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Forms

arXiv:math.GT/0306194 v1 11 Jun 2003

David Bachman

California Polytechnic State University

E-mail address: dbachman@calpoly.edu

For the Instructor

The present work is not meant to contain any new material about diļ¬erential

forms. There are many good books out there which give nice, complete treatments of

the subject. Rather, the goal here is to make the topic of diļ¬erential forms accessible

to the sophomore level undergraduate. The target audience for this material is

primarily students who have completed three semesters of calculus, although the

later sections will be of interest to advanced undergraduate and beginning graduate

students. At many institutions a course in linear algebra is not a prerequisite for

vector calculus. Consequently, these notes have been written so that the earlier

chapters do not require many concepts from linear algebra.

What follows began as a set of lecture notes from an introductory course in

diļ¬erential forms, given at Portland State University, during the summer of 2000.

The notes were then revised for subsequent courses on multivariable calculus and

vector calculus at California Polytechnic State University. At some undetermined

point in the future this may turn into a full scale textbook, so any feedback would

be greatly appreciated!

I thank several people. First and foremost, I am grateful to all those students

who survived the earlier versions of this book. I would also like to thank several of

my colleagues for giving me helpful comments. Most notably, Don Hartig had several

comments after using an earlier version of this text for a vector calculus course. John

Etnyre and Danny Calegari gave me feedback regarding Chapter 6. Alvin Bachman

had good suggestions regarding the format of this text. Finally, the idea to write this

text came from conversations with Robert Ghrist while I was a graduate student at

the University of Texas at Austin. He also deserves my gratitude.

Prerequisites. Most of the text is written for students who have completed three

semesters of calculus. In particular, students are expected to be familiar with partial

derivatives, multiple integrals, and parameterized curves and surfaces.

3

4 FOR THE INSTRUCTOR

Concepts from linear algebra are kept to a minimum, although it will be important

that students know how to compute the determinant of a matrix before delving into

this material. Many will have learned this in secondary school. In practice they will

only need to know how this works for nĆ—n matrices with n ā¤ 3, although they should

know that there is a way to compute it for higher values of n. It is crucial that they

understand that the determinant of a matrix gives the volume of the parallelepiped

spanned by its row vectors. If they have not seen this before the instructor should,

at least, prove it for the 2 Ć— 2 case.

The idea of a matrix as a linear transformation is only used in Section 2 of

Chapter 5, when we deļ¬ne the pull-back of a diļ¬erential form. Since at this point

the students have already been computing pull-backs without realizing it, little will

be lost by skipping this section.

The heart of this text is Chapters 2 through 5. Chapter 1 is purely motivational.

Nothing from it is used in subsequent chapters. Chapter 7 is only intended for

advanced undergraduate and beginning graduate students.

For the Student

It often seems like there are two types of students of mathematics: those who

prefer to learn by studying equations and following derivations, and those who like

pictures. If you are of the former type this book is not for you. However, it is the

opinion of the author that the topic of diļ¬erential forms is inherently geometric, and

thus, should be learned in a very visual way. Of course, learning mathematics in this

way has serious limitations: how can you visualize a 23 dimensional manifold? We

take the approach that such ideas can usually be built up by analogy from simpler

cases. So the ļ¬rst task of the student should be to really understand the simplest

case, which CAN often be visualized.

Figure 1. The faces of the n-dimensional cube come from connecting

up the faces of two copies of an (n ā’ 1)-dimensional cube.

