Features an introduction to advanced calculus and highlights its inherent concepts from linear algebraAdvanced Calculus reflects the unifying role of linear algebra in an effort to smooth readers' transition to advanced mathematics. The book fosters the development of complete theorem-proving skills through abundant exercises while also promoting a sound approach to the study. The traditional theorems of elementary differential and integral calculus are rigorously established, presenting the foundations of calculus in a way that reorients thinking toward modern analysis.Following an introduction dedicated to writing proofs, the book is divided into three parts:Part One explores foundational one-variable calculus topics from the viewpoint of linear spaces, norms, completeness, and linear functionals.Part Two covers Fourier series and Stieltjes integration, which are advanced one-variable topics.Part Three is dedicated to multivariable advanced calculus, including inverse and implicit function theorems and Jacobian theorems for multiple integrals.Numerous exercises guide readers through the creation of their own proofs, and they also put newly learned methods into practice. In addition, a "Test Yourself" section at the end of each chapter consists of short questions that reinforce the understanding of basic concepts and theorems. The answers to these questions and other selected exercises can be found at the end of the book along with an appendix that outlines key terms and symbols from set theory.Guiding readers from the study of the topology of the real line to the beginning theorems and concepts of graduate analysis, Advanced Calculus is an ideal text for courses in advanced calculus and introductory analysis at the upper-undergraduate and beginning-graduate levels. It also serves as a valuable reference for engineers, scientists, and mathematicians.
Author(s): Leonard F. Richardson
Edition: 1
Publisher: Wiley-Interscience
Year: 2008
Language: English
Pages: 414
Tags: Математика;Функциональный анализ;
Contents......Page 7
Preface......Page 13
Acknowledgments......Page 19
Introduction......Page 21
PART I ADVANCED CALCULUS IN ONE VARIABLE......Page 27
1.1 The Real Number System......Page 29
1.2 Limits of Sequences & Cauchy Sequences......Page 34
1.3 The Completeness Axiom and Some Consequences......Page 39
1.4 Algebraic Combinations of Sequences......Page 45
1.5 The Bolzano…Weierstrass Theorem......Page 48
1.6 The Nested Intervals Theorem......Page 50
1.7 The Heine…Borel Covering Theorem......Page 53
1.8 Countability of the Rational Numbers......Page 57
1.9 Test Yourself......Page 63
2.1 Limits of Functions......Page 65
2.2 Continuous Functions......Page 72
2.3 Some Properties of Continuous Functions......Page 76
2.4 Extreme Value Theorem and Its Consequences......Page 81
2.5 The Banach Space C[a,b]......Page 87
2.6 Test Yourself......Page 93
3.1 Definition and Basic Properties......Page 95
3.2 The Darboux Integrability Criterion......Page 102
3.3 Integrals of Uniform Limits......Page 109
3.4 The Cauchy…Schwarz Inequality......Page 116
3.5 Test Yourself......Page 121
4.1 Derivatives and Differentials......Page 125
4.2 The Mean Value Theorem......Page 131
4.3 The Fundamental Theorem of Calculus......Page 136
4.4 Uniform Convergence and the Derivative......Page 140
4.5 Cauchy's Generalized Mean Value Theorem......Page 143
4.6 Taylor's Theorem......Page 148
4.7 Test Yourself......Page 152
5.1 Series of Constants......Page 153
5.2 Convergence Tests for Positive Term Series......Page 160
5.3 Absolute Convergence and Products of Series......Page 164
5.4 The Banach Space l1 and Its Dual Space......Page 174
5.5 Series of Functions: The Weierstrass M-Test......Page 180
5.6 Power Series......Page 184
5.7 Real Analytic Functions and C(infinity) Functions......Page 188
5.8 Weierstrass Approximation Theorem......Page 195
5.9 Test Yourself......Page 200
PART II ADVANCED TOPICS IN ONE VARIABLE......Page 203
6 Fourier Series......Page 205
6.1 The Vibrating String and Trigonometric Series......Page 206
6.2 Euler's Formula and the Fourier Transform......Page 210
6.3 Bessel's Inequality and l2......Page 218
6.4 Uniform Convergence & Riemann Localization......Page 223
6.5 L2-Convergence & the Dual of l2......Page 231
6.6 Test Yourself......Page 238
7 The Riemann…Stieltjes Integral......Page 241
7.1 Functions of Bounded Variation......Page 242
7.2 Riemann…Stieltjes Sums and Integrals......Page 249
7.3 Riemann…Stieltjes Integrability Theorems......Page 254
7.4 The Riesz Representation Theorem......Page 257
7.5 Test Yourself......Page 267
PART III ADVANCED CALCULUS IN SEVERAL VARIABLES......Page 269
8.1 Euclidean Space as a Complete Normed Vector Space......Page 271
8.2 Open Sets and Closed Sets......Page 278
8.3 Compact Sets......Page 282
8.4 Connected Sets......Page 285
8.5 Test Yourself......Page 289
9.1 Limits of Functions......Page 291
9.2 Continuous Functions......Page 296
9.3 Continuous Image of a Compact Set......Page 300
9.4 Continuous Image of a Connected Set......Page 304
9.5 Test Yourself......Page 306
10.1 Linear Transformations and Norms......Page 309
10.2 Differentiable Functions......Page 315
10.3 The Chain Rule in Euclidean Space......Page 324
10.3.1 The Mean Value Theorem......Page 326
10.3.2 Taylor's Theorem......Page 327
10.4 Inverse Functions......Page 331
10.5 Implicit Functions......Page 337
10.6 Tangent Spaces and Lagrange Multipliers......Page 348
10.7 Test Yourself......Page 354
11.1 Definition of the Integral......Page 357
11.2 Lebesgue Null Sets and Jordan Null Sets......Page 364
11.3 Lebesgue's Criterion for Riemann Integrability......Page 368
11.4 Fubini's Theorem......Page 372
11.5 Jacobian Theorem for Change of Variables......Page 377
11.6 Test Yourself......Page 383
A.1 Terminology and Symbols......Page 385
A.2 Paradoxes......Page 389
Problem Solutions......Page 391
References......Page 405
Index......Page 407
