A first course in real analysis:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York ; Berlin ; Heidelberg ; Barcelona ; Budapest ; Hong Kong ; London ; Milan ; Paris ; Santa Clara ; Singapore ; Tokyo
Springer
1997
|
| Ausgabe: | 2. ed., corr. 5. printing |
| Schriftenreihe: | Undergraduate texts in mathematics
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XVIII, 534 S. 143 graph. Darst. |
| ISBN: | 3540974377 0387974377 |
Internformat
MARC
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| 245 | 1 | 0 | |a A first course in real analysis |c Murray H. Protter ; Charles B. Morrey, Jr. |
| 250 | |a 2. ed., corr. 5. printing | ||
| 264 | 1 | |a New York ; Berlin ; Heidelberg ; Barcelona ; Budapest ; Hong Kong ; London ; Milan ; Paris ; Santa Clara ; Singapore ; Tokyo |b Springer |c 1997 | |
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Datensatz im Suchindex
| _version_ | 1804138151411712000 |
|---|---|
| adam_text | Contents
Preface to the Second Edition vii
Preface to the First Edition xi
CHAPTER 1
The Real Number System 1
1.1 Axioms for a Field 1
1.2 Natural Numbers and Sequences 9
1.3 Inequalities 15
1.4 Mathematical Induction 25
CHAPTER 2
Continuity and Limits 30
2.1 Continuity 30
2.2 Limits 35
2.3 One-Sided Limits 42
2.4 Limits at Infinity; Infinite Limits 48
2.5 Limits of Sequences 55
CHAPTER 3
Basic Properties of Functions on R1 59
3.1 The Intermediate-Value Theorem 59
3.2 Least Upper Bound; Greatest Lower Bound 62
3.3 The Bolzano-Weierstrass Theorem 68
3.4 The Boundedness and Extreme-Value Theorems 70
3.5 Uniform Continuity 72
3.6 The Cauchy Criterion 75
3.7 The Heine-Borel and Lebesgue Theorems 77
xv
xvi Contents
CHAPTER 4
Elementary Theory of Differentiation 83
4.1 The Derivative in R1 83
4.2 Inverse Functions in R1 94
CHAPTER 5
Elementary Theory of Integration 98
5.1 The Darboux Integral for Functions on R1 98
5.2 The Riemann Integral 111
5.3 The Logarithm and Exponential Functions 117
5.4 Jordan Content and Area 122
CHAPTER 6
Elementary Theory of Metric Spaces 130
6.1 The Schwarz and Triangle Inequalities; Metric Spaces 130
6.2 Elements of Point Set Topology 136
6.3 Countable and Uncountable Sets 145
6.4 Compact Sets and the Heine-Borel Theorem 150
6.5 Functions on Compact Sets 157
6.6 Connected Sets 161
6.7 Mappings from One Metric Space to Another 164
CHAPTER 7
Differentiation in UN 173
7.1 Partial Derivatives and the Chain Rule 173
7.2 Taylor s Theorem; Maxima and Minima 178
7.3 The Derivative in W 188
CHAPTER 8
Integration in UN 194
8.1 Volume in R* 194
8.2 The Darboux Integral in UN 197
8.3 The Riemann Integral in R 203
CHAPTER 9
Infinite Sequences and Infinite Series 211
9.1 Tests for Convergence and Divergence 211
9.2 Series of Positive and Negative Terms; Power Series 216
9.3 Uniform Convergence of Sequences 222
9.4 Uniform Convergence of Series; Power Series 230
9.5 Unordered Sums 241
9.6 The Comparison Test for Unordered Sums; Uniform Convergence 250
9.7 Multiple Sequences and Series 254
Contents xvii
CHAPTER 10
Fourier Series 263
10.1 Expansions of Periodic Functions 263
10.2 Sine Series and Cosine Series; Change of Interval 270
10.3 Convergence Theorems 275
CHAPTER 11
Functions Defined by Integrals; Improper Integrals 285
11.1 The Derivative of a Function Defined by an Integral; the Leibniz Rule 285
11.2 Convergence and Divergence of Improper Integrals 290
11.3 The Derivative of Functions Defined by Improper Integrals;
the Gamma Function 295
CHAPTER 12
The Riemann-Stieltjes Integral and Functions of Bounded Variation 305
12.1 Functions of Bounded Variation 305
12.2 The Riemann-Stieltjes Integral 316
