vickersa 11.9999 Normal 0 false false false 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 inR; Continuity onR; Differentiability onR;Integrability onR;Infinite Series of Real Numbers; Infinite Series of Functions; MULTIDIMENSIONAL THEORY; Euclidean Spaces; Convergence inR n ;Metric Spaces; Differentiability onR n ;Integration onR n ;Fundamental Theorems of Vector Calculus; Fourier Series For all readers interested in analysis.

**William Wade** received his PhD in harmonic analysis from the University of California—Riverside. 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.

Wade’s 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.

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