This text prepares readers for fluency with analytic ideas, such as real and complex analysis, partial and ordinary differential equations, numerical analysis, fluid mechanics, and differential geometry. This book is designed to challenge advanced readers while encouraging and helping readers with weaker skills. Offering readability, practicality and flexibility, Wade presents fundamental theorems and ideas from a practical viewpoint, showing readers the motivation behind the mathematics and enabling them to construct their own proofs.
ONE-DIMENSIONAL THEORY; The Real Number System; Sequences in R; Continuity on R; Differentiability on R; Integrability on R; Infinite Series of Real Numbers; Infinite Series of Functions; MULTIDIMENSIONAL THEORY; Euclidean Spaces; Convergence in Rn; Metric Spaces; Differentiability on Rn; Integration on Rn; Fundamental Theorems of Vector Calculus; Fourier Series
For all readers interested in analysis.
| Preface |
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PART I ONE-DIMENSIONAL THEORY |
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1 | (40) |
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1 | (4) |
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5 | (11) |
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16 | (7) |
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23 | (6) |
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Inverse Functions and Images |
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29 | (6) |
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Countable and Uncountable Sets |
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35 | (6) |
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41 | (27) |
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41 | (5) |
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46 | (7) |
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Bolzano-Weierstrass Theorem |
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53 | (5) |
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58 | (3) |
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Limits Supremum and Infimum |
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61 | (7) |
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68 | (30) |
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68 | (8) |
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One-Sided Limits and Limits at Infinity |
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76 | (7) |
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83 | (9) |
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92 | (6) |
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98 | (32) |
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98 | (7) |
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Differentiability Theorems |
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105 | (4) |
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109 | (8) |
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Taylor's Theorem and 1' Hopital's Rule |
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117 | (8) |
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Inverse Function Theorems |
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125 | (5) |
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130 | (54) |
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130 | (11) |
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141 | (11) |
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The Fundamental Theorem of Calculus |
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152 | (11) |
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Improper Riemann Integration |
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163 | (7) |
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Functions of Bounded Variation |
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170 | (5) |
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175 | (9) |
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Infinite Series of Real Numbers |
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184 | (38) |
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184 | (8) |
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Series with Nonnegative Terms |
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192 | (6) |
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198 | (11) |
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209 | (5) |
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214 | (5) |
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219 | (3) |
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Infinite Series of Functions |
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222 | (45) |
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Uniform Convergence of Sequences |
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222 | (8) |
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Uniform Convergence of Series |
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230 | (7) |
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237 | (12) |
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249 | (12) |
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261 | (6) |
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PART II MULTIDIMENSIONAL THEORY |
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267 | (36) |
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267 | (12) |
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Planes and Linear Transformations |
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279 | (9) |
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288 | (9) |
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Interior, Closure, and Boundary |
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297 | (6) |
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303 | (39) |
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303 | (4) |
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307 | (5) |
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312 | (9) |
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321 | (6) |
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327 | (3) |
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330 | (12) |
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342 | (41) |
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342 | (8) |
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350 | (5) |
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Interior, Closure, and Boundary |
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355 | (6) |
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361 | (6) |
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367 | (5) |
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372 | (5) |
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Stone-Weierstrass Theorem |
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377 | (6) |
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383 | (66) |
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Partial Derivatives and Partial Integrals |
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383 | (11) |
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The Definition of Differentiability |
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394 | (9) |
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Derivatives, Differentials, and Tangent Planes |
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403 | (9) |
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412 | (4) |
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The Mean Value Theorem and Taylor's Formula |
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416 | (8) |
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The Inverse Function Theorem |
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424 | (11) |
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435 | (14) |
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449 | (74) |
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449 | (13) |
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Riemann Integration on Jordan Regions |
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462 | (14) |
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476 | (14) |
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490 | (13) |
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503 | (11) |
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The Gamma Function and Volume |
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514 | (9) |
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Fundamental Theorems of Vector Calculus |
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523 | (61) |
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523 | (13) |
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536 | (8) |
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544 | (11) |
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555 | (10) |
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Theorems of Green and Gauss |
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565 | (10) |
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575 | (9) |
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584 | (35) |
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584 | (7) |
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Summability of Fourier Series |
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591 | (7) |
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Growth of Fourier Coefficients |
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598 | (8) |
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Convergence of Fourier Series |
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606 | (6) |
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612 | (7) |
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619 | (27) |
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619 | (5) |
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624 | (5) |
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Matrices and Determinants |
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629 | (8) |
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637 | (4) |
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Vector Calculus and Physics |
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641 | (3) |
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644 | (2) |
| References |
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646 | (1) |
| Answers and Hints to Selected Exercises |
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647 | (20) |
| Subject Index |
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667 | (12) |
| Notation Index |
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679 | |
William Wade received his PhD in harmonic analysis from the University of CaliforniaRiverside. He has been a professor of the Department of Mathematics at the University of Tennessee for more than forty years. During that time, he has received multiple awards including two Fulbright Scholarships, the Chancellor's Award for Research and Creative Achievements, the Dean's Award for Extraordinary Service, and the National Alumni Association Outstanding Teaching Award.
Wades research interests include problems of uniqueness, growth and dyadic harmonic analysis, on which he has published numerous papers, two books and given multiple presentations on three continents. His current publication, An Introduction to Analysis,is now in its fourth edition.
In his spare time, Wade loves to travel and take photographs to document his trips. He is also musically inclined, and enjoys playing classical music, mainly baroque on the trumpet, recorder, and piano.
