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Full Description
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.
Contents
PrefacePartI.ONE-DIMENSIONAL THEORY1. The Real Number System1.1 Introduction1.2 Ordered field axioms1.3 Completeness Axiom1.4 Mathematical Induction1.5 Inverse functions and images1.6 Countable and uncountable sets2. Sequences in R2.1 Limits of sequences2.2 Limit theorems2.3 Bolzano-Weierstrass Theorem2.4 Cauchy sequences*2.5 Limits supremum and infimum3. Continuity on R3.1 Two-sided limits3.2 One-sided limits and limits at infinity3.3 Continuity3.4 Uniform continuity4. Differentiability on R4.1 The derivative4.2 Differentiability theorems4.3 The Mean Value Theorem4.4 Taylor's Theorem and l'Hopital's Rule4.5 Inverse function theorems5 Integrability on R5.1 The Riemann integral5.2 Riemann sums5.3 The Fundamental Theorem of Calculus5.4 Improper Riemann integration*5.5 Functions of bounded variation*5.6 Convex functions6. Infinite Series of Real Numbers6.1 Introduction6.2 Series with nonnegative terms6.3 Absolute convergence6.4 Alternating series*6.5 Estimation of series*6.6 Additional tests7. Infinite Series of Functions7.1 Uniform convergence of sequences7.2 Uniform convergence of series7.3 Power series7.4 Analytic functions*7.5 ApplicationsPart II. MULTIDIMENSIONAL THEORY8. Euclidean Spaces8.1 Algebraic structure8.2 Planes and linear transformations8.3 Topology of Rn8.4 Interior, closure, boundary9. Convergence in Rn9.1 Limits of sequences9.2 Heine-Borel Theorem9.3 Limits of functions9.4 Continuous functions*9.5 Compact sets*9.6 Applications10. Metric Spaces10.1 Introduction10.2 Limits of functions10.3 Interior, closure, boundary10.4 Compact sets10.5 Connected sets10.6 Continuous functions10.7 Stone-Weierstrass Theorem11. Differentiability on Rn11.1 Partial derivatives and partial integrals11.2 The definition of differentiability11.3 Derivatives, differentials, and tangent planes11.4 The Chain Rule11.5 The Mean Value Theorem and Taylor's Formula11.6 The Inverse Function Theorem*11.7 Optimization12. Integration on Rn12.1 Jordan regions12.2 Riemann integration on Jordan regions12.3 Iterated integrals12.4 Change of variables*12.5 Partitions of unity*12.6 The gamma function and volume13. Fundamental Theorems of Vector Calculus13.1 Curves13.2 Oriented curves13.3 Surfaces13.4 Oriented surfaces13.5 Theorems of Green and Gauss13.6 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 UniquenessReferencesAnswers and Hints to ExercisesSubject IndexSymbol Index*Enrichment section
