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.

Preface Part I. ONE-DIMENSIONAL THEORY 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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