Two-Dimensional Calculus (2011)

Preface

In 1958 there appeared a treatment of advanced calculus by Nickerson, Spencer, and Steenrod. In their preface they observed that “the standard treatises on this subject, at any rate those available in English, tend to be omnibus collections of seemingly unrelated topics. The presentation of vector analysis often degenerates! into a list of formulas and manipulative exercises, and the student is not brought to grips with the underlying mathematical ideas.”

This view of the situation was widely shared by those who were familiar with the available textbooks. The most important parts of the subject were often obscured in a jumble of disconnected topics. In particular, the theory of functions of several variables, which is a clearly delineated unit, was frequently scattered piecemeal among other topics in elementary and advanced calculus.

There were several reasons for this disorder. To some degree it can be ascribed to historical accident in the development of the undergraduate mathematics curriculum. But more fundamentally, it stems from certain inherent difficulties. The main problem is that without a knowledge of linear algebra it is possible to attain a purely manipulative mastery of the subject, but no real understanding. On a more elementary level, it is possible to discuss functions of several variables without using vector methods, but many important insights will be missed.

The solution proposed by Nickerson, Spencer, and Steenrod, and since adopted by a number of others, was to present first a full-scale treatment of linear algebra, and then to use it freely in their development of higher-dimensional calculus. This approach certainly solves some of the basic problems, but it also creates others.

This text presents an alternative approach. The decision to restrict the entire discussion to two variables was based on several considerations, the most important of which are:

1. The basic problem concerning linear algebra is easily solved. The subjects of quadratic forms and linear transformations in two dimensions are presented in a relatively few pages. They appear as a natural part of the general theory of differentiable functions and transformations and are then used to give further insight into that theory.

2. Geometric intuition operates as a powerful aid to understanding the fundamental concepts, and it also casts light on the reasons for introducing those concepts. Geometric visualization is clearest in two dimensions, still possible in three dimensions, and absent in higher dimensions. Indeed, the later development of geometric insight in n dimensions depends on a thorough familiarity with the lower-dimensional models.

3. The conceptual understanding acquired in the two-dimensional case allows the student to move rapidly through the higher dimensional cases, distinguishing the underlying ideas from the technical complications which arise in the transition from two to higher dimensions.

4. A thorough familiarity with two-dimensional calculus is an ideal preparation for the theory of functions of a complex variable. Indeed, complex functions are interpreted geometrically as plane transformations. Chapter Three of this book concentrates on the ability to visualize these transformations and to work with them effectively.

5. The unity achieved by detaching functions of several variables from the rest of advanced calculus is further consolidated by the elimination of dimension hopping. Unless the student is ready for a general treatment of n-dimensional calculus, he is probably more confused than edified by the usual proliferation of cases—one function of three variables, two functions of four variables, etc.

It should be made clear that the idea of a unified presentation of two-dimensional calculus is certainly not original with the author. Just such a course has been given for a number of years at Stanford. Experience has shown that a student is then prepared to go very rapidly through the classical treatment of three-dimensional calculus, or else to go directly to the general n-dimensional case if his over-all mathematical level is sufficiently advanced. There are a number of good treatments available for each of these alternatives. (See the list of references on page 430.)

The two-dimensional aspect of this book should be viewed in perspective as subsidiary to the principal goal—a presentation of the fundamental ideas that distinguish several-variable from one-variable calculus. It is the clearest possible exposition of these underlying ideas, and wherever relevant their geometric or physical content, that has motivated the selection of topics, their order, and the accompanying examples.

Another, and related goal, is a treatment that is sufficiently elementary to be accessible to a student with a minimum of preparation, but sufficiently rich in insights to be stimulating for a more advanced student who may have seen some of the topics presented, perhaps in a more superficial way, as part of a basic course in calculus.

The prerequisite for reading this book is a knowledge of elementary one-variable calculus. Karel de Leeuw’s Calculus, which precedes this book in the Harbrace College Mathematics Series, is an excellent, concise treatment of all that is needed in the way of preparation.

Since some students will have had more extensive training in elementary calculus, this book is designed so that it may be used at various levels.

For a brief introduction to functions of several variables, it would suffice to cover the material in the first two chapters, together with selected sections in Chapters 3 and 4. The first few exercises at the end of each section are generally of a more routine nature, and may be used to provide a check on elementary comprehension of the text.

For students who already have some familiarity with the background material in Chapter 1, the main body of the text could be completed in a standard course, lasting roughly one semester.

Finally, those students who may have been previously exposed to some version of several-variable calculus, and who wish to deepen their understanding, will find in addition to the topics discussed in the main text, a large number of applications and elaborations in the exercise sections.

The exercises form an important component of the book. They include various topics which could have been presented in the body of the text but are relegated to the exercises in order not to disrupt the main flow of ideas. One example is the treatment of simple connectivity, which uses some of the material in the exercises to Section 25 and appears as an addendum to that section. Another is the discussion of some of the more subtle properties of mappings, connected with the notion of the degree of a map; these may be handled using special methods available in two dimensions. Thus, winding numbers, the argument principle, and Rouché's theorem are seen in a more general setting which puts into perspective their usual treatment involving functions of a complex variable.

The importance of the exercises makes it advisable to read entirely through each exercise section, at least summarily. The exercises often contain facts worth having seen and may occasionally provide just the insight needed to illuminate a part of the text. Those exercises which appear to be specially relevant may be explored in greater detail. A few exercises are starred to indicate that they may for one reason or another pose greater difficulties than the rest.

It is a pleasure to acknowledge my gratitude to Blaine Lawson, Charles Micchelli, and Tom Savits for their assistance in the preparation of the exercises; to Rosemarie Stampfel and Gail Lemmond for their excellent typing services; to Karel de Leeuw, who read through two entire versions of the manuscript and made innumerable helpful suggestions; to my son, Paul, for his diligent aid with the index; and to my wife, Maria, who good humoredly bore the brunt of months of writing, revising, and preoccupation.