Last edition Elsevier For one- or two-semester junior or senior level courses in Advanced Calculus, Analysis I, or Real Analysis. This text prepares students for future courses that use 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 students while encouraging and helping weaker students. Offering readability, practicality and flexibility, Wade presents fundamental theorems and ideas from a practical viewpoint, showing students the motivation behind the mathematics and enabling them to construct their own proofs.
Last Edition
ISBN 13: 9781292039329
Imprint: Pearson Education
Language: English
Authors: William R. Wade
Pub Date: 11/2013
Pages: 656
Illus: Illustrated
Weight: 1,200.000 grams
Size: h 216 x 279 mm
Product Type: Softcover
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- • Flexible presentation, with uniform writing style and notation, covers the material in small sections, allowing instructors to adapt this book to their syllabus.
- • The practical focus explains assumptions so that students learn the motivation behind the mathematics and are able to construct their own proofs.
- • Early introduction of the fundamental goals of analysis Refers and examines how a limit operation interacts with algebraic operation.
- • Optional appendices and enrichment sections enables students to understand the material and allows instructors to tailor their courses.
- • An alternate chapter on metric spaces allows instructors to cover either chapter independently without mentioning the other.
- • More than 200 worked examples and 600 exercises encourage students to test comprehension of concepts, while using techniques in other contexts.
- • Separate coverage of topology and analysis presents purely computational material first, followed by topological material in alternate chapters.
- • Rigorous presentation of integers provides shorter presentations while focusing on analysis.
- • Reorganized coverage of series separates series of constants and series of functions into separate chapters.
- • Consecutive numbering of theorems, definitions and remarks allows students and instructors to find citations easily.
- William R. Wade
- 1. The Real Number System
- 1.1 Introduction 1.2 Ordered field axioms 1.3 Completeness Axiom 1.4 Mathematical Induction 1.5 Inverse functions and images 1.6 Countable and uncountable sets
- 2. Sequences in R
- 2.1 Limits of sequences 2.2 Limit theorems 2.3 Bolzano-Weierstrass Theorem 2.4 Cauchy sequences *2.5 Limits supremum and infimum
- 3. Continuity on R
- 3.1 Two-sided limits 3.2 One-sided limits and limits at infinity 3.3 Continuity 3.4 Uniform continuity
- 4. Differentiability on R
- 4.1 The derivative 4.2 Differentiability theorems 4.3 The Mean Value Theorem 4.4 Taylor's Theorem and l'H?pital's Rule 4.5 Inverse function theorems
- 5 Integrability on R
- 5.1 The Riemann integral 5.2 Riemann sums 5.3 The Fundamental Theorem of Calculus 5.4 Improper Riemann integration *5.5 Functions of bounded variation *5.6 Convex functions
- 6. Infinite Series of Real Numbers
- 6.1 Introduction 6.2 Series with nonnegative terms 6.3 Absolute convergence 6.4 Alternating series *6.5 Estimation of series *6.6 Additional tests
- 7. Infinite Series of Functions
- 7.1 Uniform convergence of sequences 7.2 Uniform convergence of series 7.3 Power series 7.4 Analytic functions *7.5 Applications
- Part II. MULTIDIMENSIONAL THEORY
- 8. Euclidean Spaces
- 8.1 Algebraic structure 8.2 Planes and linear transformations 8.3 Topology of Rn 8.4 Interior, closure, boundary
- 9. Convergence in Rn
- 9.1 Limits of sequences 9.2 Heine-Borel Theorem 9.3 Limits of functions 9.4 Continuous functions *9.5 Compact sets *9.6 Applications
- 10. Metric Spaces
- 10.1 Introduction 10.2 Limits of functions 10.3 Interior, closure, boundary 10.4 Compact sets 10.5 Connected sets 10.6 Continuous functions 10.7 Stone-Weierstrass Theorem
- 11. Differentiability on Rn
- 11.1 Partial derivatives and partial integrals 11.2 The definition of differentiability 11.3 Derivatives, differentials, and tangent planes 11.4 The Chain Rule 11.5 The Mean Value Theorem and Taylor's Formula 11.6 The Inverse Function Theorem *11.7 Optimization
- 12. Integration on Rn
- 12.1 Jordan regions 12.2 Riemann integration on Jordan regions 12.3 Iterated integrals 12.4 Change of variables *12.5 Partitions of unity *12.6 The gamma function and volume
- 13. Fundamental Theorems of Vector Calculus
- 13.1 Curves 13.2 Oriented curves 13.3 Surfaces 13.4 Oriented surfaces 13.5 Theorems of Green and Gauss 13.6 Stokes's Theorem
- *14. Fourier Series
- *14.1 Introduction *14.2 Summability of Fourier series *14.3 Growth of Fourier coefficients *14.4 Convergence of Fourier series *14.5 Uniqueness
- References Answers and Hints to Exercises Subject Index Symbol Index
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