One of the best and traditional introductions to mathematical reasoning for college students is a rigorous introduction to real analysis, which usually comes after the intuitive and operational calculus sequence offered to first-year students. Over the years we have come to an almost universal agreement on the contents of such an Advanced Calculus course. Only minor points of disagreement remain: Should we start by constructing the field of real numbers? If so, should we do it via Dedekind cuts or using Cauchy sequences of rational numbers? Or should we start by listing the field axioms of R and focus the discussion on the order properties to emphasize the completeness of the real field?
There has always been a constant supply of excellent textbooks on (Advanced) Calculus that generations of mathematics students have profited from. To mention just a few of the ones that are still in print, we have Landau’s Differential and Integral Calculus (AMS-Chelsea, 2001) of ice-cold beauty, Hardy’s A Course of Pure Mathematics (Cambridge, 2008) almost a manifesto of a creed, Courant’s various takes on Calculus, from his two volume set Differential and Integral Calculus, Vol. I, Vol. II (Wiley, 1988) to his and John’s Introduction to Calculus and Analysis (Springer 1999 and 2000), and modern classics such as Spivak’s Calculus (Publish or Perish, 4th Edition, 2008). Other common choices among older books are the Advanced Calculus books by Buck and Kaplan and Apostol’s Calculus.
Nowadays we have in print several of the classical textbooks, but in addition we have a new supply of texts that follow the tradition of their illustrious predecessors, such as Schumacher’s Closer and Closer: Introducing Real Analysis (Jones and Bartlett, 2009), Ross’ Elementary Analysis: The Theory of Calculus (Springer 1980, 14th printing 2003) Morgan’s minimalist Real Analysis (AMS, 2005) or Abbot’s dangerous Understanding Analysis (Springer, 2001), to mention a few from the new harvest.
This brings us to the book by Fitzpatrick, first published by Thompson Brooks/Cole in 1995 and now in its Second (Revised) Edition. Perhaps the most important feature one needs in an introduction to formal reasoning in mathematics is a balance between intuition and abstraction. This balance is so delicate that the seesaw sometimes falls on one side or the other. Fitzpatrick book does not give a construction of the real field, relegates the field and order axioms and their consequences to an appendix; he starts the first chapter discussing the notions of bounded sets to formulate the least intuitive and more important axiom for R, the completeness axiom, and to obtain from it, for example, the Archimedean property.
Chapters 2 to 10 provide a clear, concise and motivated rigorous presentation of one-variable calculus, starting with the convergence of sequences, boundedness of convergent sequences, and monotone sequences, introducing along the way some properties of the real field such as that there are no holes in R, i.e., the Nested Interval Property, and the Heine-Borel Theorem. Next come continuity and limits of real-valued functions, the intermediate value theorem and uniform continuity. Chapters 4 and 6 develop the concepts of differentiation and integration culminating with the Fundamental Theorems relating both concepts. Skipping chapters 5 and 7 of optional topics, we have two chapters devoted to sequences an series of functions, that go from Taylor polynomials to power series, developing the essential ideas.
The remaining chapters, from 10 to 20, study functions of several variables, starting with the topology of the Euclidean space Rn, convergence of sequences, continuity and limits of functions on Rn, partial and directional derivatives, differentials, the inverse and implicit function theorems, integration of functions of several variables including Fubini’s Theorem and the change of variables theorem, concluding with the Integral Theorems of Stokes, working with parameterized paths and surfaces to avoid the complications related to working in the general setting of manifolds.
The examples in the book are chosen to illustrate or motivate the concepts and theorems being discussed. The proofs offer a fair balance between giving all the details and just sketching the main ideas. This is a good teaching aid that encourages the student to get involved from the very beginning by filling the details as needed before he/she attempts to solve the exercises at the end of each section. It could be used for a two-semester course on Advanced Calculus or for self-study by motivated mathematics students.
Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is firstname.lastname@example.org.