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Preface xi

One way I have chosen to stress the geometric is by concentrating on what happens

in two and three dimensions, where we can construct—with the help of a computer

algebra system as needed—illustrations that help us “see” theorems. And this is not

a bad thing: the words theorem and theatre stem from the same Greek root θ εα ,

“the act of seeing.” In a literal sense, a theorem is “that which is seen.” But the eye,

and the mind’s eye not less, can play tricks. To be certain a theorem is true, we know

we must test what we see. Here is where proof comes in:to prove means “to test.”

The cognate formto probe makes this more evident; probate tests the validity of a

will. Ordinary language supports this meaning, too: yeast is “proofed” before it is

used to leaven bread dough, “the proof of the pudding is in the eating,” and “the

exception proves the rule” because it tests how widely the rule applies.

In much of mathematical exposition, provingis given more weight than seeing .

Jean Dieudonn´e’s seminal Foundations of Modern Analysis [4] is a good example. In

the preface he argues for the “necessity of a strict adherence to axiomatic methods,

with no appeal to ‘geometric intuition’, at least in the formal proofs: a necessity

which we have emphasized by deliberately abstaining from introducing any diagram

in the book.” As prevalent as it is, the axiomatic tradition is not the only one. Ren´e

Thom, a contemporary of Dieudonn´e and Bourbaki, followed a distinctly different

geometric tradition in framing the study of map singularities, a study whose outlines

have guided the development of this book. Although proof may be given a different

weight in the geometric tradition, it still has a crucial role. I believe that a student

who sees a theorem more fully has all the more reason to test its validity.

But there is, of course, usually no reason to restrict the proofs themselves to

low dimensions. For example, my proof of the inverse function theorem (Chapter 5,

p. 169ff.) is for maps on R

n

. It elaborates upon Serge Lang’s proof for maps on

infinite-dimensional Banach spaces [10, 11]. Incidentally, Lang points out that, in

finite dimensions, the inverse function theorem is often proven using the implicit

function theorem, but that does not work in infinite dimensions. Lang gives the

proofs the other way around, and I do the same. Furthermore, because there is so

much instructive geometry associated with implicit functions, I provide not just a

general proof but a sequence of more gradually complicated ones (Chapter 6) that

fold in the growing geometric complexity that additional variables entail. I think

the student benefits from seeing all this put together. Other important examples of

n -dimensional proofs of theorems that are visualized primarily in R

2

are Taylor’s

theorem (Chapter 3), the chain rule (Chapter 4), and Morse’s lemma (Chapter 7).

The definition of the derivative gets the same kind of treatment as the proof of

the implicit function theorem, and for the same reason. Unlike the other topics,

integral proofs are mainly restricted to two dimensions. One reason is that the many

technical details about Jordan content are easiest to see there. Another reason is that

the extension to higher dimensions is straightforward and can be carried out by the

student.

At a couple of points in the text, I provide brief Mathematicacommands that

generate certain 3D images. Because programs like Mathematicaare always being

updated (and the Mathematica 5code I have used in the text has already been superceded), details are

bound to change. My aim has simply been to indicate how

easy it is to generate useful images. I have also included a simple BASICprogram

that calculates a Riemann sum for a particular double integral. Again, it is not my

aim to advocate for a particular computational tool. Nevertheless, I do think it is

important for students to see that programs do have a role—integrals arise out of

computations—and that even a simple program can increase our power to estimate

the value of an integral.

To help keep the focus on geometry, I have excluded proofs of nearly all the

theorems that are associated with introductory real analysis (e.g., those concerning

uniform continuity, convergence of sequences of functions, or equality of mixed par-

tial derivatives). I consider real analysis to be a different course, one that is treated

thoroughly and well in a variety of texts at different levels, including the classics of

Rudin [15] and Protter and Morrey [14]. To be sure, I am recalibrating the balance

here between that which is seen and that which is tested.

This book does not attempt to be an exhaustive treatment of advanced calculus. Even

so, it has plenty of material for a year-long course, and it can be used for a variety

of semester courses. (As I was writing, it occurred to me that a course is like a walk

in the woods—a personal excursion—but a text must be like a map of the whole

woodland, so that others can take walks of their own choosing.) My own course

goes through the basics in Chapters 2–4 and then draws mainly on Chapters 9–11.

A rather different one could go from the basics to inverse and implicit functions

(Chapters 5 and 6), in preparation for a study of differentiable manifolds. The pace

of the book, with its numerous visual examples to introduce new ideas and topics,

is particularly suited for independent study. From start to finish, illustrations carry

the same weight as text and the two are thoroughly interwoven. The eye has an

important role to play.

In addition to theCUPM Proceedings[12] that contain the lectures of Gleason

and Steenrod, I have been strongly influenced by the content and tone of the beauti-

ful three-volume Introduction to Calculus and Analysis [3] by Richard Courant and

Fritz John. In particular, I took their approach to integration via Jordan content. At

a different level of detail, I adopted their phrase order of vanishing as a replace-

ment for the less aptorder of magnitude for vanishing quantities. For the theorems

connecting Riemann and Darboux integrals in Chapter 8, I relied on Protter and

Morrey [14]; my own contribution was a number of figures to illustrate their proofs.

It was Gleason who argued that the Morse lemma has a place in the undergraduate

advanced calculus course. I was fully persuaded after my student Stephanie Jakus

(Smith ’05) wrote her senior honors thesis on the subject.

The Feynman Lectures on Physics [6] have had a pervasive influence on this

book. First of all, Feynman’s vision of his subject, and his flair for explanation, is

awe-inspiring. I felt I could find no better introduction to surface integrals than the

context of fluid flux. Because physics works with two-dimensional surfaces in R

3

,

I also felt justified in concentrating my treatment of surface integrals on this case.

I believe students will have learned all they need in order to deal with the integral

of a k -form over a k -dimensional parametrized surface patch in R

n

, for arbitrary

k < n . In providing a physical basis for the curl, the Lectures prodded me to try to

understand it geometrically. The result is a discussion of the curl (in Chapter 11)

that—like the discussion of the Morse lemma—has not previously appeared in an

advanced calculus text, as far as I am aware.

I thank my students over the last decade for their curiosity, their perseverance,

their interest in the subject, and their support. I especially thank Anne Watson

(Smith ’09), who worked with me to produce and check exercises. My editor at

Springer, Kaitlin Leach, makes the rough places smooth; I am most fortunate to

have worked with her. I am grateful to Smith College for its generous sabbatical

policy; I wrote much of the book while on sabbatical during the 2005–2006 aca-demic year. My deepest debt is to my teacher, Linus Richard Foy, who stimulated

my interest in both mathematics and teaching. In his advanced calculus course, I

often caught myself trying to follow him along two tracks simultaneously: what he

was saying, and how he was saying it