Introductory Analysis The Theory of Calculus

Name: Introductory Analysis The Theory of Calculus
Price: 7.77 USD
Availability: InStock
ISBN: 9780155018457

ISBN-10: 0155018450

ISBN-13: 9780155018457

Edition: 1987

Authors: John A. Fridy

List price: $82.00

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Book details

List price: $82.00
Copyright year: 1987
Publisher: Saunders College Publishing
Binding: Hardcover
Pages: 254
Size: 7.00" wide x 9.75" long x 1.00" tall
Weight: 1.540
Language: English

Introduction: Mathematical Statements and Proofs
Types of Mathematical Statements	p. 1
The Structure of Proofs	p. 2
Ordering of the Real Numbers
The Order Axiom	p. 4
Least Upper Bounds	p. 7
The Density of the Rational Numbers	p. 9
Sequence Limits
Convergent Sequences	p. 11
Algebraic Combinations of Sequences	p. 16
Infinite Limits	p. 19
Subsequences and Limit Points	p. 20
Monotonic Sequences	p. 23
Completeness of the Real Numbers
The Bolzano-Weierstrass Theorem	p. 25
Cauchy Sequences	p. 26
The Nested Intervals Theorem	p. 29
The Heine-Borel Covering Theorem	p. 30
Continuous Functions
Continuity	p. 33
The Sequential Criterion for Continuity	p. 37
Combinations of Continuous Functions	p. 40
One-Sided Continuity	p. 42
Function Limits	p. 43
The Sequential Criterion for Function Limits	p. 46
Variations of Function Limits	p. 48
Consequences of Continuity
The Range of a Continuous Function	p. 50
The Intermediate Value Property	p. 52
Uniform Continuity	p. 54
The Sequential Criterion for Uniform Continuity	p. 59
The Derivative
Difference Quotients	p. 62
The Chain Rule	p. 68
The Law of the Mean	p. 70
Cauchy Law of the Mean	p. 75
Taylor's Formula with Remainder	p. 77
L'Hopital's Rule	p. 79
The Riemann Integral
Riemann Sums and Integrable Functions	p. 87
Basic Properties	p. 91
The Darboux Criterion for Integrability	p. 97
Integrability of Continuous Functions	p. 103
Products of Integrable Functions	p. 106
The Fundamental Theorem of Calculus	p. 110
Improper Integrals
Types of Improper Integrals	p. 115
Integrals over Unbounded Domains	p. 115
Integrals of Unbounded Functions	p. 120
The Gamma Function	p. 123
The Laplace Transform	p. 130
Infinite Series
Convergent and Divergent Series	p. 135
Comparison Tests	p. 139
The Cauchy Condensation Test	p. 141
Elementary Tests	p. 143
Delicate Tests	p. 145
Absolute and Conditional Convergence	p. 149
Regrouping and Rearranging Series	p. 152
Multiplication of Series	p. 156
The Riemann-Stieltjes Integral
Functions of Bounded Variation	p. 163
The Total Variation Function	p. 167
Riemann-Stieltjes Sums and Integrals	p. 169
Integration by Parts	p. 176
Integrability of Continuous Functions	p. 177
Function Sequences
Pointwise Convergence	p. 180
Uniform Convergence	p. 183
Sequences of Continuous Functions	p. 187
Sequences of Integrable Functions	p. 189
Sequences of Differentiable Functions	p. 192
The Weierstrass Approximation Theorem	p. 196
Function Series	p. 200
Power Series
Convergence of Power Series	p. 205
Integration and Differentiation of Power Series	p. 209
Taylor Series	p. 213
The Remainder Term	p. 216
Taylor Series of Some Elementary Functions	p. 219
Metric Spaces and Euclidean Spaces
Metric Spaces	p. 223
Euclidean n-Space	p. 227
Metric Space Topology	p. 230
Connectedness	p. 236
Point Sequences	p. 239
Completeness of E[superscript n]	p. 243
Dense Subsets of E[superscript n]	p. 247
Continuous Transformations
Transformations and Functions	p. 250
Criteria for Continuity	p. 253
The Range of a Continuous Transformation	p. 256
Continuity in E[superscript n]	p. 258
Linear Transformations	p. 261
Differential Calculus in Euclidean Spaces
Patrial Derivatives and Directional Derivatives	p. 267
Differentials and the Approximation Property	p. 270
The Chain Rule	p. 274
The Law of the Mean	p. 278
Mixed Partial Derivatives	p. 279
The Implicit Function Theorem	p. 282
Area and Integration in E[superscript 2]
Integration on a Bounded Set	p. 288
Inner and Outer Area	p. 291
Properties of the Double Integral	p. 297
Line Integrals	p. 299
Independence of Path and Exact Differentials	p. 303
Green's Theorem	p. 308
Analogs of Green's Theorem	p. 312
Mathematical Induction	p. 315
Countable and Uncountable Sets	p. 317
Infinite Products	p. 321
List of Mathematicians	p. 324
Glossary of Symbols	p. 326
Index	p. 331
Table of Contents provided by Syndetics. All Rights Reserved.