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Elementary classical analysis
Marsden, Jerrold E
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| 자료유형 | 단행본 |
|---|---|
| 서명/저자사항 | Elementary classical analysis / Jerrold E. Marsden, Michael J. Hoffman. |
| 개인저자 | Marsden, Jerrold E. Hoffman, Michael J. |
| 판사항 | 2nd ed. |
| 발행사항 | New York : W.H. Freeman, c1993. |
| 형태사항 | xiv, 738 p. : ill. ; 25 cm. |
| ISBN | 0716721058 9780716721055 |
| 서지주기 | Includes bibliographical references (p. [677]-682) and index. |
| 일반주제명 | Mathematical analysis. |
| 언어 | 영어 |
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| No. | 등록번호 | 청구기호 | 소장처 | 도서상태 | 반납예정일 | 예약 | 매체정보 |
|---|---|---|---|---|---|---|---|
| 1 | 00021133494 | 515 93d | [신촌]도서관/과학기술자료실(중도4층)/ | 대출가능 | |||
| 2 | 00021215019 | 515 93d c.2 | [신촌]도서관/과학기술자료실(중도4층)/ | 대출중 | 2022-03-22 | ||
| 3 | 00021266101 | 515 93d c.3 | [신촌]도서관/과학기술자료실(중도4층)/ | 대출가능 | |||
| 4 | 00098004820 | 수학과 515 93e | [신촌]수학과도서실/과학관226호(수학과전용)/ | 대출가능 | |||
| 5 | 00092012273 | 국제학 515 93a | [신촌]국제학도서관/새천년관3층/ | 대출가능 |
목차
목차 일부
[영문] CONTENTS
PREFACE = ⅸ
PREFACE TO THE FIRST EDITION = xi
INTRODUCTION: SETS AND FUNCTIONS = 1
Supplement on the Axioms of Set Theory = 7
Worked Examples = 18
Exercises = 20
1 THE REAL LINE AND E...
PREFACE = ⅸ
PREFACE TO THE FIRST EDITION = xi
INTRODUCTION: SETS AND FUNCTIONS = 1
Supplement on the Axioms of Set Theory = 7
Worked Examples = 18
Exercises = 20
1 THE REAL LINE AND E...
목차 전체
[영문] CONTENTS
PREFACE = ⅸ
PREFACE TO THE FIRST EDITION = xi
INTRODUCTION: SETS AND FUNCTIONS = 1
Supplement on the Axioms of Set Theory = 7
Worked Examples = 18
Exercises = 20
1 THE REAL LINE AND EUCLIDEAN SPACE = 25
1.1 Ordered Fields and the Number System = 25
1.2 Completeness and the Real Number System = 35
1.3 Least Upper Bounds = 45
1.4 Cauchy Sequences = 49
1.5 Cluster Points; lim inf and lim sup = 52
1.6 Euclidean Space = 57
1.7 Norms, Inner Products, and Metrics = 64
1.8 The Complex Numbers = 70
Theorem Proofs = 79
Worked Examples = 95
Exercises = 97
2 THE TOPOLOGY OF EUCLIDEAN SPACE = 103
2.1 Open Sets = 104
2.2 Interior of a Set = 108
2.3 Closed Sets = 110
2.4 Accumulation Points = 113
2.5 Closure of a Set = 116
2.6 Boundary of a Set = 118
2.7 Sequences = 120
2.8 Completeness = 123
2.9 Series of Real Numbers and Vectors = 125
Theorem Proofs = 130
Worked Examples = 140
Exercises = 143
3 COMPACT AND CONNECTED SETS = 151
3.1 Compactness = 151
3.2 The Heine-Borel Theorem = 155
3.3 Nested Set Property = 157
3.4 Path-Connected Sets = 160
3.5 Connected Sets = 163
Theorem Proofs = 165
Worked Examples = 170
Exercises = 172
4 CONTINUOUS MAPPINGS = 177
4.1 Continuity = 177
4.2 Images of Compact and Connected Sets = 182
4.3 Operations on Continuous Mappings = 184
4.4 The Boundedness of Continuous Functions on Compact Sets = 188
4.5 The Intermediate Value Theorem = 191
4.6 Uniform Continuity = 194
4.7 Differentiation of Functions of One Variable = 196
4.8 Integration of Functions of One Variable = 204
Theorem Proofs = 211
Worked Examples = 227
Exercises = 231
5 UNIFORM CONVERGENCE = 237
5.1 Pointwise and Uniform Convergence = 237
5.2 The Weierstrass M Test = 244
5.3 Integration and Differentiation of Series = 247
5.4 The Elementary Functions = 254
5.5 The Space of Continuous Functions = 268
5.6 The Arzela-Ascoli Theorem = 272
5.7 The Contraction Mapping Principle and Its Applications = 275
5.8 The Stone-Weierstrass Theorem = 283
