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Multivariable advanced calculus PDF

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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

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