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A Course in Analysis, Volume II: Differentiation and Integration of Functions of Several Variables, Vector Calculus

Niels Jacob and Kristian P. Evans
World Scientific
Publication Date: 
Number of Pages: 
[Reviewed by
Jason M. Graham
, on

A Course in Analysis by Niels Jacob and Kristian P. Evans is a planned seven volume series of texts on mathematical analysis. The first two volumes of the series, which are the only volumes discussed here, provide a mathematically rigorous treatment of the calculus (analysis) of functions of one and several real variables.

According to the preface, the content of the texts form the syllabus for the first three semesters of calculus (analysis) taught to budding mathematicians at Swansea University. It is clear that the texts are meant for students in a mathematics program intent on studying advanced pure and applied mathematics, as there are no applications as usually found in a text for a mixed-audience calculus sequence. In other words, these two volumes are not well suited for students of engineering or the natural sciences. That is not to say that students of those subjects would not benefit from the texts, only that supplements discussing applications would be desirable for some audiences.

As the authors point out in their introduction to Vol. I, there have been calculus textbooks in print for over 300 years. Of course today there are a variety of texts on calculus and analysis, some of which are very popular, or at least well-known. Examples aimed at the undergraduate level are the calculus books of Stewart, Apostol, Spivak, and Courant and John; the advanced calculus books of Buck, Taylor and Mann, and Loomis and Sternberg; and the analysis books of Dieudonné, Abbott, Ross, Rudin, and Bartle and Sherbert. I would argue that volumes I and II of A Course in Analysis lie somewhere between Apostol’s Calculus and Rudin’s Principles of Mathematical Analysis in terms of level of mathematical maturity required on the part of the reader.

What I find interesting and appealing about Jacob and Evans’s book is the philosophy or spirit of mathematical curriculum that the authors promote. Specifically, the authors argue against a highly compartmentalized mathematical curriculum. In their preface, the authors write:

The modular approach to teaching combined with examination pressure has prevented students from seeing crucial connections between topics being taught in different modules, even when prerequisites and dependencies are emphasised.

They go on to state:

All this has led to a situation where topics such as analysis of several variables, vector calculus, differential geometry of curves and surfaces are seen by students as rather unrelated topics.

Thus, part of the impetus for A Course in Analysis, and I presume the mathematics curriculum at Swansea, is to provide a more connected or unified treatment of many of the most important topics in mathematics. I personally find the approach of the Jacob and Evans to be smooth, natural and valuable.

Volume I begins with some basic naïve set theory, subsets of \(\mathbb{R}\), and functions. The authors discuss in appendices related topics such as elementary logic and the completeness of \(\mathbb{R}\). Throughout both of the first two volumes of A Course in Analysis the authors follow the definition, theorem, proof format. I do not find the books to be overly abstract, however. The authors give many examples, illustrations and exercises to help students digest the theory and they employ use of clear and neat notation throughout. I really appreciate their selection of exercises, since many of the problems develop simple techniques to be used later in the book or make connections of analysis with other parts of mathematics. There are also solutions to all of the exercises in the back of the book. I think that certainly the first volume and perhaps both volumes I and II could be used as the primary text for a sequence in honors calculus for mathematics majors at a typical American college or university.

Volume II of A Course in Analysis is almost entirely concerned with functions of several real variables, although it begins with an introduction to metric spaces. The authors do a nice job in their treatment of this material, balancing rigour and intuition. As in the first volume there are some real gems in volume II. For example, after a discussion of multi-variable integration by parts the authors give a quick derivation of a simple version of Poincaré’s inequality. A Course in Analysis seems to be full of these little gems where the authors use the material or ask the readers to use the material to obtain results or examples that the reader will certainly see again in another context later in their studies of mathematics. Generally, the quality of exposition in both of the first two volumes is very high. I recommend these books.

Jason M. Graham is an assistant professor in the department of mathematics at the University of Scranton, Scranton, Pennsylvania. His current professional interests are in teaching applied mathematics and mathematical biology, and collaborating with biologists specializing in the collective behavior of groups of organisms.

  • Part 3: Differentiation of Functions of Several Variables:
    • Metric Spaces
    • Convergence and Continuity in Metric Spaces
    • More on Metric Spaces and Continuous Functions
    • Continuous Mappings Between Subsets of Euclidean Spaces
    • Partial Derivatives
    • The Differential of a Mapping
    • Curves in ℝn
    • Surfaces in ℝ3. A First Encounter
    • Taylor Formula and Local Extreme Values
    • Implicit Functions and the Inverse Mapping Theorem
    • Further Applications of the Derivatives
    • Curvilinear Coordinates
    • Convex Sets and Convex Functions in ℝn
    • Spaces of Continuous Functions as Banach Spaces
    • Line Integrals
  • Part 4: Integration of Functions of Several Variables:
    • Towards Volume Integrals in the Sense of Riemann
    • Parameter Dependent and Iterated Integrals
    • Volume Integrals on Hyper-Rectangles
    • Boundaries in ℝn and Jordan Measurable Sets
    • Volume Integrals on Bounded Jordan Measurable Sets
    • The Transformation Theorem : Result and Applications
    • Improper and Parameter Dependent Integrals
  • Part 5: Vector Calculus:
    • The Scope of Vector Calculus
    • The Area of a Surface in ℝ3 and Surface Integrals
    • Gauss' Theorem in ℝ3
    • Stokes Theorem in ℝ2 and ℝ3
    • Gauss' Theorem for Domains in ℝn
  • Appendix I : Vector Spaces and Linear Mappings
  • Appendix II : Two Postponed Proofs of Part 3
  • Solutions to Problems of Part 3
  • Solutions to Problems of Part 4
  • Solutions to Problems of Part 5
  • References
  • Mathematicians Contributing to Analysis (Continued)
  • Subject Index