Stokes Theorem - B.McKay.pdf

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BenjaminMcKay
Stokes’sTheorem
April24,2013
Preface
This course proves Stokes’s theorem, starting from a background of rigorous
calculus. Chapters 9 and 10 of Rudin [ 6 ] cover the same ground we will, as does
Spivak [ 7 ] and Hubbard and Hubbard [ 4 ]. I encourage you to read those books.
All of them, including these notes, are attempts to rewrite Milnor’s little book
[ 5 ], and try to lead up to Bott and Tu [ 2 ] and Guillemin and Pollack [ 3 ].
There is one abstract idea in this book: dierential forms. Let’s consider 5
motivations for pursuing this abstract idea.
1. Look at an integral, like
Z
x 2 dx.
What is dx ? Recall
Z
X f ( x i ) x,
f ( x ) dx =
lim
x ! 0
so dx plays the role of x . Since x ! 0, we think of dx as being
“infinitesimal” (infinitely small). But that is nonsense. Physicists play
with infinitesimals as if they made sense. We will give dx a meaning in
this book, so that we can play with the physicists. This dx is the simplest
example of a dierential form.
2. We want to carry out integrals over geometric objects: curves and surfaces.
To calculate those integrals we have to parameterize those objects, by
mapping pieces of the real number line or the plane to the curves or surfaces.
The integral will make sense once we check that the result doesn’t depend
on how we parameterize. Dierential forms yield reparameterization
invariant integrals.
3. Many integrals depend on the direction we integrate. For example, in
single variable calculus, R 0
1 means − R 1
0 , which is the only definition that
makes all of the theory hold identically no matter which direction you
integrate. Similar phenomena occur in higher dimensions; we need to
keep track of signs. The notation of dierential forms keeps track of the
signs for us. For multidimensional integrals, this involves a little algebra.
4. Dierential forms yield a bridge between problems in topology and prob-
lems in calculus. We will use them to prove Brouwer’s fixed point theorem.
5. Dierential forms arise naturally in the theory of electromagnetic fields;
we won’t pursue this direction.
iii
Contents
1 Euclidean Space
1
2 Dierentiation
11
3 The Contraction Mapping Principle
21
4 The Inverse Function Theorem
29
5 The Implicit Function Theorem
33
6 Manifolds
37
7 Integration
45
8 Vector Fields
49
9 Dierential Forms
53
10 Dierentiating Dierential Forms
59
11 Integrating Dierential Forms
65
12 Stokes’s Theorem
73
13 The Brouwer Fixed Point Theorem
81
14 Manifolds from Inside
87
15 Volumes of Manifolds
93
16 Sard’s Theorem
101
17 Homotopy and Degree
107
Hints
117
Bibliography
129
List of Notation
131
Index
133
iv
Chapter 1
Euclidean Space
We recall definitions from previous courses and discuss compactness. Closed bounded
sets of points are called compact. Compactness has some subtle consequences.
Maps
We use the usual terminology and notation of sets without introduction. We
writeRto mean the set of all real numbers. A map or function f : X ! Y is
a rule associating to any point x in some set X a point y in the set Y . We
assume the reader is familiar with composition of functions, inverse functions,
and what it means to say that a function is 1-1 (also known as injective), or is
onto (also known as surjective).
Suppose that f : X ! Y is a map between sets and S X is a subset. The
image f ( S ) of S is the set of all points f ( x ) for all x 2 S . The image of f is
f ( X ). Similarly if T Y is a subset, the preimage , f − 1 T , of T is the set of
points x 2 X for which f ( x ) 2 T . It will often be convenient to avoid choosing
a name for a function, for example writing x 7! x 2 sin x. to mean the function
f :R ! R ,f ( x ) = x 2 sin x.
Euclidean space
The setR n is the set of all n -tuples
0
@
1
A
x 1
x 2
.
x n
x =
of real numbers x 1 ,x 2 ,...,x n 2 R. Following standard practice, we will often
be lazy and write such a tuple horizontally as
x = ( x 1 ,x 2 ,...,x n ) .
1.1 For X and Y subsets of points in the following sets, how might we try to
draw a picture to describe a map f : X ! Y ?
(a) X R ,Y R,
(b) X R 2 ,Y R,
1

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