For example, suppose one wants to understand the combinatorics of the n- di-

mensional cube. We can visualize a 1-D cube (i.e. an interval), and see just from our

mental picture that it has two boundary points. Next, we can visualize a 2-D cube

5

6 FOR THE STUDENT

(a square), and see from our picture that this has 4 intervals on its boundary. Fur-

thermore, we see that we can construct this 2-D cube by taking two parallel copies of

our original 1-D cube and connecting the endpoints. Since there are two endpoints,

we get two new intervals, in addition to the two we started with (see Fig. 1). Now,

to construct a 3-D cube, we place two squares parallel to each other, and connect

up their edges. Each time we connect an edge of one to an edge of the other, we get

a new square on the boundary of the 3-D cube. Hence, since there were 4 edges on

the boundary of each square, we get 4 new squares, in addition to the 2 we started

with, making 6 in all. Now, if the student understands this, then it should not be

hard to convince him/her that every time we go up a dimension, the number of lower

dimensional cubes on the boundary is the same as in the previous dimension, plus 2.

Finally, from this we can conclude that there are 2n (n-1)-dimensional cubes on the

boundary of the n-dimensional cube.

Note the strategy in the above example: we understand the āsmallā cases visually,

and use them to generalize to the cases we cannot visualize. This will be our approach

in studying diļ¬erential forms.

Perhaps this goes against some trends in mathematics of the last several hundred

years. After all, there were times when people took geometric intuition as proof,

and later found that their intuition was wrong. This gave rise to the formalists, who

accepted nothing as proof that was not a sequence of formally manipulated logical

statements. We do not scoļ¬ at this point of view. We make no claim that the

above derivation for the number of (n-1)-dimensional cubes on the boundary of an

n-dimensional cube is actually a proof. It is only a convincing argument, that gives

enough insight to actually produce a proof. Formally, a proof would still need to be

given. Unfortunately, all too often the classical math book begins the subject with

the proof, which hides all of the geometric intuition which the above argument leads

to.

Contents

For the Instructor 3

For the Student 5

Chapter 1. Introduction 9

1. So what is a Diļ¬erential Form? 9

2. Generalizing the Integral 10

3. Interlude: A review of single variable integration 11

4. What went wrong? 11

5. What about surfaces? 14

Chapter 2. Forms 17

1. Coordinates for vectors 17

2. 1-forms 19

3. Multiplying 1-forms 22

4. 2-forms on Tp R3 (optional) 27

5. n-forms 29

Chapter 3. Diļ¬erential Forms 33

1. Families of forms 33

2. Integrating Diļ¬erential 2-Forms 35

3. Orientations 42

4. Integrating n-forms on Rm 45

5. Integrating n-forms on parameterized subsets of Rn 48

6. Summary: How to Integrate a Diļ¬erential Form 52

Chapter 4. Diļ¬erentiation of Forms. 57

1. The derivative of a diļ¬erential 1-form 57

2. Derivatives of n-forms 60

7

8 CONTENTS

3. Interlude: 0-forms 61

4. Algebraic computation of derivatives 63

Chapter 5. Stokesā™ Theorem 65

1. Cells and Chains 65

2. Pull-backs 67

3. Stokesā™ Theorem 70

4. Vector calculus and the many faces of Stokesā™ Theorem 74

Chapter 6. Applications 81

1. Maxwellā™s Equations 81

2. Foliations and Contact Structures 82

3. How not to visualize a diļ¬erential 1-form 86

Chapter 7. Manifolds 91

1. Forms on subsets of Rn 91

2. Forms on Parameterized Subsets 92

3. Forms on quotients of Rn (optional) 93

4. Deļ¬ning Manifolds 96

5. Diļ¬erential Forms on Manifolds 97

6. Application: DeRham cohomology 99

Appendix A. Non-linear forms 103

1. Surface area and arc length 103

CHAPTER 1

Introduction

1. So what is a Diļ¬erential Form?

A diļ¬erential form is simply this: an integrand. In other words, itā™s a thing

you can integrate over some (often complicated) domain. For example, consider the

1

x2 dx. This notation indicates that we are integrating x2 over the

following integral:

0

interval [0, 1]. In this case, x2 dx is a diļ¬erential form. If you have had no exposure to

this subject this may make you a little uncomfortable. After all, in calculus we are

taught that x2 is the integrand. The symbol ādxā is only there to delineate when the

integrand has ended and what variable we are integrating with respect to. However,

as an object in itself, we are not taught any meaning for ādxā. Is it a function? Is it

an operator on functions? Some professors call it an āinļ¬nitesimalā quantity. This is

1 n

2

x2 āx,

x dx is deļ¬ned to be the limit, as n ā’ ā, of

very tempting... after all, i

i=1

0

where {xi } are n evenly spaced points in the interval [0, 1], and āx = 1/n. When we

take the limit, the symbol ā ā becomes ā ā, and the symbol āāxā becomes ādxā.

