## 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.

**CourseSmart textbooks do not include any media or print supplements that come packaged with the bound book.**

## Table of Contents

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'Hôpital'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 *R ^{n}*

8.4 Interior, closure, boundary

**9. Convergence in R^{n}**

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 R^{n}**

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 ***R ^{n}*

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

Appendices

A. Algebraic laws

B. Trigonometry

C. Matrices and determinants

D. Quadric surfaces

E. Vector calculus and physics

F. Equivalence relations

References

Answers and Hints to Exercises

Subject Index

Symbol Index

*Enrichment section

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Introduction to Analysis, Coursesmart eTextbook, 4th Edition

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$72.99 | ISBN-13: 978-0-321-69097-5