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 *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 |