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A Geometric Approach to Di¬erential
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 di¬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 di¬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
di¬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 n—n matrices with n ¤ 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 — 2 case.
The idea of a matrix as a linear transformation is only used in Section 2 of
Chapter 5, when we de¬ne the pull-back of a di¬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 di¬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 ¬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 ’ 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 “small” cases visually,
and use them to generalize to the cases we cannot visualize. This will be our approach
in studying di¬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 sco¬ 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 Di¬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. Di¬erential Forms 33
1. Families of forms 33
2. Integrating Di¬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 Di¬erential Form 52

Chapter 4. Di¬erentiation of Forms. 57
1. The derivative of a di¬erential 1-form 57
2. Derivatives of n-forms 60
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8 CONTENTS

3. Interlude: 0-forms 61
4. Algebraic computation of derivatives 63

Chapter 5. Stokes™ Theorem 65
1. Cells and Chains 65
2. Pull-backs 67
3. Stokes™ Theorem 70
4. Vector calculus and the many faces of Stokes™ Theorem 74

Chapter 6. Applications 81
1. Maxwell™s Equations 81
2. Foliations and Contact Structures 82
3. How not to visualize a di¬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. De¬ning Manifolds 96
5. Di¬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 Di¬erential Form?

A di¬erential form is simply this: an integrand. In other words, it™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 di¬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 “dx” 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 “dx”. Is it a function? Is it
an operator on functions? Some professors call it an “in¬nitesimal” quantity. This is
1 n
2
x2 ∆x,
x dx is de¬ned to be the limit, as n ’ ∞, of
very tempting... after all, i
i=1
0
where {xi } are n evenly spaced points in the interval [0, 1], and ∆x = 1/n. When we
take the limit, the symbol “ ” becomes “ ”, and the symbol “∆x” becomes “dx”.
This implies that dx = lim∆x’0 ∆x, which is absurd. lim∆x’0 ∆x = 0!! We are not
trying to make the argument that the symbol “dx” 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 di¬erential forms. In fact, in
most calculus classes we learn how to calculate arc length, which involves an integrand
which is not a di¬erential form. Di¬erential forms are just very natural objects to
integrate, and also the ¬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 ¬rst object to this, since linear functions are a very restrictive class.
On the other hand, eventually we learn that any di¬erentiable function (a much more
general class) can be locally approximated by a linear function. Hence, in some sense,


9
10 1. INTRODUCTION

the linear functions are the most important ones. In the same way, one can make
the argument that di¬erential forms are the most important integrands.



2. Generalizing the Integral

Let™s begin by studying a simple example, and trying to ¬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. Let™s restrict the function f to the
domain, M, and try to integrate it. Here we encounter our ¬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 aren™t. But there are
lots of parameterizations of M. Here are two which we shall use:

φ1 (a) = (a, 1 ’ a2 ), where ’1 ¤ a ¤ 1, and
φ2 (t) = (cos(t), sin(t)), where 0 ¤ t ¤ π.
OK, now here™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 φ1 to translate this problem
into an integral over the interval [’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, shouldn™t we get the same
thing as if we added up all the numbers f (φ1 (a)), for all the points, a, of [’1, 1]?
(see Fig. 1)

3/4
f
f —¦φ
M
φ

1
’1 0



Figure 1. Shouldn™t the integral of f over M be the same as the
integral of f —¦ φ over [’1, 1]?
4. WHAT WENT WRONG? 11

1 ’ a2 ), so f (φ1 (a)) = 1’a2 . Hence, we are saying that
Let™s try it. φ1 (a) = (a,
1
1 ’ a2 da. Using a little calculus,
the integral of f over M should be the same as
’1
we can determine that this evaluates to 4/3.
Let™s try this again, this time using φ2 . By the same argument, we have that the
π π
sin2 (t)dt = π/2.
integral of f over M should be the same as f (φ2 (t))dt =
0 0
But hold on! The problem was stated before we chose any parameterizations.
Shouldn™t the answer be independent of which one we picked? It wouldn™t be a very
meaningful problem if two people could get di¬erent correct answers, depending on
how they went about solving it. Something strange is going on!

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