Stokes's Theorem

306 Pages·2021·103.137 MB·English

by  Benjamin McKay

Benjamin McKay
Stokes’s Theorem
November 11, 2021
ThisworkislicensedunderaCreativeCommonsAttribution-ShareAlike4.0UnportedLicense.
iii

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 lead up to Milnor’s little book [5], and to Bott and Tu [2]
and Guillemin and Pollack [3].
There is one abstract idea in this book: differential forms. Let’s consider 5
motivations for pursuing this abstract idea.
a. Look at an integral, like
Z
2
x dx.
What is dx? Recall
Z
X
f(x)dx= lim f(x )∆x,
i
∆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
thisbook,sothatwecanplaywiththephysicists. Thisdxisthesimplest
example of a differential form.
b. Wewanttocarryoutintegralsovergeometricobjects: curvesandsurfaces.
To calculate those integrals we have to parameterise 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 parameterise. Integrals of differential forms
are reparameterisation invariant.
c. Many integrals depend on the direction we integrate. For example, in
single variable calculus, R0 means −R1, which is the only definition that
1 0
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 differential forms keeps track of the
signs for us. For multidimensional integrals, this involves a little algebra.
d. Differential forms yield a bridge between problems in topology and prob-
lemsincalculus. WewillusethemtoproveBrouwer’sfixedpointtheorem.
e. Differential forms arise naturally in the theory of electromagnetic fields;
we won’t pursue this direction.
v
Contents
1 Euclidean Space 1
2 Compact Sets 13
3 Differentiation 29
4 Maxima and Minima 37
5 Integration 47
6 The Contraction Mapping Theorem 53
7 The Inverse Function Theorem 61
8 The Implicit Function Theorem 67
9 Manifolds 73
10 Boundaries and Corners 85
11 Vector Fields 93
12 Differential Forms 99
13 Differentiating Differential Forms 105
14 Integrating Differential Forms 113
15 Stokes’s Theorem 121
16 The Brouwer Fixed Point Theorem 127
17 Change of Variable in Integrals 133
18 Stokes’s Theorem 143
19 Advanced Linear Algebra 149
20 The Brouwer Fixed Point Theorem 161
21 Manifolds from Inside 167
22 Lagrange Multipliers 175
23 The Gradient 185
24 Length, Area and Volume 191
25 Immersed Submanifolds 205
26 Calibrations 211
27 Sard’s Theorem 219
28 Brackets of Vector Fields 227
Hints 237
Bibliography 291
vi
Contents vii
List of notation 293
Index 295

Chapter 1
Euclidean Space
We recall definitions from previous courses and discuss closed and open sets.
Maps
We use the usual terminology and notation of sets without introduction. We
write R to mean the set of all real numbers. A map or function f: X →Y is a
ruleassociatingtoanypointxinsomesetX apointy =f(x)inthesetY. 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 ∈ S. The image of
f is f(X). Similarly, even if f doesn’t have an inverse, if T ⊆ Y is a subset,
the preimage, f−1T, of T is the set of points x ∈ X for which f(x) ∈ T. It
will often be convenient to avoid choosing a name for a function, for example
writing x7→x2sinx to mean the function
f: R →R,f(x)..=x2sinx.
The set Rn is the set of all n-tuples
 
x1
x2
x= . 
 .. 
 
x
n
of real numbers x1,x2,...,xn ∈R. Following standard practice, we will often
be lazy and write such a tuple horizontally as
x=(x1,x2,...,xn).
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 ⊆R2, Y ⊆R,
1
2 Euclidean Space
c. X ⊆R, Y ⊆R2,
d. X ⊆R3, Y ⊆R,
e. X ⊆R, Y ⊆R3,
f. X ⊆R2, Y ⊆R2
1.2 Whichplotdrawswhichcurve? x2−y2 =1,x2+y2 =1,y =x2,x4+y4 =1,
(x,y)=(e−tcost,e−tsint), (x/3)2+(y/2)2 =1
1.3 Write each point of the plane as (x,y), and let r =px2+y2. Which plot
plots which function?
r2e−r2, r2, e−r2, −r2e−r2, −e−r2, −r2, log(cid:0)1+r2(cid:1), 1, r
1.4 Which contour map draws which function: x,x2+y2,x2−y2,−x2−y2
If X,W ⊆ R2 are subsets of the plane, we can draw a picture of a map
f: X →W by colouring X with some pattern, and then colouring each point
(u,v)=f(x,y) by the same colour as (x,y):