Multivariable advanced calculus

450 Pages·2016·2.575 MB·English

by  Kuttler K

Multivariable Advanced Calculus
Kenneth Kuttler
February 7, 2016
2
Contents
1 Introduction 9
2 Some Fundamental Concepts 11
2.1 Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.2 The Schroder Bernstein Theorem . . . . . . . . . . . . . . . . . . 13
2.1.3 Equivalence Relations . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2 limsup And liminf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 Double Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3 Basic Linear Algebra 25
3.1 Algebra in Fn, Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Subspaces Spans And Bases . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3 Linear Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.4 Block Multiplication Of Matrices . . . . . . . . . . . . . . . . . . . . . . 38
3.5 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.5.1 The Determinant Of A Matrix . . . . . . . . . . . . . . . . . . . 39
3.5.2 The Determinant Of A Linear Transformation . . . . . . . . . . 50
3.6 Eigenvalues And Eigenvectors Of Linear Transformations . . . . . . . . 51
3.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.8 Inner Product And Normed Linear Spaces . . . . . . . . . . . . . . . . . 54
3.8.1 The Inner Product In Fn . . . . . . . . . . . . . . . . . . . . . . 54
3.8.2 General Inner Product Spaces . . . . . . . . . . . . . . . . . . . . 55
3.8.3 Normed Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . 57
3.8.4 The p Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.8.5 Orthonormal Bases . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.8.6 The Adjoint Of A Linear Transformation . . . . . . . . . . . . . 61
3.8.7 Schur’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.9 Polar Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4 Sequences 73
4.1 Vector Valued Sequences And Their Limits . . . . . . . . . . . . . . . . 73
4.2 Sequential Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.3 Closed And Open Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.4 Cauchy Sequences And Completeness . . . . . . . . . . . . . . . . . . . 81
4.5 Shrinking Diameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3
4 CONTENTS
5 Continuous Functions 87
5.1 Continuity And The Limit Of A Sequence . . . . . . . . . . . . . . . . . 90
5.2 The Extreme Values Theorem . . . . . . . . . . . . . . . . . . . . . . . . 91
5.3 Connected Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.4 Uniform Continuity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.5 Sequences And Series Of Functions . . . . . . . . . . . . . . . . . . . . . 96
5.6 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.7 Sequences Of Polynomials, Weierstrass Approximation . . . . . . . . . . 101
5.7.1 The Tietze Extension Theorem . . . . . . . . . . . . . . . . . . . 106
5.8 The Operator Norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.9 Ascoli Arzela Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6 The Derivative 123
6.1 Limits Of A Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.2 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.3 The Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.4 The Matrix Of The Derivative . . . . . . . . . . . . . . . . . . . . . . . 128
6.5 A Mean Value Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.6 Existence Of The Derivative, C1 Functions . . . . . . . . . . . . . . . . 132
6.7 Higher Order Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.8 Ck Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.8.1 Some Standard Notation. . . . . . . . . . . . . . . . . . . . . . . 137
6.9 The Derivative And The Cartesian Product . . . . . . . . . . . . . . . . 138
6.10 Mixed Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . 141
6.11 Implicit Function Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 142
6.11.1 More Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
6.11.2 The Case Of Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
6.12 Taylor’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.12.1 Second Derivative Test . . . . . . . . . . . . . . . . . . . . . . . . 150
6.13 The Method Of Lagrange Multipliers . . . . . . . . . . . . . . . . . . . . 152
6.14 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
7 Measures And Measurable Functions 159
7.1 Compact Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
7.2 An Outer Measure On P(R) . . . . . . . . . . . . . . . . . . . . . . . . 161
7.3 General Outer Measures And Measures . . . . . . . . . . . . . . . . . . 163
7.3.1 Measures And Measure Spaces . . . . . . . . . . . . . . . . . . . 163
7.4 The Borel Sets, Regular Measures . . . . . . . . . . . . . . . . . . . . . 164
7.4.1 Definition of Regular Measures . . . . . . . . . . . . . . . . . . . 164
7.4.2 The Borel Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
7.4.3 Borel Sets And Regularity . . . . . . . . . . . . . . . . . . . . . . 165
7.5 Measures And Outer Measures . . . . . . . . . . . . . . . . . . . . . . . 171