CHAPTER 13
Contraction Mappings, Newton s Method, and
Differential Equations 329
13.1 A Fixed Point Theorem and Newton s Method 329
13.2 Application of the Fixed Point Theorem to Differential Equations 335
CHAPTER 14
Implicit Function Theorems and Lagrange Multipliers 341
14.1 The Implicit Function Theorem for a Single Equation 341
14.2 The Implicit Function Theorem for Systems 348
14.3 Change of Variables in a Multiple Integral 359
14.4 The Lagrange Multiplier Rule 369
CHAPTER 15
Functions on Metric Spaces; Approximation 374
15.1 Complete Metric Spaces 374
15.2 Convex Sets and Convex Functions 381
15.3 Arzela s Theorem; the Tietze Extension Theorem 393
15.4 Approximations and the Stone-Weierstrass Theorem 403
CHAPTER 16
Vector Field Theory; the Theorems of Green and Stokes 413
16.1 Vector Functions on Rl 413
16.2 Vector Functions and Fields on R 423
xviii Contents
16.3 Line Integrals in Us 434
16.4 Green s Theorem in the Plane 445
16.5 Surfaces in (R3; Parametric Representation 455
16.6 Area of a Surface in U3; Surface Integrals 461
16.7 Orientable Surfaces 471
16.8 The Stokes Theorem 477
16.9 The Divergence Theorem 486
Appendixes 495
Appendix 1 Absolute Value 495
Appendix 2 Solution of Algebraic Inequalities 499
Appendix 3 Expansions of Real Numbers in Any Base 503
Appendix 4 Vectors in EN 507
Answers to Odd-Numbered Problems 515
Index 529
|
| adam_txt |
Contents
Preface to the Second Edition vii
Preface to the First Edition xi
CHAPTER 1
The Real Number System 1
1.1 Axioms for a Field 1
1.2 Natural Numbers and Sequences 9
1.3 Inequalities 15
1.4 Mathematical Induction 25
CHAPTER 2
Continuity and Limits 30
2.1 Continuity 30
2.2 Limits 35
2.3 One-Sided Limits 42
2.4 Limits at Infinity; Infinite Limits 48
2.5 Limits of Sequences 55
CHAPTER 3
Basic Properties of Functions on R1 59
3.1 The Intermediate-Value Theorem 59
3.2 Least Upper Bound; Greatest Lower Bound 62
3.3 The Bolzano-Weierstrass Theorem 68
3.4 The Boundedness and Extreme-Value Theorems 70
3.5 Uniform Continuity 72
3.6 The Cauchy Criterion 75
3.7 The Heine-Borel and Lebesgue Theorems 77
xv
xvi Contents
CHAPTER 4
Elementary Theory of Differentiation 83
4.1 The Derivative in R1 83
4.2 Inverse Functions in R1 94
CHAPTER 5
Elementary Theory of Integration 98
5.1 The Darboux Integral for Functions on R1 98
5.2 The Riemann Integral 111
5.3 The Logarithm and Exponential Functions 117
5.4 Jordan Content and Area 122
CHAPTER 6
Elementary Theory of Metric Spaces 130
6.1 The Schwarz and Triangle Inequalities; Metric Spaces 130
6.2 Elements of Point Set Topology 136
6.3 Countable and Uncountable Sets 145
6.4 Compact Sets and the Heine-Borel Theorem 150
6.5 Functions on Compact Sets 157
6.6 Connected Sets 161
6.7 Mappings from One Metric Space to Another 164
CHAPTER 7
Differentiation in UN 173
7.1 Partial Derivatives and the Chain Rule 173
7.2 Taylor's Theorem; Maxima and Minima 178
7.3 The Derivative in W 188
CHAPTER 8
Integration in UN 194
8.1 Volume in R* 194
8.2 The Darboux Integral in UN 197
8.3 The Riemann Integral in R" 203
CHAPTER 9
Infinite Sequences and Infinite Series 211
9.1 Tests for Convergence and Divergence 211
9.2 Series of Positive and Negative Terms; Power Series 216
9.3 Uniform Convergence of Sequences 222
9.4 Uniform Convergence of Series; Power Series 230
9.5 Unordered Sums 241
9.6 The Comparison Test for Unordered Sums; Uniform Convergence 250
9.7 Multiple Sequences and Series 254
Contents xvii
CHAPTER 10
Fourier Series 263
10.1 Expansions of Periodic Functions 263
10.2 Sine Series and Cosine Series; Change of Interval 270
10.3 Convergence Theorems 275
