Mathematical Analysis: An Introduction to Functions of Several Variables

361 Pages·2009·3.38 MB·English

by  Mariano Giaquinta

Mariano Giaquinta
Giuseppe Modica
Mathematical Analysis
An Introduction to
Functions of Several Variables
Birkha¨user
Boston • Basel • Berlin
MarianoGiaquinta GiuseppeModica
ScuolaNormaleSuperiore DipartimentodiMatematicaApplicata
Piazza dei Cavalieri, 7 Universita`diFirenze
I-56100 Pisa, Italy Via S. Marta, 3
[email protected] I-50139 Firenze, Italy
[email protected]
LibraryofCongressControlNumber:2009922164
ISBN 978-0-8176-4509-0 (hardcover) e-ISBN 978-0-8176-4612-7
ISBN 978-0-8176-4507-6 (softcover)
DOI 10.1007/978-0-8176-4612-7
MathematicsSubjectClassification(2000):00A35,32A10, 42B05, 49J40, 34D99
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Preface
This book1 introduces the main ideas and fundamental methods of differ-
ential and integral calculus for functions of several variables.
In Chapter 1 we discuss differential calculus for functions of several
variableswithashortexcursionintodifferentialcalculusinBanachspaces.
In Chapter 2 we present some of the most relevant results of the
Lebesgue integration theory, including the limit and approximation theo-
rems, Fubini’s theorem, the area and coarea theorems, and Gauss–Green
formulas. The aim is to provide the reader with all that is needed to use
the power of Lebesgue integration. For this reason some details as well as
some proofs concerning the formulation of the theory are skipped, as we
think they are more appropriatein the generalcontext of measuretheory.
In Chapter 3 we deal with potentials and integration of differential
1-forms, focusing on solenoidal and irrotational fields.
Chapter 4 provides a sufficiently wide introduction to the theory of
holomorphicfunctions ofonecomplexvariable.Wepresentthefundamen-
tal theorems and discuss singularities and residues as well as Riemann’s
theoremonconformalrepresentationandtherelatedSchwarzandPoisson
formulas and Hilbert’s transform.
In Chapter 5, we discuss the notions of immersed and embedded sur-
face in Rn, and we present the implicit function theorem and some of its
applications to vector fields, constrained minimization, and functional de-
pendence. The chapter ends with the study of some notions of the local
theory of curves and surfaces, such as of curvature, first variation of area,
the Laplace–Beltrami operator, and distance function.
In Chapter 6, after a few preliminaries about systems of linear ordi-
nary differential equations, we discuss a few results concerning the sta-
bility of nonlinear systems and the Poincar´e–Bendixson theorem in or-
der to show that dynamical systems with one degree of freedom do not
present chaos, in contrast with the one-dimensional discrete dynamics or
the higher-dimensional continuous dynamics.
1 This book is a translated and revised edition of M. Giaquinta, G. Modica, Analisi
Matematica, IV. Funzioni di piu` variabili,PitagoraEd.,Bologna,2005.
vi Preface
Thestudyofthisvolumerequiresastrongereffortcomparedtothatof
[GM1],[GM2],and [GM3]2 both because of intrinsic difficulties and broad
scope of the themes we present. We think, in fact, that it is useful for the
readertohaveawidespectrumofcontextsinwhichtheseideasplayanim-
portantroleandwhereineventhetechnicalandformalaspectsplayarole.
However, we have tried to keep the same spirit, always providing exam-
ples, illustrations, andexercises to clarify the main presentation,omitting
severaltechnicalitiesordevelopmentsthatwethoughttobetooadvanced.
We are greatly indebted to Cecilia Conti for her help in polishing our
first draft and we warmly thank her. We would like to thank also Paolo
Acquistapace, Timoteo Carletti, Giulio Ciraolo, Roberto Conti, Giovanni
Cupini, Matteo Focardi, Pietro Majer, and Stefano Marmi for their com-
ments and their invaluable help in catching errors and misprints and Ste-
fan Hildebrandt for his comments and suggestions concerning especially
the choice of illustrations. Our special thanks also go to all members of
the editorial and technical staff of Birkha¨user for the excellent quality of
their work and especially to Rebecca Biega and the executive editor Ann
Kostant.
Note: We have tried to avoidmisprints and errors.But, as most authors,
weareimperfect.Wewillbeverygratefultoanybodywhowantstoinform
us about errors or just misprints, or wants to express criticism or other
comments. Our e-mail addresses are
[email protected] [email protected]
We shall try to maintain any errata and corrigenda at the following web
pages:
http://www.sns.it/~giaquinta
http://www.dma.unifi.it/~modica
Mariano Giaquinta
Giuseppe Modica
Pisa and Firenze
July 2007
2 We shall refer to the following sources as [GM1], [GM2], and [GM3], respectively:
[GM1]:M.Giaquinta,G.Modica,MathematicalAnalysis,FunctionsofOneVariable,
Birkh¨auser,Boston,2003;[GM2]:M.Giaquinta,G.Modica,Mathematical Analysis,
Approximation and Discrete Processes, Birkh¨auser, Boston, 2004; [GM3]: M. Gia-
quinta, G. Modica,Mathematical Analysis, Linear and Metric Structures and Con-
tinuity,Birkh¨auser,Boston,2007.
Contents
Preface ..................................................... v
1. Differential Calculus .................................... 1
1.1 Differential Calculus of Scalar Functions................. 1
1.1.1 Directionalandpartialderivatives,andthedifferential 1
a. Directional derivatives ........................ 1
b. The differential............................... 2
c. The gradient vector........................... 5
d. Direction of steepest ascent .................... 6
1.1.2 Directional derivatives and differential in coordinates 7
a. Partial derivatives ............................ 7
b. Jacobian matrix.............................. 8
c. The differential in the dual basis ............... 8
d. The gradient vector in coordinates.............. 9
e. The tangent plane ............................ 9
f. The orthogonalto the tangent space ............ 10
g. The tangent map............................. 11
h. Differentiability and blow-up................... 12
1.2 Differential Calculus for Vector-valued Functions ......... 12