7.5.1 Measures From Outer Measures . . . . . . . . . . . . . . . . . . . 171
7.5.2 Completion Of Measure Spaces . . . . . . . . . . . . . . . . . . . 175
7.6 One Dimensional Lebesgue Stieltjes Measure . . . . . . . . . . . . . . . 178
7.7 Measurable Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
7.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
CONTENTS 5
8 The Abstract Lebesgue Integral 187
8.1 Definition For Nonnegative Measurable Functions . . . . . . . . . . . . . 187
8.1.1 Riemann Integrals For Decreasing Functions. . . . . . . . . . . . 187
8.1.2 The Lebesgue Integral For Nonnegative Functions . . . . . . . . 188
8.2 The Lebesgue Integral For Nonnegative Simple Functions . . . . . . . . 189
8.3 The Monotone Convergence Theorem . . . . . . . . . . . . . . . . . . . 190
8.4 Other Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
8.5 Fatou’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
8.6 The Righteous Algebraic Desires Of The Lebesgue Integral . . . . . . . 192
8.7 The Lebesgue Integral, L1 . . . . . . . . . . . . . . . . . . . . . . . . . . 193
8.8 Approximation With Simple Functions . . . . . . . . . . . . . . . . . . . 197
8.9 The Dominated Convergence Theorem . . . . . . . . . . . . . . . . . . . 199
8.10 Approximation With C (Y) . . . . . . . . . . . . . . . . . . . . . . . . . 201
c
8.11 The One Dimensional Lebesgue Integral . . . . . . . . . . . . . . . . . . 203
8.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
9 The Lebesgue Integral For Functions Of p Variables 213
9.1 π Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
9.2 p Dimensional Lebesgue Measure And Integrals . . . . . . . . . . . . . . 214
9.2.1 Iterated Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
9.2.2 p Dimensional Lebesgue Measure And Integrals . . . . . . . . . . 215
9.2.3 Fubini’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
9.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
9.4 Lebesgue Measure On Rp . . . . . . . . . . . . . . . . . . . . . . . . . . 222
9.5 Mollifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
9.6 The Vitali Covering Theorem . . . . . . . . . . . . . . . . . . . . . . . . 231
9.7 Vitali Coverings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
9.8 Change Of Variables For Linear Maps . . . . . . . . . . . . . . . . . . . 237
9.9 Change Of Variables For C1 Functions . . . . . . . . . . . . . . . . . . . 242
9.10 Change Of Variables For Mappings Which Are Not One To One . . . . 248
9.11 Spherical Coordinates In p Dimensions . . . . . . . . . . . . . . . . . . . 249
9.12 Brouwer Fixed Point Theorem . . . . . . . . . . . . . . . . . . . . . . . 253
9.13 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
10 Degree Theory, An Introduction 265
10.1 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
10.2 Definitions And Elementary( Prop)erties . . . . . . . . . . . . . . . . . . . 267
10.2.1 The Degree For C2 Ω;Rn . . . . . . . . . . . . . . . . . . . . . 268
10.2.2 Definition Of The Degree For Continuous Functions . . . . . . . 274
10.3 Borsuk’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
10.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
10.5 The Product Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
10.6 Integration And The Degree . . . . . . . . . . . . . . . . . . . . . . . . . 293
10.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
11 Integration Of Differential Forms 303
11.1 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
11.2 Some Important Measure Theory . . . . . . . . . . . . . . . . . . . . . . 306
11.2.1 Eggoroff’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 306
11.2.2 The Vitali Convergence Theorem . . . . . . . . . . . . . . . . . . 308
11.3 The Binet Cauchy Formula . . . . . . . . . . . . . . . . . . . . . . . . . 310
11.4 The Area Measure On A Manifold . . . . . . . . . . . . . . . . . . . . . 311
11.5 Integration Of Differential Forms On Manifolds . . . . . . . . . . . . . . 314
6 CONTENTS
11.5.1 The Derivative Of A Differential Form . . . . . . . . . . . . . . . 317
11.6 Stoke’s Theorem And The Orientation Of ∂Ω . . . . . . . . . . . . . . . 317
11.7 Green’s Theorem, An Example . . . . . . . . . . . . . . . . . . . . . . . 321
11.7.1 An Oriented Manifold . . . . . . . . . . . . . . . . . . . . . . . . 321
11.7.2 Green’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
11.8 The Divergence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 325
11.9 Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328
11.10Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332
12 The Laplace And Poisson Equations 335
12.1 Balls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
12.2 Poisson’s Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
12.2.1 Poisson’s Problem For A Ball . . . . . . . . . . . . . . . . . . . . 341
12.2.2 Does It Work In Case f =0? . . . . . . . . . . . . . . . . . . . . 343
12.2.3 The Case Where f ̸=0, Poisson’s Equation . . . . . . . . . . . . 345
