Several Real Variables

315 Pages·2016·3.185 MB·English

by  Shmuel Kantorovitz (auth.)

Springer Undergraduate Mathematics Series
Shmuel Kantorovitz
Several
Real
Variables
Springer Undergraduate Mathematics Series
Advisory Board
M.A.J. Chaplain, University of St. Andrews, Dundee, UK
K. Erdmann, University of Oxford, Oxford, England, UK
A. MacIntyre, Queen Mary University of London, London, England, UK
E. Süli, University of Oxford, Oxford, England, UK
M.R. Tehranchi, University of Cambridge, Cambridge, England, UK
J.F. Toland, University of Cambridge, Cambridge, England, UK
More information about this series at http://www.springer.com/series/3423
Shmuel Kantorovitz
Several Real Variables
123
Shmuel Kantorovitz
Bar-Ilan University
Ramat Gan
Israel
ISSN 1615-2085 ISSN 2197-4144 (electronic)
SpringerUndergraduate MathematicsSeries
ISBN978-3-319-27955-8 ISBN978-3-319-27956-5 (eBook)
DOI 10.1007/978-3-319-27956-5
LibraryofCongressControlNumber:2015959583
MathematicsSubjectClassification: 26B05,26B10,26B12,26B15,26B20
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Preface
This book is based on the lecture notes of a course I gave half a dozen times to
second-year undergraduates at Bar Ilan University. The prerequisites are two
semesters on one variable Differential and Integral Calculus and a semester in
LinearAlgebra.SomefamiliaritywiththelanguageofelementaryAbstractAlgebra
is assumed. Otherwise, the presentation is self-contained, but several of the more
sophisticated proofs are omitted in order to keep the book within the appropriate
constraints of width and depth. A description of the course follows.
Chapter 1 deals with the concept of continuity of functions of several real
variables, or equivalently, functions on the k-dimensional space Rk. This requires
theintroductionofametriconRk.WediscusstheEuclideanmetricinducedbythe
Euclideannorm,whichisitselfinducedbythestandardinnerproductonRk.More
generally, we define inner product spaces and normed spaces, and study in par-
ticular the p-norms on Rk and the metric topology induced on Rk by any one
oftheseequivalentnorms.Wethendefinecompactnessandshowthatcompactsets
are closed bounded sets in any metric space. The validity of the converse in Rk is
then proved. Completeness is defined for metric spaces, and Rk is shown to be
complete.Thesebasictoolsarethenappliedtothestudyofcontinuityoffunctions
betweenmetricspaces.Theopensetscharacterizationofcontinuityyieldseasilyto
the fundamental properties of continuous functions: the image of a compact set
(connected set) by a continuous function is compact (connected, respectively), the
inversefunctionof a bijective continuousfunctionon a compactset is continuous,
and the Intermediate Value Theorem is valid for any continuous real valued
function on a connected metric space. The uniform continuity of continuous
functions on a compact metric space and the equivalence of various definitions of
connectedness of open sets are the closing subjects of Chap. 1.
The concept of differentiability for real or vector valued functions on Rk is
introduced by means of linear approximation of the change of the function with
respect to the change of the variable. We find sufficient conditions for differentia-
bility and express the differential by means of the Jacobian matrix of the function
with respect to the variable. We then prove the standard theorems on derivation
v
vi Preface
offunctionsofseveralrealvariables:thechainrule,Taylor’stheoremandsufficient
conditions for local extrema.
In Chap. 3, we prove the Implicit Function Theorem for real (or vector) valued
functions by applying the Banach Fixed Point Theorem for contractions on a
completemetricspace.TherelevantspaceisaclosedballintheBanachspaceofall
continuousreal(orvector)valuedfunctionsonacellinRk.Thistheoryisappliedto
extrema with constraints (Lagrange multipliers). Some important properties of the
Banach algebra of continuous functions on a compact subset of Rk (or on more
generalcompactmetricspaces)arestudiedinthelastsectionsofthechapter:these
include the Weierstrass Approximation Theorem and its Stone–Weierstrass gen-
eralization to compact metric spaces, and the Arzela–Ascoli Theorem on the
Bolzano–Weierstrass Property.
Standard geometric applications in R3 are discussed at a slower pace, using
intentionallysometoolsofthebasictheoryofsystemsoflinearequations,inorder
to stress again the role played by Linear Algebra.
The Banach Fixed Point Theorem is also applied to obtain one of the versions
of the Existence and Uniqueness Theorem for systems of ordinary differential
equations. Consequences of the latter for linear systems are elaborated, with many
details relegated to the Exercises section.
The chapter on integration begins with partial integrals (or integrals depending
on parameters) of real functions of several real variables. We prove Leibniz’s
formula for derivation under the integral sign, and a theorem on the change of
integration order.
The treatment of Riemann integration on a closed bounded domain in Rk does
notseekgreatergenerality because integrationismore efficiently treatedbymeans
oftheLebesgueintegralinmoreadvancedtexts.Wedefinethe(Jordan)contentof
bounded closed domains, and the Riemann integral of bounded real functions
defined on them. The basic properties of Riemann integrable functions on such
domainsareproved.Fornormaldomainsindimension2or3,theintegralisshown
to reduce to an iterated (partial) integral. The change of variables formula is stated
foranydimension,buttheproofisomitted.Thestandardapplicationsindimension
