Differential forms : theory and practice

398 Pages·2014·2.118 MB·English

by  Steven H Weintraub

Differential Forms
Theory and Practice
Differential Forms
Theory and Practice
Second edition
by
Steven H. Weintraub
Lehigh University
Bethlehem, PA, USA
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Library of Congress Cataloging-in-Publication Data
Weintraub, Steven H., author.
Differential forms: theory and practice / by Steven H. Weintraub. -- [2] edition.
pages cm
Includes index.
ISBN 978-0-12-394403-0 (alk. paper)
1. Differential forms. I. Title.
QA381.W45 2014
515’.37--dc23
2013035820
British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library
For information on all Academic Press publications
visit our web site at store.elsevier.com
Printed and bound in USA
14 15 16 17 10 9 8 7 6 5 4 3 2 1
ISBN: 978-0-12-394403-0
To my mother and the memory of my father
Preface
Differential forms are a powerful computational and theoretical tool.
They play a central role in mathematics, in such areas as analysis
on manifolds and differential geometry, and in physics as well, in
such areas as electromagnetism and general relativity. In this book,
we present a concrete and careful introduction to differential forms,
attheupper-undergraduateorbeginninggraduatelevel,designedwith
the needs of both mathematicians and physicists (and other users of
the theory) in mind.
On the one hand, our treatment is concrete. By that we mean that
we present quite a bit of material on how to do computations with
differential forms, so that the reader may effectively use them.
On the other hand, our treatment is careful. By that we mean that
we present precise definitions and rigorous proofs of (almost) all of
the results in this book.
We begin at the beginning, defining differential forms and show-
ing how to manipulate them. First we show how to do algebra with
them, and then we show how to find the exterior derivative dϕ of
a differential form ϕ. We explain what differential forms really are:
Roughly speaking, a k-form is a particular kind of function on
k-tuples of tangent vectors. (Of course, in order to make sense of
this we must first make sense of tangent vectors.) We carry on to
our main goal, the Generalized Stokes’s Theorem, one of the central
theorems of mathematics. This theorem states:
Theorem (Generalized Stokes’s Theorem (GST)). Let M be an
orientedsmoothk-manifoldwithboundary∂M (possiblyempty)and
let ∂M be given the induced orientation. Let ϕ be a (k −1)-form on
M with compact support. Then
(cid:2) (cid:2)
dϕ = ϕ.
M ∂M
x Preface
This goal determines our path. We must develop the notion of an
orientedsmoothmanifoldandshowhowtointegratedifferentialforms
onthese.Oncewehavedoneso,wecanstateandprovethistheorem.
The theory of differential forms was first developed in the early
twentieth century by Elie Cartan, and this theory naturally led to
de Rham cohomology, which we consider in our last chapter.
One thing we call the reader’s attention to here is the theme of
“naturality” that pervades the book. That is, everything commutes
withpull-backs–thiscrypticstatementwillbecomeclearuponreading
thebook—andthisenablesustodoallourcalculationsonsubsetsof
Rn, which is the only place we really know how to do calculus.
This book is an outgrowth of the author’s earlier book Differential
Forms:AComplementtoVectorCalculus.Inthatbookweintroduced
differentialformsatalowerlevel,thatofthirdsemestercalculus.The
point there was to show how the theory of differential forms unified
and clarified the material in multivariable calculus: the gradient of
a function, and the curl and divergence of a vector field (in R3) are
all “really” special cases of the exterior derivative of a differential
form, and the classical theorems of Green, Stokes, and Gauss are all
“really” special cases of the GST. By “really” we mean that we must
firstrecasttheseresultsintermsofdifferentialforms,andthisisdone
by what we call the “Fundamental Correspondence.”