1.2.1 Differentiability................................. 14
a. Jacobian matrix.............................. 14
b. The tangent space ............................ 16
1.2.2 The calculus ................................... 20
1.2.3 Differentiation of compositions.................... 21
1.2.4 Calculus for matrix-valued maps .................. 22
1.3 Theorems of Differential Calculus....................... 24
1.3.1 Maps with continuous derivatives ................. 24
a. Functions of class C1(A) ...................... 24
b. Functions of class C1(A) ...................... 25
c. Functions of class C2(A) ...................... 26
d. Functions of classes Ck(A) and C∞(A).......... 28
1.3.2 Mean value theorem............................. 29
a. Scalar functions .............................. 29
b. Vector-valued functions ....................... 31
1.3.3 Taylor’s formula ................................ 32
a. Taylor’s formula of second order................ 33
viii Contents
b. Taylor formulas of higher order................. 34
c. Real analytic functions........................ 36
d. A converse of Taylor’s theorem................. 37
1.3.4 Critical points .................................. 38
1.3.5 Some classical partial differential equations......... 42
1.4 Invertibility of Maps Rn →Rn ......................... 46
1.4.1 Banach’s fixed point theorem..................... 47
1.4.2 Local invertibility ............................... 48
1.4.3 A few examples................................. 51
1.4.4 A variational proof of the inverse function theorem.. 55
1.4.5 Global invertibility.............................. 56
1.5 Differential Calculus in Banach Spaces .................. 57
1.5.1 Gaˆteaux and Fr´echet differentials ................. 57
a. Gradient .................................... 59
b. Mean value theorem .......................... 60
c. Higher order derivatives and Taylor’s formula .... 61
1.5.2 Local invertibility in Banach spaces ............... 62
1.6 Exercises ............................................ 62
2. Integral Calculus ....................................... 67
2.1 Lebesgue’s Integral ................................... 67
2.1.1 Definitions and properties: a short summary........ 67
a. Lebesgue’s measure........................... 68
b. Measurable functions ......................... 70
c. Lebesgue’s integral ........................... 71
d. Basic properties of Lebesgue’s integral .......... 72
e. The integral as area of the subgraph ............ 74
f. Chebyshev’s inequality ........................ 74
g. Negligible sets and the integral................. 74
h. Riemann integrable functions .................. 75
2.1.2 Fubini’s theorem and reduction to iterated integrals . 76
2.1.3 Change of variables ............................. 78
2.1.4 Differentiation and primitives..................... 78
2.2 Convergence Theorems................................ 81
a. Monotone convergence ........................ 81
b. Dominated convergence ....................... 83
c. Absolute continuity of the integral.............. 86
d. Differentiation under the integral sign........... 86
2.3 Mollifiers and Approximations ......................... 89
a. C0-approximations and Lusin’s theorem......... 89
b. Mollifying in Rn.............................. 91
c. Mollifying in Ω............................... 94
2.4 Calculus of Integrals .................................. 96
2.4.1 Calculus of multiple integrals..................... 96
a. Normal sets.................................. 97
b. Rotational figures ............................ 99
c. Changes of coordinates........................100
Contents ix
d. Measure of the n-dimensional ball ..............104
e. Isodiametric inequality ........................105
f. Euler’s Γ function ............................106
g. Tetrahedrons ................................108
2.4.2 Monte Carlo method ............................110
2.4.3 Differentiation under the integral sign .............111
2.5 Measure and Area ....................................114
2.5.1 Hausdorff’s measures ............................114
2.5.2 Area formula ...................................116
a. Calculus of the area of a surface................119
2.5.3 The coarea formula .............................121
2.6 Gauss–Green Formulas................................123
a. Two simple situations.........................124
b. Admissible sets...............................126
c. Decomposition of unity........................127
d. Gauss–Green formulas ........................128
e. Integration by parts ..........................130
f. The divergence theorem .......................130
g. Geometrical meaning of the divergence..........130
h. Divergence and transport of volume.............131
2.7 Exercises ............................................132
3. Curves and Differential Forms ..........................137
3.1 Differential Forms, Vector Fields, and Work..............137
a. Vector fields and differential forms..............137
b. Curves ......................................138
c. Integration along a curve and work .............140
3.2 Conservative Fields and Potentials......................142
a. Exact differential forms .......................142
3.3 Closed Forms and Irrotational Fields....................145
a. Closed forms.................................145
b. Poincar´e lemma ..............................147
c. Homotopic curves and work....................148
d. Simply connected subsets and closed forms ......150
3.3.1 Pull back of a differential form....................151
3.3.2 Homotopy formula ..............................153
a. Stokes’s theorem in a square ...................153
b. Homotopy formula............................155
3.4 Stokes’s Formula in the Plane..........................156
3.5 Exercises ............................................158
4. Holomorphic Functions .................................159
4.1 Functions from C to C ................................159
a. Complex numbers ............................159
b. Complex derivative ...........................159
c. Cauchy–Riemann equations....................160
4.2 The Fundamental Theorem of Calculus on C.............163