动手学高级微积分：问题驱动课程

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"《基于问题的高级微积分课程》是一本旨在鼓励学生参与数学实践的教材，强调通过解决实际问题来学习高级微积分知识。书中的附录包含了被视为预备知识的概要，如对可数与不可数集合的区分等。这本书适应不同背景的学生，无论是对预备知识非常熟悉还是不甚了解的学生，都能找到适合自己的学习路径。" 《基于问题的高级微积分课程》是约翰·M·厄德曼所著的一本专为本科生设计的教材，属于美国数学会的"纯与应用本科教材系列"第32卷。这本书的核心理念是通过问题解决来教授高级微积分，呼应了数学并非"观赏性运动"的观点，提倡学生积极参与数学活动而非仅靠阅读和被动听讲。在内容结构上，书中的附录被视作逻辑上的起点，它们汇总了学习高级微积分前一般被认为必要的预备知识。这些知识包括但不限于集合论的基本概念，如可数与不可数集合的区分。对于已经熟悉这些内容的学生，附录可以作为参考和记号手册；而对基础知识掌握不足的学生，可以通过深入学习这些部分来提升自己。大多数学生可能处于两者之间，可以根据自己的需求灵活选择何时学习这些预备知识。书中的问题导向方法使学生能够在实际操作中学习和理解微积分的概念，这与学习网球类似，需要大量的实践。这种教学方式有助于提高学生的理解力和解决问题的能力，而不只是积累知识。书中涵盖的微积分主题可能包括极限、导数、积分、多元函数微积分、级数、泰勒公式以及泛函分析的基础等内容，这些主题在科学和工程领域有着广泛的应用。书的编写考虑到了学生背景的多样性，因此提供了不同的学习路径。例如，如果学生对可数与不可数集合的区别不熟悉，他们可以选择立即学习这部分，以备后续使用，或者等到在阅读过程中遇到相关概念时再进行学习。《基于问题的高级微积分课程》是一本注重实践和探索的教材，它鼓励学生积极投身于数学的海洋，通过解决问题来深化理解和掌握高级微积分的知识。对于那些希望在大学阶段深入学习微积分的学生来说，这本书提供了一种独特且有效的学习途径。

Preface xv

(because they unify the material conceptually and greatly simplify subsequent ar-

guments). But the important thing is that deﬁnitions in the single variable case

and the metric space case are the same. Students do not have to unlearn material

as they go to more general settings. Similarly, the diﬀerential calculus is eventually

developed in its natural habitat of normed linear spaces. But here again, the stu-

dent who has mastered the one-dimensional case, which occurs earlier in the text,

will encounter deﬁnitions and theorems and proofs that are virtually identical to

the ones with which he/she is already familiar. There is nothing to unlearn.

In the process of writing this document, I have rethought the proofs of many

standard theorems. Although some, perhaps most, results in advanced calculus

have reached a ﬁnal, optimal form, there are many others that, despite dozens of

diﬀerent approaches over the years, have proofs that are genuinely confusing to most

students. To mention just one example, there is the theorem concerning change of

order of diﬀerentiation whose conclusion is

∂

∂x∂y

∂

∂y ∂x

This well-known result to this day still receives clumsy, impenetrable, and even

incorrect proofs. I make an attempt, here and elsewhere in the text, to lead students

to proofs that are conceptually clear, even when they may not be the shortest or

most elegant. And throughout I try very hard to show that mathematics is about

ideas and not about manipulation of symbols.

I extend my gratitude to the many students over the years who have endured

various versions of this text and to colleagues and reviewers who have generously

nudged me when they found me napping. I want especially to thank Dan Streeter,

who provided me with much help in the technical aspects of getting this document

produced. The text was prepared using A

S-L

X. For the diagrams I used

the macro package X

-pic by Kristoﬀer H. Rose and Ross Moore supplemented by

additional macros in the diagxy package by Michael Barr.

Finally it remains only to say that I will be delighted to receive, and

will consider, any comments, suggestions, or error reports. My e-mail address is

erdmanj@comcast.net.

John M. Erdman

Portland State University

Portland, Oregon

For students:

How to use this book

Some years ago at a national meeting of mathematicians, many of the convention-

eers went about wearing pins which proclaimed, “Mathematics is Not a Spectator

Sport”. It is hard to overemphasize the importance of this observation. The idea

behind it has been said in many ways by many people; perhaps it was said best by

Paul Halmos [Hal67]: The only way to learn mathematics is to do mathematics.