However, in the (many) years since that book appeared, we have
received a steady stream of emails from students and teachers who
used this book, but almost invariably at a higher level. We have thus
decided to rewrite it at a higher level, in order to address the needs
of the actual readers of the book. Our previous book had minimal
prerequisites,butforthisbookthereaderwillhavetobefamiliarwith
thebasicsofpoint-settopology,andtohavehadagoodundergraduate
course in linear algebra. We use additional linear algebra material,
oftennotcoveredinsuchacourse,andwedevelopitwhenweneedit.
We would like to take this opportunity to correct two historical
errorswemadeinourearlierbook.Oneofthemotivationsfordevelop-
ing vector calculus was, as we wrote, Maxwell’s equations in
electromagnetism. We wrote that Maxwell would have recognized
vector calculus. In fact, the (common) expression of those equations
in vector calculus terms was not due to him, but rather to Heaviside.
Preface xi
Butit isindeed the case that thisisanineteenth century formulation,
and there is an illuminating reformulation of Maxwell’s equations
in terms of differential forms (which we urge the interested reader to
investigate).Also,Poincaré’sworkincelestialmechanicswasanother
important precursor of the theory of differential forms, and in partic-
ular he proved a result now known as Poincaré’s Lemma. However,
thereisconsiderabledisagreementamongmodernauthorsastowhat
this lemma is (some say it is a given statement, others its converse).
In our earlier book we wrote that the statement in one direction was
Poincaré’s Lemma, but we believe we got it backwards then (and
correct now). See Remark 1.4.2.
We conclude with some remarks about notation and language.
Results in this book have three-level numbering, so that, for exam-
ple, Theorem 1.2.7 is the 7th numbered item in Chapter 1, Section 2.
The ends of proofs are marked by the symbol (cid:2). The statements of
theorems,corollaries,etc.,areinitalics,soareclearlydelineated.But
the statements of definitions, remarks, etc., are in ordinary type, so
there is nothing to delineate them. We thus mark their ends by the
symbol♦.Weuse A ⊆ B tomeanthat Aisasubsetof B,and A ⊂ B
to mean that A is a proper subset of B. We use the term “manifold”
to mean precisely that, i.e., a manifold without boundary. The term
“manifoldwithboundary”isageneralizationoftheterm“manifold,”
i.e.,itincludesthecasewhentheboundaryisempty,inwhichcaseit
is simply a manifold.
Steven H. Weintraub
Bethlehem, PA, USA
May, 2013
1
Rn
Differential Forms in , I
InthischapterweintroducedifferentialformsinRn asformalobjects
and show how to do algebra with them. We then introduce the oper-
ation of exterior differentiation and discuss closed and exact forms.
Our discussion here is completely general, but for the sake of clarity
andsimplicity,wewillbedrawingmostofourexamplesfromR1,R2,
or R3.
The special case n=3 has some particularly interesting features
mathematically, and corresponds to the world we live in physically.
In Section 1.3 of this chapter, we show the correspondence between
differential forms and functions/vector fields in R3.
1.0 Euclidean spaces, tangent spaces, and tangent
vector fields
We begin by establishing some notation and by making some subtle
but important distinctions that are often ignored.
We let R denote the set of real numbers.
Definition 1.0.1. For a positive integer n, Rn is
Rn = {(x ,...,x ) | x ∈ R}. (cid:3)
1 n i
In dealing with R1, R2, andR 3, we will often use (x), (x,y), and
(x,y,z) rather than (x ), (x ,x ), and (x ,x ,x ), respectively, as
1 1 2 1 2 3
coordinates.
We are about to introduce tangent spaces. For some purposes, it is
convenient to have a single vector space that serves as a “model” for
each tangent space, and we introduce that first.
DifferentialForms,SecondEdition.http://dx.doi.org/10.1016/B978-0-12-394403-0.00001-3
©2014ElsevierInc.Allrightsreserved.
2 DifferentialForms
Definition 1.0.2. For a positive integer n, Rn is the vector space
⎧ ⎡ ⎤ ⎫
⎨ a ⎬
1
.
v = ⎣ .. ⎦ | a ∈ R .
