener of my students would read some of Part Two for themselves and might even ask to do a reading course on the topic.

I have not yet taught with Friedman's book (I'm on sabbatical this year and not slated to teach advanced calculus next year), but I plan to, and I expect it will an interesting experience. Each author has his or her own voice that can only be found through a careful reading or, better yet, careful use as a textbook. From what I've seen thus far, Friedman is concise and precise and covers all the appropriate topics for more than one course. If his voice is able to speak to me and my students, I may just have found a long sought commodity — a good analysis text that is easily affordable.

Blair Madore is Associate Professor of Mathematics at SUNY Potsdam.
The table of contents is not available.