Preface

**Part**** I.**** ONE-DIMENSIONAL THEORY**

Â

**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

Â

**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

Â

**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

Â

**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'HÃ´pital's Rule

4.5 Inverse function theorems

Â

**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

Â

**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

Â

**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

Â

**Part II. MULTIDIMENSIONAL THEORY**

Â

**8. Euclidean Spaces**

8.1 Algebraic structure

8.2 Planes and linear transformations

8.3 Topology of *R*^{n}

8.4 Interior, closure, boundary

Â

**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

Â

**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

Â

**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

Â

**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

Â

**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

Â

***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

Â

Appendices

A. Algebraic laws

B. Trigonometry

C. Matrices and determinants

D. Quadric surfaces

E. Vector calculus and physics

F. Equivalence relations

Â

References

Answers and Hints to Exercises

Subject Index

Symbol Index

Â

*Enrichment section