A Problems Based Course in Advanced Calculus

This is a text for an inquiry based learning (IBL) course in undergraduate real analysis, covering not only single and multivariable theory but also quite a bit of metric space topology. In addition to 16 appendices discussing background material (basic logic and proof, functions and composition, field axioms, order properties of the real numbers, least upper bounds, countable and uncountable sets, etc.), there are thirty chapters of text. They can be conveniently grouped as follows: the first two chapters discuss the basic topology of the real line (open and closed sets, interior, etc.), chapters 3 through 7 are concerned with various aspects of limits (sequential limits, continuous functions, the intermediate value and extreme value theorems), and chapter 8 introduces differentiation (of real-valued functions of a real variable).

This is then followed by 15 chapters (9 through 23) on various aspects of metric space topology, including compactness, connectedness, completeness, normed vector spaces and function spaces. (Two chapters in this group are exclusively linear algebraic, covering vector spaces, linear transformations and matrices.) Uniform convergence is introduced via the “sup norm” on the space of bounded real-valued functions defined on an interval. Some fairly sophisticated material (including the contraction mapping principle and its application to integral equations, and the Stone–Weierstrass theorem), are also discussed.

Chapter 24 begins with a section on uniform continuity and then discusses integration (more about this chapter a little later). Chapters 25–27 and 29–30 discuss multivariable calculus from a general (Banach space) point of view, focusing on issues relating to differentiation such as the implicit function and inverse function theorems. Sandwiched between these two groups of chapters is chapter 28, on infinite series. This is also done from a general point of view, and in fact Banach algebras are defined and used in this chapter.

Having discussed what the book covers, it is appropriate to briefly address some things that are omitted. Iterated integrals are the subject of a brief section in chapter 26; a theorem there, left as a problem with copious hints, asks the student to prove that the order of integration does not matter if the integrand is continuous. More general results like Fubini’s theorem are not discussed. Neither are line and surface integrals, or differential forms. Fourier series are not discussed in the text, and neither is the Lebesgue integral.

The exposition has some interesting and unusual features. The author’s discussion of differentiation in chapter 8, for example, does not proceed along standard lines. Erdman eschews the traditional “limit of a difference quotient” definition of derivative in favor of a more conceptual approach that basically amounts to the definition of the derivative in the multi-variable case (i.e., the existence of a linear differential), specialized to the simpler case of a real-valued function of a real variable. (Indeed, when Erdman gets to the definition of the more general derivative in chapter 25, the discussion tracks almost verbatim the discussion in chapter 8.) This approach has advantages and disadvantages; it does give the student a better sense of the true significance of the derivative, but at the same time it might be a bit more difficult for a beginning student to assimilate.

Likewise, Erdman’s discussion of integration in chapter 24 is done in a rather general context (the functions considered here take their values in a Banach space, not just the set of real numbers) and also does not cover the traditional Riemann integral. Instead, it discusses the Cauchy integral, defined here in terms of uniform limits of integrals of step functions. The set of “regulated functions” for which the Cauchy integral is defined is defined as the closure of a certain subset of the normed vector space of bounded functions from a closed bounded subset of the real line to a Banach space.It is a proper subset of the set of Riemann-integrable functions. When both the Cauchy and Riemann integral exist for a function, they give the same result. (This is neither proved in the book nor left to the reader; it is a result beyond the scope of the text.)

It seems to me that there is a potential problem with deferring integration until so late in the text. This is a topic that is typically covered in a first-semester analysis course, but metric space ideas like completeness, compactness, and connectedness often are not. The order of presentation in this text reverses this traditional approach. And because of the generality in which the integral is discussed here, it doesn’t seem possible to just skip the intervening chapters and proceed directly to chapter 24. Likewise, deferring a discussion of uniform continuity until this late in the text also may raise concerns for instructors wanting to cover this topic in the first semester. (Of course, in many universities, the first semester of undergraduate analysis is also the only semester, so putting things like uniform continuity and integration off may simply not be an option.)

But of course the really interesting feature of the book — in fact, its defining feature — relates to the comment made in the first sentence of this review, that this is a book for an inquiry based learning course. IBL is a pedagogical method derived from the old “Moore method”, pursuant to which students are given definitions and statements of theorems but are expected to do the proofs by themselves, and then present them. The idea is that learning this way imparts a deeper understanding of the material.

This program is followed in this text. Although some proofs are given, the vast majority are not, and are, along with the task of filling in details of many of the examples, left to the reader as either “exercises” or “problems”. The exercises have solutions publicly available online at the AMS webpage for the book, but the problems are not accompanied by solutions. (There is apparently no password-protected solutions manual for instructors.) For some of these exercises and problems (such as the exercise asking for a proof of the contraction mapping theorem), generous hints are provided.

Doing an IBL book right requires more, of course, than just going through a traditional book and throwing away all the proofs. Erdman does it right. He has arranged things in the text so that results are carefully teased out of the student in appropriate steps, with good examples and enough exposition to be helpful within the guidelines of a problems-based course. This is an ambitious and thoughtful attempt by the author, one that is clearly the result of a considerable expenditure of time and thought.

Does it work? The answer to that question, I think, depends largely on the extent to which you are willing to buy into the rationale behind Inquiry Based Learning. I remain an agnostic on this question. On the one hand, it does seem reasonable to believe that having students learn something by discovering it for themselves will give them a better grasp of the material, and certainly the IBL method has many supporters. On the other hand, it takes a lot more time to do things this way, and my own experience teaching analysis is that the semester is not really long enough to cover everything I want even if the traditional lecture format is employed. People who have, to quote Liam Neeson in the movie Taken, “a very particular set of skills”, may be better than I am when it comes to making this method work in class and overcoming these time constraints.

I suppose it would be possible to use this as a text for a more traditional lecture course, perhaps with details supplied in class, but why bother? The principal characteristic of this book is the use of problems to advance the theory, and if you don’t like that idea, it seems silly to use a text that is essentially defined by it. So if you are a traditionalist, this is likely not the book for you. But if you like the idea of using IBL and can figure out a way to cover the material in a reasonable amount of time, this book certainly warrants a close look.

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Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University.