5.9 The Dirichlet and Abel Tests = 287
5.10 Power Series and Cesaro and Abel Summability = 289
Theorem Proofs = 294
Worked Examples = 313
Exercises = 316
6 DIFFERENTIABLE MAPPINGS = 327
6.1 Definition of the Derivative = 327
6.2 Matrix Representation = 331
6.3 Continuity of Differentiable Mappings; Differentiable Paths = 334
6.4 Conditions for Differentiability = 340
6.5 The Chain Rule = 345
6.6 Product Rule and Gradients = 349
6.7 The Mean Value Theorem = 353
6.8 Taylor's Theorem and Higher Derivatives = 355
6.9 Maxima and Minima = 362
Theorem Proofs = 367
Worked Examples = 380
Exercises = 383
7 THE INVERSE AND IMPLICIT FUNCTION THEOREMS AND RELATED TOPICS = 391
7.1 Inverse Function Theorem = 392
7.2 Implicit Function Theorem = 397
7.3 The Domain-Straightening Theorem = 401
7.4 Further Consequences of the Implicit Function Theorem = 403
7.5 An Existence Theorem for Ordinary Differential Equations = 407
7.6 The Morse Lemma = 411
7.7 Constrained Extrema and Lagrange Multipliers = 414
Theorem Proofs = 420
Worked Examples = 435
Exercises = 438
8 INTEGRATION = 445
8.1 Integrable Functions = 445
8.2 Volume and Sets of Measure Zero = 451
8.3 Lebesgue's Theorem = 454
8.4 Properties of the Integral = 457
8.5 Improper Integrals = 459
8.6 Some Convergence Theorems = 466
8.7 Introduction to Distributions = 469
Theorem Proofs = 472
Worked Examples = 487
Exercises = 490
9 FUBINI'S THEOREM AND THE CHANGE OF VARIABLES FORMULA = 497
9.1 Introduction = 497
9.2 Fubini's Theorem = 500
9.3 Change of Variables Theorem = 505
9.4 Polar Coordinates = 508
9.5 Spherical and Cylindrical Coordinates = 510
9.6 A Note on the Lebesgue Integral = 513
9.7 Interchange of Limiting Operations = 514
Theorem Proofs = 521
Worked Examples = 531
Exercises = 535
10 FOURIER ANALYSIS = 543
10.1 Inner Product Spaces = 545
10.2 Orthogonal Families of Functions = 551
10.3 Completeness and Convergence Theorems = 560
10.4 Functions of Bounded Variation and Fej<TEX>$$\acute e$$</TEX>r Theory (Optional) = 570
10.5 Computation of Fourier Series = 573
10.6 Further Convergence Theorems = 587
10.7 Applications = 593
10.8 Fourier Integrals = 605
10.9 Quantum Mechanical Formalism = 610
Theorem Proofs = 618
Worked Examples = 644
Exercises = 650
APPENDIX A: MISCELLANEOUS EXERCISES = 663
APPENDIX B: REFERENCES AND SUGGESTIONS FOR FURTHER STUDY = 677
APPENDIX C: ANSWERS AND SUGGESTIONS FOR SELECTED ODD-NUMBERED EXERCISES = 683
INDEX = 729
PREFACE = ⅸ
PREFACE TO THE FIRST EDITION = xi
INTRODUCTION: SETS AND FUNCTIONS = 1
Supplement on the Axioms of Set Theory = 7
Worked Examples = 18
Exercises = 20
1 THE REAL LINE AND EUCLIDEAN SPACE = 25
1.1 Ordered Fields and the Number System = 25
1.2 Completeness and the Real Number System = 35
1.3 Least Upper Bounds = 45
1.4 Cauchy Sequences = 49
1.5 Cluster Points; lim inf and lim sup = 52
1.6 Euclidean Space = 57
1.7 Norms, Inner Products, and Metrics = 64
1.8 The Complex Numbers = 70
Theorem Proofs = 79
Worked Examples = 95
Exercises = 97
2 THE TOPOLOGY OF EUCLIDEAN SPACE = 103
2.1 Open Sets = 104
2.2 Interior of a Set = 108
2.3 Closed Sets = 110
2.4 Accumulation Points = 113
2.5 Closure of a Set = 116
2.6 Boundary of a Set = 118
2.7 Sequences = 120
2.8 Completeness = 123
2.9 Series of Real Numbers and Vectors = 125
Theorem Proofs = 130
Worked Examples = 140
Exercises = 143
3 COMPACT AND CONNECTED SETS = 151
3.1 Compactness = 151
3.2 The Heine-Borel Theorem = 155
