Introductory Analysis The Theory of Calculus

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 Statementsp. 1
The Structure of Proofsp. 2
Ordering of the Real Numbers
The Order Axiomp. 4
Least Upper Boundsp. 7
The Density of the Rational Numbersp. 9
Sequence Limits
Convergent Sequencesp. 11
Algebraic Combinations of Sequencesp. 16
Infinite Limitsp. 19
Subsequences and Limit Pointsp. 20
Monotonic Sequencesp. 23
Completeness of the Real Numbers
The Bolzano-Weierstrass Theoremp. 25
Cauchy Sequencesp. 26
The Nested Intervals Theoremp. 29
The Heine-Borel Covering Theoremp. 30
Continuous Functions
Continuityp. 33
The Sequential Criterion for Continuityp. 37
Combinations of Continuous Functionsp. 40
One-Sided Continuityp. 42
Function Limitsp. 43
The Sequential Criterion for Function Limitsp. 46
Variations of Function Limitsp. 48
Consequences of Continuity
The Range of a Continuous Functionp. 50
The Intermediate Value Propertyp. 52
Uniform Continuityp. 54
The Sequential Criterion for Uniform Continuityp. 59
The Derivative
Difference Quotientsp. 62
The Chain Rulep. 68
The Law of the Meanp. 70
Cauchy Law of the Meanp. 75
Taylor's Formula with Remainderp. 77
L'Hopital's Rulep. 79
The Riemann Integral
Riemann Sums and Integrable Functionsp. 87
Basic Propertiesp. 91
The Darboux Criterion for Integrabilityp. 97
Integrability of Continuous Functionsp. 103
Products of Integrable Functionsp. 106
The Fundamental Theorem of Calculusp. 110
Improper Integrals
Types of Improper Integralsp. 115
Integrals over Unbounded Domainsp. 115
Integrals of Unbounded Functionsp. 120
The Gamma Functionp. 123
The Laplace Transformp. 130
Infinite Series
Convergent and Divergent Seriesp. 135
Comparison Testsp. 139
The Cauchy Condensation Testp. 141
Elementary Testsp. 143
Delicate Testsp. 145
Absolute and Conditional Convergencep. 149
Regrouping and Rearranging Seriesp. 152
Multiplication of Seriesp. 156
The Riemann-Stieltjes Integral
Functions of Bounded Variationp. 163
The Total Variation Functionp. 167
Riemann-Stieltjes Sums and Integralsp. 169
Integration by Partsp. 176
Integrability of Continuous Functionsp. 177
Function Sequences
Pointwise Convergencep. 180
Uniform Convergencep. 183
Sequences of Continuous Functionsp. 187
Sequences of Integrable Functionsp. 189
Sequences of Differentiable Functionsp. 192
The Weierstrass Approximation Theoremp. 196
Function Seriesp. 200
Power Series
Convergence of Power Seriesp. 205
Integration and Differentiation of Power Seriesp. 209
Taylor Seriesp. 213
The Remainder Termp. 216
Taylor Series of Some Elementary Functionsp. 219
Metric Spaces and Euclidean Spaces
Metric Spacesp. 223
Euclidean n-Spacep. 227
Metric Space Topologyp. 230
Connectednessp. 236
Point Sequencesp. 239
Completeness of E[superscript n]p. 243
Dense Subsets of E[superscript n]p. 247
Continuous Transformations
Transformations and Functionsp. 250
Criteria for Continuityp. 253
The Range of a Continuous Transformationp. 256
Continuity in E[superscript n]p. 258
Linear Transformationsp. 261
Differential Calculus in Euclidean Spaces
Patrial Derivatives and Directional Derivativesp. 267
Differentials and the Approximation Propertyp. 270
The Chain Rulep. 274
The Law of the Meanp. 278
Mixed Partial Derivativesp. 279
The Implicit Function Theoremp. 282
Area and Integration in E[superscript 2]
Integration on a Bounded Setp. 288
Inner and Outer Areap. 291
Properties of the Double Integralp. 297
Line Integralsp. 299
Independence of Path and Exact Differentialsp. 303
Green's Theoremp. 308
Analogs of Green's Theoremp. 312
Mathematical Inductionp. 315
Countable and Uncountable Setsp. 317
Infinite Productsp. 321
List of Mathematiciansp. 324
Glossary of Symbolsp. 326
Indexp. 331
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