⎩ i ⎭
a
n
with the operations
⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤
a b a +b a ca
1 1 1 1 1 1
. . . . .
⎣ .. ⎦+⎣ .. ⎦ = ⎣ .. ⎦ and c⎣ .. ⎦ = ⎣ .. ⎦. (cid:3)
a b a +b a ca
n n n n n n
Definition 1.0.3. Let p=(x ,...,x )∈Rn. The tangent space
1 n
Rn = T Rn to Rn at p is
p p
⎧ ⎫
⎡ ⎤
⎪ ⎪
⎨ a ⎬
1
.
v = ⎣ .. ⎦ | a ∈ R . (cid:3)
⎪ p i ⎪
⎩ ⎭
a
n p
Let us carefully discuss the distinction here. Elements of Rn are
points, while for each fixed point p of Rn, elements v of T Rn are
p p
vectors. For a fixed point p of Rn, ifv ∈ T Rn, we say that v is a
p p p
tangentvectorbasedatthatpoint,sothat T Rn isthevectorspaceof
p
tangent vectors to Rn at p. This is indeed a vector space as we have
the operations of vector addition and scalar multiplication given by
⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤
a b a +b a ca
1 1 1 1 1 1
. . . . .
⎣ .. ⎦ +⎣ .. ⎦ = ⎣ .. ⎦ and c⎣ .. ⎦ = ⎣ .. ⎦ .
a b a +b a ca
n p n p n n p n p n p
WesaythatRn servesasamodelforeverytangentspaceRn aswe
p
have an isomorphism from Rn to Rn given by
p
⎡ ⎤ ⎡ ⎤
a a
1 1
. .
⎣ .. ⎦ (cid:4)→ ⎣ .. ⎦ .
a a
n n p
Geometrically, it makes sense to add tangent vectors based at the
samepointofRn,andthatiswhatwehavedone.Butitdoesnotmake
DifferentialFormsinRn,I 3
geometricsensetoaddtangentvectorsbasedatdifferentpointsofRn,
nor does it make geometric sense to add points of Rn, so we do not
attempt to define such operations.
Let us emphasize the distinctions we have made. Often one sees
points and vectors identified, but they are really different kinds of
objects. Also, often one sees tangent vectors at different points iden-
tified but they are really different objects.
What we have just said is mathematically precise, but also makes
sensephysically.Consider,forexample,abodyunderthegravitational
influence of a star. The body has some position, i.e., it is located at
some point p of R3, and the gravity of the star exerts a force on the
body,thisforcebeinggivenbyavectorv whosedirectionistoward
p
the star and whose magnitude is given by Newton’s law of universal
gravitation. Furthermore v is indeed based at p as this is where the
p
body is located, i.e., this is the point at which the force is acting.
Thusweseeacleardistinctionbetweentheposition(point)andforce
(vectorbasedatthatpoint).Ifourbody,locatedatthepoint p,isunder
the influence of a binary star system, with v the force vector for the
p
gravitational attraction of the first star and w the force vector for
p
thegravitationalattractionofthesecondstar,thentheirsumv +w
p p
(given by the “parallelogram law”) is the net gravitational force on
the body, and this indeed makes sense. On the other hand, if v is a
p
tangent vector based at p and w is a tangent vector based at some
q
different point q, their sumv + w is not defined, and would not
p q
make physical sense either, as what could this represent? (A body
can’t be in two different places at the same time!)
Nextwecometothecloselyrelatednotionofatangentvectorfield.
Definition 1.0.4. A tangent vector field v on Rn is a function v
that associates to every point p ∈ Rn a tangent vector to Rn based at
p, i.e., a tangent vector v ∈ T Rn. Thus we may write v(p) = v
p p p
for every p ∈ Rn. (cid:3)
Example 1.0.5. Foranyn-tupleofrealnumbers(a ,...,a ),we
1 n
have the constant vector field
⎡ ⎤
a
1
.
v = ⎣ .. ⎦
a
n