3.3 Nested Set Property = 157
3.4 Path-Connected Sets = 160
3.5 Connected Sets = 163
Theorem Proofs = 165
Worked Examples = 170
Exercises = 172
4 CONTINUOUS MAPPINGS = 177
4.1 Continuity = 177
4.2 Images of Compact and Connected Sets = 182
4.3 Operations on Continuous Mappings = 184
4.4 The Boundedness of Continuous Functions on Compact Sets = 188
4.5 The Intermediate Value Theorem = 191
4.6 Uniform Continuity = 194
4.7 Differentiation of Functions of One Variable = 196
4.8 Integration of Functions of One Variable = 204
Theorem Proofs = 211
Worked Examples = 227
Exercises = 231
5 UNIFORM CONVERGENCE = 237
5.1 Pointwise and Uniform Convergence = 237
5.2 The Weierstrass M Test = 244
5.3 Integration and Differentiation of Series = 247
5.4 The Elementary Functions = 254
5.5 The Space of Continuous Functions = 268
5.6 The Arzela-Ascoli Theorem = 272
5.7 The Contraction Mapping Principle and Its Applications = 275
5.8 The Stone-Weierstrass Theorem = 283
5.9 The Dirichlet and Abel Tests = 287
5.10 Power Series and Cesaro and Abel Summability = 289
Theorem Proofs = 294
Worked Examples = 313
Exercises = 316
6 DIFFERENTIABLE MAPPINGS = 327
6.1 Definition of the Derivative = 327
6.2 Matrix Representation = 331
6.3 Continuity of Differentiable Mappings; Differentiable Paths = 334
6.4 Conditions for Differentiability = 340
6.5 The Chain Rule = 345
6.6 Product Rule and Gradients = 349
6.7 The Mean Value Theorem = 353
6.8 Taylor's Theorem and Higher Derivatives = 355
6.9 Maxima and Minima = 362
Theorem Proofs = 367
Worked Examples = 380
Exercises = 383
7 THE INVERSE AND IMPLICIT FUNCTION THEOREMS AND RELATED TOPICS = 391
7.1 Inverse Function Theorem = 392
7.2 Implicit Function Theorem = 397
7.3 The Domain-Straightening Theorem = 401
7.4 Further Consequences of the Implicit Function Theorem = 403
7.5 An Existence Theorem for Ordinary Differential Equations = 407
7.6 The Morse Lemma = 411
7.7 Constrained Extrema and Lagrange Multipliers = 414
Theorem Proofs = 420
Worked Examples = 435
Exercises = 438
8 INTEGRATION = 445
8.1 Integrable Functions = 445
8.2 Volume and Sets of Measure Zero = 451
8.3 Lebesgue's Theorem = 454
8.4 Properties of the Integral = 457
8.5 Improper Integrals = 459
8.6 Some Convergence Theorems = 466
8.7 Introduction to Distributions = 469
Theorem Proofs = 472
Worked Examples = 487
Exercises = 490
9 FUBINI'S THEOREM AND THE CHANGE OF VARIABLES FORMULA = 497
9.1 Introduction = 497
9.2 Fubini's Theorem = 500
9.3 Change of Variables Theorem = 505
9.4 Polar Coordinates = 508
9.5 Spherical and Cylindrical Coordinates = 510
9.6 A Note on the Lebesgue Integral = 513
9.7 Interchange of Limiting Operations = 514
Theorem Proofs = 521
Worked Examples = 531
Exercises = 535
10 FOURIER ANALYSIS = 543
10.1 Inner Product Spaces = 545
10.2 Orthogonal Families of Functions = 551
10.3 Completeness and Convergence Theorems = 560
10.4 Functions of Bounded Variation and Fej<TEX>$$\acute e$$</TEX>r Theory (Optional) = 570
10.5 Computation of Fourier Series = 573
10.6 Further Convergence Theorems = 587
10.7 Applications = 593
10.8 Fourier Integrals = 605
10.9 Quantum Mechanical Formalism = 610
Theorem Proofs = 618
Worked Examples = 644
Exercises = 650
APPENDIX A: MISCELLANEOUS EXERCISES = 663
APPENDIX B: REFERENCES AND SUGGESTIONS FOR FURTHER STUDY = 677
APPENDIX C: ANSWERS AND SUGGESTIONS FOR SELECTED ODD-NUMBERED EXERCISES = 683
INDEX = 729
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