This implies that dx = limāxā’0 āx, which is absurd. limāxā’0 āx = 0!! We are not

trying to make the argument that the symbol ādxā should be done away with. It

does have meaning. This is one of the many mysteries that this book will reveal.

One word of caution here: not all integrands are diļ¬erential forms. In fact, in

most calculus classes we learn how to calculate arc length, which involves an integrand

which is not a diļ¬erential form. Diļ¬erential forms are just very natural objects to

integrate, and also the ļ¬rst that one should study. As we shall see, this is much like

beginning the study of all functions by understanding linear functions. The naive

student may at ļ¬rst object to this, since linear functions are a very restrictive class.

On the other hand, eventually we learn that any diļ¬erentiable function (a much more

general class) can be locally approximated by a linear function. Hence, in some sense,

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10 1. INTRODUCTION

the linear functions are the most important ones. In the same way, one can make

the argument that diļ¬erential forms are the most important integrands.

2. Generalizing the Integral

Letā™s begin by studying a simple example, and trying to ļ¬gure out how and what

to integrate. The function f (x, y) = y 2 maps R2 to R. Let M denote the top half

of the circle of radius 1, centered at the origin. Letā™s restrict the function f to the

domain, M, and try to integrate it. Here we encounter our ļ¬rst problem: I have

given you a description of M which is not particularly useful. If M were something

more complicated, it would have been much harder to describe it in words as I have

just done. A parameterization is far easier to communicate, and far easier to use to

determine which points of R2 are elements of M, and which arenā™t. But there are

lots of parameterizations of M. Here are two which we shall use:

ā

Ļ1 (a) = (a, 1 ā’ a2 ), where ā’1 ā¤ a ā¤ 1, and

Ļ2 (t) = (cos(t), sin(t)), where 0 ā¤ t ā¤ Ļ.

OK, now hereā™s the trick: Integrating f over M is hard. It may not even be so

clear as to what this means. But perhaps we can use Ļ1 to translate this problem

into an integral over the interval [ā’1, 1]. After all, an integral is a big sum. If we add

up all the numbers f (x, y) for all the points, (x, y), of M, shouldnā™t we get the same

thing as if we added up all the numbers f (Ļ1 (a)), for all the points, a, of [ā’1, 1]?

(see Fig. 1)

3/4

f

f ā—¦Ļ

M

Ļ

1

ā’1 0

Figure 1. Shouldnā™t the integral of f over M be the same as the

integral of f ā—¦ Ļ over [ā’1, 1]?

4. WHAT WENT WRONG? 11

ā

1 ā’ a2 ), so f (Ļ1 (a)) = 1ā’a2 . Hence, we are saying that

Letā™s try it. Ļ1 (a) = (a,

1

1 ā’ a2 da. Using a little calculus,

the integral of f over M should be the same as

ā’1

we can determine that this evaluates to 4/3.

Letā™s try this again, this time using Ļ2 . By the same argument, we have that the

Ļ Ļ

sin2 (t)dt = Ļ/2.

integral of f over M should be the same as f (Ļ2 (t))dt =

0 0

But hold on! The problem was stated before we chose any parameterizations.

Shouldnā™t the answer be independent of which one we picked? It wouldnā™t be a very

meaningful problem if two people could get diļ¬erent correct answers, depending on

how they went about solving it. Something strange is going on!

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