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Multivariate Calculus and Geometry

Seán Dineen
Publisher: 
Springer
Publication Date: 
2014
Number of Pages: 
257
Format: 
Paperback
Edition: 
3
Series: 
Springer Undergraduate Mathematics Series
Price: 
49.99
ISBN: 
9781447164180
Category: 
Textbook
BLL Rating: 

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Mark Hunacek
, on
01/2/2015
]

This book offers an interesting take on a standard subject — or, I should say, several standard subjects. Just about all university mathematics departments offer courses in multivariable calculus and real analysis, and some (but probably not as many as when I was an undergraduate in the 1970s) offer a semester of classical differential geometry (i.e., curves and surfaces in the plane and three-space). Few, however, offer a combined look at these subjects. This book does: it uses each subject to reinforce the others, and also provides a nice illustration of how linear algebra can be used profitably in this context.

At the outset, it should be noted that, notwithstanding the title of this book, it is not intended as a text for the typical third-semester multivariable calculus course. It explicitly lists prior courses in partial derivatives and linear algebra as a prerequisite, for example, and is written at a considerably more theoretical level than is the usual Calculus III book. As such, the book may suffer from an audience problem, but a discussion of that issue requires, first, a quick look at what can be found here.

The first four chapters look at differentiation in several variables, done the “right way,” with the derivative of a function from n-space to m-space being defined as a linear map. Topics discussed here include the implicit function theorem (stated, but not proved), Lagrange multipliers, and critical points and relative extrema of real-valued functions defined on an open set in n-dimensional Euclidean space.

The author then turns to geometry; over the course of three chapters (punctuated by one other chapter on line integrals) we see directed curves defined precisely and, in the plane and three-space, explored geometrically via the Frenet-Serret equations. After another analysis-based chapter (on multiple integrals), the focus is again on geometry with a chapter introducing parameterized surfaces in three-space and another discussing the area of such surfaces in terms of the three functions E, F and G that will later be seen to play a role in the curvature of surfaces.

There follow several chapters on multivariable surface theory, covering triple integrals, surface integrals, the divergence theorem, and Stokes’ theorem. From here it’s back to classical differential geometry, with the final three chapters of the book addressing the various curvatures (principal, normal, Gaussian, geodesic) of a surface and their geometric significance. The famous theorema egregium is mentioned, and some indication of its significance provided, but it is not proved.

With this summary of the contents in hand, we can turn to the question posed earlier about whether this book may have an audience problem. Two obvious questions arise: Is this book suitable as a text? If so, as a text for what?

The answer to the first question, I think, is: yes, providing you have the right students. The author’s writing style is clear, but by no means easy. Dineen does not waste words, and he states things precisely, sometimes so precisely that it may cause trouble for typical undergraduates on this side of the Pond. Consider for example, his definition of “directed curve”; I won’t reproduce the lengthy definition here in full detail, but suffice it to say that he defines this to be an ordered quadruple \((\Gamma, A, B, \mathbf{v})\), where \(\Gamma\) is a set of points in \(\mathbb{R}^n\), \(A\) and \(B\) are points in \(\mathbb{R}^n\) (the initial and terminal points of the curve) and \(\mathbf{v}\) is the direction vector of the curve, defined here as a vector satisfying four precise criteria. Needless to say, this is not what most students have in mind when they think of the word “curve”, so some degree of mathematical maturity is clearly necessary when reading this text.

On the plus side, there are a good number of examples done in detail, and there are also lots of exercises at the end of each chapter. Solutions to quite a few of these exercises appear in a 20+ page Appendix at the end of the book; depending on your particular mindset, the easy availability of solutions may or may not be an attractive feature.

So, overall, I think that a good, mathematically mature, undergraduate who is willing to put in some serious effort should be able to derive quite a lot from this text. This in turn leads to the second question: for what kind of course is this book appropriate?

Iowa State University (in common with, I assume, any number of other universities) offers a two-semester analysis sequence. The first semester consists of the canonical single-variable topics: limits, continuity, differentiation, integration. The second semester is a bit more flexible. It is supposed to cover some multivariable theory, but in recent years several instructors have taken to doing some differential geometry as well. Last spring, for example, the book Elementary Differential Geometry by O’Neill was used, and this spring Spivak’s Calculus on Manifolds will be.

My initial thought was that the book under review would serve nicely as a text for this course, but after looking at the book I see some potential problems with its selection. At times it seems that Dineen is assuming a background not only in the typical “Calculus III” course but also in multivariable real analysis or advanced calculus as well. Several topics that an instructor might well want to cover in detail in a second-semester analysis course (such as the implicit and inverse function theorems, and the fact that continuous functions defined on a compact subset of n-space take on their maxima and minima) are all stated without proof here, perhaps because the author feels the student reader may already be familiar with these ideas.

Of course, one could possibly use this as a text for a one-semester course in classical differential geometry by simply focusing on the relevant chapters covering curves and surfaces, but there may not be enough material here for a full semester course and, in any event, this kind of cannibalization would seem to defeat the underlying purpose of the book, which is to present a kind of holistic approach to the subject.

So, in the final analysis, I suspect that there aren’t a lot of undergraduate courses for which this book would readily serve as a standalone text. That, of course, is hardly a fatal flaw; an interesting point of view justifies a text even in the absence of a ready-made course for which it is appropriate. For students who are not learning this material for the first time and who already have some mathematical maturity, the rigorous, geometric point of view adopted in this book may well help to cement these ideas and offer new insight.


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