CHAPTER 11
Functions Defined by Integrals; Improper Integrals 285
11.1 The Derivative of a Function Defined by an Integral; the Leibniz Rule 285
11.2 Convergence and Divergence of Improper Integrals 290
11.3 The Derivative of Functions Defined by Improper Integrals;
the Gamma Function 295
CHAPTER 12
The Riemann-Stieltjes Integral and Functions of Bounded Variation 305
12.1 Functions of Bounded Variation 305
12.2 The Riemann-Stieltjes Integral 316
CHAPTER 13
Contraction Mappings, Newton's Method, and
Differential Equations 329
13.1 A Fixed Point Theorem and Newton's Method 329
13.2 Application of the Fixed Point Theorem to Differential Equations 335
CHAPTER 14
Implicit Function Theorems and Lagrange Multipliers 341
14.1 The Implicit Function Theorem for a Single Equation 341
14.2 The Implicit Function Theorem for Systems 348
14.3 Change of Variables in a Multiple Integral 359
14.4 The Lagrange Multiplier Rule 369
CHAPTER 15
Functions on Metric Spaces; Approximation 374
15.1 Complete Metric Spaces 374
15.2 Convex Sets and Convex Functions 381
15.3 Arzela's Theorem; the Tietze Extension Theorem 393
15.4 Approximations and the Stone-Weierstrass Theorem 403
CHAPTER 16
Vector Field Theory; the Theorems of Green and Stokes 413
16.1 Vector Functions on Rl 413
16.2 Vector Functions and Fields on R" 423
xviii Contents
16.3 Line Integrals in Us 434
16.4 Green's Theorem in the Plane 445
16.5 Surfaces in (R3; Parametric Representation 455
16.6 Area of a Surface in U3; Surface Integrals 461
16.7 Orientable Surfaces 471
16.8 The Stokes Theorem 477
16.9 The Divergence Theorem 486
Appendixes 495
Appendix 1 Absolute Value 495
Appendix 2 Solution of Algebraic Inequalities 499
Appendix 3 Expansions of Real Numbers in Any Base 503
Appendix 4 Vectors in EN 507
Answers to Odd-Numbered Problems 515
Index 529 |
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| spelling | Protter, Murray H. 1918-2008 Verfasser (DE-588)1102930148 aut A first course in real analysis Murray H. Protter ; Charles B. Morrey, Jr. 2. ed., corr. 5. printing New York ; Berlin ; Heidelberg ; Barcelona ; Budapest ; Hong Kong ; London ; Milan ; Paris ; Santa Clara ; Singapore ; Tokyo Springer 1997 XVIII, 534 S. 143 graph. Darst. txt rdacontent n rdamedia nc rdacarrier Undergraduate texts in mathematics Mathematical analysis Analysis (DE-588)4001865-9 gnd rswk-swf 1\p (DE-588)4151278-9 Einführung gnd-content Analysis (DE-588)4001865-9 s DE-604 Morrey, Charles B. 1907-1984 Verfasser (DE-588)172264332 aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016834565&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Protter, Murray H. 1918-2008 Morrey, Charles B. 1907-1984 A first course in real analysis Mathematical analysis Analysis (DE-588)4001865-9 gnd |
| subject_GND | (DE-588)4001865-9 (DE-588)4151278-9 |
| title | A first course in real analysis |
| title_auth | A first course in real analysis |
| title_exact_search | A first course in real analysis |
| title_exact_search_txtP | A first course in real analysis |
| title_full | A first course in real analysis Murray H. Protter ; Charles B. Morrey, Jr. |
| title_fullStr | A first course in real analysis Murray H. Protter ; Charles B. Morrey, Jr. |
| title_full_unstemmed | A first course in real analysis Murray H. Protter ; Charles B. Morrey, Jr. |
| title_short | A first course in real analysis |
| title_sort | a first course in real analysis |
| topic | Mathematical analysis Analysis (DE-588)4001865-9 gnd |
| topic_facet | Mathematical analysis Analysis Einführung |
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