12.3 Properties Of Harmonic Functions . . . . . . . . . . . . . . . . . . . . . 348
12.4 Laplace’s Equation For General Sets . . . . . . . . . . . . . . . . . . . . 351
12.4.1 Properties Of Subharmonic Functions . . . . . . . . . . . . . . . 351
12.4.2 Poisson’s Problem Again. . . . . . . . . . . . . . . . . . . . . . . 356
13 The Jordan Curve Theorem 359
14 Line Integrals 371
14.1 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
14.1.1 Length. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
14.1.2 Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373
14.2 The Line Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376
14.3 Simple Closed Rectifiable Curves . . . . . . . . . . . . . . . . . . . . . . 387
14.3.1 The Jordan Curve Theorem . . . . . . . . . . . . . . . . . . . . . 389
14.3.2 Orientation And Green’s Formula . . . . . . . . . . . . . . . . . 393
14.4 Stoke’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398
14.5 Interpretation And Review . . . . . . . . . . . . . . . . . . . . . . . . . 402
14.5.1 The Geometric Description Of The Cross Product . . . . . . . . 402
14.5.2 The Box Product, Triple Product . . . . . . . . . . . . . . . . . . 404
14.5.3 A Proof Of The Distributive Law For The Cross Product . . . . 404
14.5.4 The Coordinate Description Of The Cross Product . . . . . . . . 405
14.5.5 The Integral Over A Two Dimensional Surface . . . . . . . . . . 405
14.6 Introduction To Complex Analysis . . . . . . . . . . . . . . . . . . . . . 407
14.6.1 Basic Theorems, The Cauchy Riemann Equations . . . . . . . . 407
14.6.2 Contour Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 409
14.6.3 The Cauchy Integral . . . . . . . . . . . . . . . . . . . . . . . . . 411
14.6.4 The Cauchy Goursat Theorem . . . . . . . . . . . . . . . . . . . 417
14.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419
15 Hausdorff Measures 429
15.1 Definition Of Hausdorff Measures . . . . . . . . . . . . . . . . . . . . . . 429
15.1.1 Properties Of Hausdorff Measure . . . . . . . . . . . . . . . . . . 430
15.1.2 Hn And m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432
n
15.2 Technical Considerations∗ . . . . . . . . . . . . . . . . . . . . . . . . . . 435
15.2.1 Steiner Symmetrization∗ . . . . . . . . . . . . . . . . . . . . . . . 437
15.2.2 The Isodiametric Inequality∗ . . . . . . . . . . . . . . . . . . . . 439
∗
15.2.3 The Proper Value Of β(n) . . . . . . . . . . . . . . . . . . . . . 439
∗
15.2.4 A Formula For α(n) . . . . . . . . . . . . . . . . . . . . . . . . 440
CONTENTS 7
15.3 Hausdorff Measure And Linear Transformations . . . . . . . . . . . . . . 442
Copyright ⃝c 2007,
8 CONTENTS
Chapter 1
Introduction
This book is directed to people who have a good understanding of the concepts of one
variable calculus including the notions of limit of a sequence and completeness of R. It
develops multivariable advanced calculus.
Inordertodomultivariablecalculuscorrectly,youmustfirstunderstandsomelinear
algebra. Therefore, a condensed course in linear algebra is presented first, emphasizing
those topics in linear algebra which are useful in analysis, not those topics which are
primarily dependent on row operations.
ManytopicscouldbepresentedingreatergeneralitythanIhavechosentodo. Ihave
also attempted to feature calculus, not topology although there are many interesting
topics from topology. This means I introduce the topology as it is needed rather than
using the possibly more efficient practice of placing it right at the beginning in more
generality than will be needed. I think it might make the topological concepts more
memorable by linking them in this way to other concepts.
After the chapter on the n dimensional Lebesgue integral, you can make a choice
between a very general treatment of integration of differential forms based on degree
theory in chapters 10 and 11 or you can follow an independent path through a proof
of a general version of Green’s theorem in the plane leading to a very good version of
Stoke’s theorem for a two dimensional surface by following Chapters 12 and 13. This
approach also leads naturally to contour integrals and complex analysis. I got this idea
from reading Apostol’s advanced calculus book. Finally, there is an introduction to
Hausdorff measures and the area formula in the last chapter.
I have avoided many advanced topics like the Radon Nikodym theorem, represen-
tation theorems, function spaces, and differentiation theory. It seems to me these are
topics for a more advanced course in real analysis. I chose to feature the Lebesgue
integral because I have gone through the theory of the Riemann integral for a function
of n variables and ended up thinking it was too fussy and that the extra abstraction of
the Lebesgue integral was worthwhile in order to avoid this fussiness. Also, it seemed
to me that this book should be in some sense “more advanced” than my calculus book
which does contain in an appendix all this fussy theory.
9
10 CHAPTER 1. INTRODUCTION