2 or 3 follow. Integration on unbounded domains closes our general discussion of
multiple integrals.
The last two sections are concerned with line and surface integrals. We define
curve length, and obtain a formula for it in the (piecewise) smooth case. We then
define line integrals of vector fields and conservative vector fields. We obtain
necessary and sufficient conditions for a field to be (locally) conservative. Green’s
theorem is proved in two dimensions. The three-dimensional case (the Divergence
Theorem)isstated buttheproofisomitted.ThegeneralizationofGreen’stheorem
to closed curves in R3 (Stokes’ formula) is also stated without proof.
Most sections conclude withexercises. Many of thelatterare routine, butsome
are rather sophisticated and complement the theory in certain important directions.
ColleagueswhotaughtparallelsectionsofthecourseatBarIlanUniversityandmy
assistants in the past three or four years contributed some of the exercises and
Preface vii
examplesinthetext,andIwishtothankthemforit.Inordertohelpstudentsusing
this text for self-study, detailed solutions of a large portion of these exercises are
giveninthe“Solutions”sectionattheendofthebook.Lastbutnotleast,Iwishto
thank my colleague Prof. Jeremy Schiff, who helped me with the use of TEX in
producing the manuscript.
Contents
1 Continuity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 The Normed Space Rk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Inner Product Space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Normed Space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Hölder’s Inequality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Minkowski’s Inequality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
The Norm jj(cid:2)jj on Rk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1
Equivalence of the Norms jj(cid:2)jj . . . . . . . . . . . . . . . . . . . . . . . 6
p
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Metric. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
The Metric Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Characterization of Closed Sets . . . . . . . . . . . . . . . . . . . . . . . . 16
Connected Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Union of Connected Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Sufficient Condition for Connectedness. . . . . . . . . . . . . . . . . . . 22
Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.2 Compact Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Open Cover and Compactness. . . . . . . . . . . . . . . . . . . . . . . . . 25
Properties of Compact Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Compactness of Closed Bounded Sets in Rk . . . . . . . . . . . . . . . 28
Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.3 Sequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Basics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Subsequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Cauchy Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
1.4 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Basics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Limits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Continuous Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Properties of Continuous Functions . . . . . . . . . . . . . . . . . . . . . 47
ix
x Contents
Uniform Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
Domains in Rk. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.1 Differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Directional Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
The Differential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
The Chain Rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
The Differential of a Vector Valued Function . . . . . . . . . . . . . . 75
Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
2.2 Higher Derivatives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Mixed Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Taylor’s Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
Local Extrema. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
The Second Differential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3 Implicit Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.1 Fixed Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
The Banach Fixed Point Theorem . . . . . . . . . . . . . . . . . . . . . . 96
The Space C(X). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.2 The Implicit Function Theorem . . . . . . . . . . . . . . . . . . . . . . . . 100
Lipschitz’ Condition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
The Implicit Function Theorem . . . . . . . . . . . . . . . . . . . . . . . . 103
Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
3.3 System of Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
The Implicit Function Theorem for Systems . . . . . . . . . . . . . . . 108
The Local Inverse Map Theorem. . . . . . . . . . . . . . . . . . . . . . . 113
The Jacobian of a Composed Map. . . . . . . . . . . . . . . . . . . . . . 113
Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
3.4 Extrema with Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
3.5 Applications in R3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
Tangent Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
3.6 Application to Ordinary Differential Equations. . . . . . . . . . . . . . 132
Existence and Uniqueness Theorem . . . . . . . . . . . . . . . . . . . . . 133
Linear ODE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
Fundamental Matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141