In most respects learning mathematics is more like learning to play tennis than

learning history. It is principally an activity, and only secondarily a body of knowl-

edge. Although no one would try to learn tennis just by watching others play the

game and perhaps reading a book or two about it, thousands of mathematics stu-

dents in American universities every year attempt to master mathematical subjects

by reading textbooks and passively watching their instructors do mathematics on

a blackboard. There are, of course, reasons why this is so, but it is unfortunate

nevertheless. This book is designed to encourage you to do mathematics.

Perhaps the most important thing to know about the structure of

this book is that, logically, the appendices come ﬁrst. The appendices

contain a summary of what is generally regarded as prerequisite information that,

ideally, one should have before venturing into advanced calculus. In the real world,

students’ backgrounds diﬀer enormously. If, at one extreme, you are quite familiar

with nearly all of the appended material, you are ready for Chapter 1. Use the

appendices for reference and notation. At the other extreme, a very poorly prepared

student may proﬁt from spending considerable time working on these sections. In

between, where most students ﬁnd themselves, you have choices. For example, if

you are unfamiliar with the distinction between “countable” and “uncountable”

sets, you can master the concepts right away, knowing that you will use them later,

or you can wait until one of the words crops up in your reading, and learn the

concepts when you need them.

xvii

xviii For students: How to use this book

When you sit down to work at mathematics it is important to have a sizeable

block of time at your disposal during which you will not be interrupted. As you

read, pay especially close attention to deﬁnitions. (After all, before you can think

about a mathematical concept you must know what it means.) Read until you

arrive at a result (results are labeled Theorem, Proposition, Example, Problem,

Lemma, etc.). Every result requires justiﬁcation. The proof of a result may appear

in the body of the text, or it may be left to you as an exercise or a problem.

When you encounter a result, stop and try to prove it. Make a serious attempt.

If a hint appears after the statement of the result, at ﬁrst do not read it. Do not try

to ﬁnd the result elsewhere; and do not ask for help. Halmos [Hal67]pointsout,

“To the passive reader, a routine computation and a miracle of ingenuity come with

equal ease, and later, when he must depend on himself, he will ﬁnd that they went

as easily as they came.” Of course, it is true that sometimes, even after considerable

eﬀort, you will not have discovered a proof. What then?

If a hint is given, and if you have tried seriously but unsuccessfully to derive the

result, then (and only then) should you read the hint. Now try again. Seriously.

What if the hint fails to help, or if there is no hint? If you are stuck on a result

whose proof is labeled Exercise, then consult the online solution. Turning to the

solution should be regarded as a last resort. Even then do not read the whole proof;

read just the ﬁrst line or two, enough to get you started. Now try to complete the

proof on your own. If you can do a few more steps, ﬁne. If you get stuck again

in midstream, read some more of the proof. Use as little of the online proof as

possible.

What if you are stuck on a result whose proof is a Problem but there is no online

solution? After a really serious attempt to solve the problem, go on. You cannot

bring your mathematical education to a halt because of one refractory problem.

Work on the next result. After a day or two go back and try again. Problems often

solve themselves; frequently, an intractably murky result, after having been allowed

to rest for a few days, will suddenly, and inexplicably become entirely clear. In the

worst case, if repeated attempts fail to produce a solution, you may have to discuss

the problem with someone else—instructor, friend, mother, . . . .

A question that students frequently ask is, “When I’m stuck and I have no idea

at all what to do next, how can I continue to work on a problem?” I know of only

one really good answer. It is advice due to P´olya, “If you can’t solve a problem,

then there is an easier problem you can solve: ﬁnd it.”

Consider examples. After all, mathematical theorems are usually generaliza-

tions of things that happen in interesting special cases. Try to prove the result in

some concrete cases. If you succeed, try to generalize your argument. Are you stuck

on a theorem about general metric spaces? Try to prove it for Euclidean n-space.

No luck? How about the plane? Can you get the result for the real line? The unit

interval?

Add hypotheses. If you cannot prove the result as stated, can you prove it

under more restrictive assumptions? If you are having no success with a theorem

concerning general matrices, can you prove it for symmetric ones? How about

diagonal ones? What about the 2 × 2case?

剩余383页未读，继续阅读

THESUMMERE

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