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9781577663027

Advanced Calculus

by
  • ISBN13:

    9781577663027

  • ISBN10:

    1577663020

  • Edition: 3rd
  • Format: Hardcover
  • Copyright: 2004-01-01
  • Publisher: Waveland Pr Inc

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Table of Contents

PREFACE ix
TO THE STUDENT xi
CHAPTER 1 SETS AND FUNCTIONS 1(70)
1.1 Introduction
1(1)
1.2 R and R"
2(9)
1.3 Distance
11(8)
1.4 Functions
19(9)
1.5 Topological Terminology
28(9)
1.6 Sequences
37(20)
1.7 Consequences of the Monotonic-Sequence Property
57(7)
1.8 Compact Sets
64(7)
CHAPTER 2 CONTINUITY 71(45)
2.1 Preview
71(1)
2.2 Basic Definitions
72(9)
2.3 Uniform Continuity
81(8)
2.4 Implications of Continuity
89(8)
2.5 Limits of Functions
97(10)
2.6 Discontinuities
107(4)
2.7 Inverses for Function; of One Variable
111(5)
CHAPTER 3 DIFFERENTIATION 116(51)
3.1 Preview
116(1)
3.2 Mean Value Theorems and L'Hospital's Rule
116(9)
3.3 Derivatives for Functions on R"
125(10)
3.4 Differentiation of Composite Functions
135(12)
3.5 Taylor's Theorem
147(7)
3.6 Extremal Problems
154(13)
CHAPTER 4 INTEGRATION 167(60)
4.1 Preview
167(1)
4.2 The Definite Integral
168(11)
4.3 Evaluation of Definite Integrals
179(22)
4.4 Substitution in Multiple Integrals
201(4)
4.5 Improper Integrals
205(22)
CHAPTER 5 SERIES 227(33)
5.1 Preview
227(1)
5.2 Infinite Series
228(8)
5.3 Conditionally Convergent Series
236(9)
5.4 Double Series
245(5)
5.5 Some Sums
250(10)
CHAPTER 6 UNIFORM CONVERGENCE 260(67)
6.1 Preview
260(1)
6.2 Series and Sequences of Functions
261(17)
6.3 Power Series
278(7)
6.4 Improper Integrals with Parameters
285(11)
6.5 The Gamma Function
296(8)
6.6 Fourier Series
304(23)
CHAPTER 7 DIFFERENTIATION OF TRANSFORMATIONS 327(48)
7.1 Preview
327(1)
7.2 Transformations
328(6)
7.3 Linear Functions and Transformations
334(7)
7.4 Differentials of Transformations
341(11)
7.5 Inverses of Transformations
352(10)
7.6 The Implicit Function Theorems
362(5)
7.7 Functional Dependence
367(8)
CHAPTER 8 APPLICATIONS TO GEOMETRY AND ANALYSIS 375(70)
8.1 Preview
375(1)
8.2 Set Functions
376(6)
8.3 Transformations of Multiple Integrals
382(17)
8.4 Curves and Arc Length
399(18)
8.5 Surfaces and Surface Area
417(22)
8.6 Integrals over Curves and Surfaces
439(6)
CHAPTER 9 DIFFERENTIAL GEOMETRY AND VECTOR CALCULUS 445(75)
9.1 Preview
445(1)
9.2 Differential Forms
446(18)
9.3 Vector Analysis
464(13)
9.4 The Theorems of Green, Gauss, and Stokes
477(20)
9.5 Exact Forms and Closed Forms
497(15)
9.6 Applications
512(8)
CHAPTER 10 NUMERICAL METHODS 520(29)
10.1 Preview
520(1)
10.2 Locating Zeros
521(6)
10.3 Fixed-Point Methods
527(8)
10.4 Extremal Problems
535(6)
10.5 Miscellaneous Approximation Methods
541(8)
APPENDIXES 549(39)
1 Logic and Set Theory
549(5)
2 Foundations of the Real Number System
554(8)
3 Linear Algebra
562(5)
4 Applications of Mathematics
567(6)
5 Introduction to Complex Analysis
573(9)
6 Further Topics in Real Analysis
582(6)
SUGGESTED READING 588(3)
HINTS AND ANSWERS 591(19)
LIST OF SYMBOLS 610(3)
INDEX 613

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