Advanced calculus: an introduction to analysis
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York u.a.
Wiley
1969
|
| Ausgabe: | 2. ed. |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XV, 597 S. graph. Darst. |
Internformat
MARC
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| 245 | 1 | 0 | |a Advanced calculus |b an introduction to analysis |
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| 264 | 1 | |a New York u.a. |b Wiley |c 1969 | |
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Datensatz im Suchindex
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| adam_text | Contents
PART I CALCULUS OF ONE VARIABLE i
Chapter 1 The Number System 3
1.1 The Peano axioms 3
1.2 Rational numbers and arithmetic 5
1.3 The real numbers: completeness 8
1.4 Geometry and the number system 12
1.5 Bounded sets 14
1.6 Some points of logic 16
1.7 Absolute value 17
Chapter 2 Functions, Sequences, and Limits 22
2.1 Mappings, functions, and sequences 22
2.2 Limits 26
2.3 Operations with limits (sequences) 33
2.4 Limits of functions 39
2.5 Operations with limits (functions) 44
2.6 Monotone sequences 48
2.7 Monotone functions 50
xi
xii Contents
Chapter 3 Continuity, Differentiability, and More Limits 54
3.1 Continuity. Uniform continuity 54
3.2 Operations with continuous functions 58
3.3 The intermediate value property 59
3.4 Inverse functions 61
3.5 Cluster points. Accumulation points 67
3.6 The Cauchy criterion 71
3.7 Limit superior and limit inferior 76
3.8 Deeper properties of continuous functions 79
3.9 The derivative. Chain rule 83
3.10 The mean value theorem 88
Chapter 4 Integration 99
4.1 Introduction 99
4.2 Preliminary lemmas 101
4.3 The Riemann integral 107
4.4 Properties of the definite integral 116
4.5 The fundamental theorem of calculus 121
4.6 Further properties of integrals 124
Chapter 5 The Elementary Transcendental Functions 129
5.1 The logarithm 129
5.2 The exponential function 132
5.3 The circular functions 136
Chapter 6 Properties of Differential Functions 148
6.1 The Cauchy mean value theorem 148
6.2 L Hospital s rule 150
6.3 Taylor s formula with remainder 755
6.4 Extreme values 161
PART II VECTOR CALCULUS 767
Chapter 7 Vectors and Curves 769
7.1 Introduction and definitions 769
7.2 Vector multiplications 775
7.3 The triple products 182
7.4 Linear independence. Bases. Orientation 186
7.5 Vector analytic geometry 790
Contents xiii
7.6 Vector spaces of other dimensions: En 192
7.7 Vector functions. Curves 196
7.8 Rectifiable curves and arc length 198
7.9 Differentiable curves 201
Chapter 8 Functions of Several Variables. Limits and Continuity 210
8.1 A little topology: open and closed sets 210
8.2 A little more topology: sequences, cluster values,
accumulation points, Cauchy criterion 214
8.3 Limits 220
8.4 Vector functions of a vector 223
8.5 Operations with limits 226
8.6 Continuity 227
8.7 Geometrical picture of a function 230
Chapter 9 Differentiable Functions 234
9.1 Partial derivatives 234
9.2 Differentiability. Total differentials 241
9.3 The gradient vector. The del operator. Directional
derivatives 248
9.4 Composite functions. The chain rule 251
9.5 The mean value theorem and Taylor s theorem for
several variables 259
9.6 The divergence and curl of a vector field 261
Chapter 10 Transformations and Implicit Functions. Extreme Values 267
10.1 Transformations. Inverse transformations 267
10.2 Linear transformations 269
10.3 The inversion theorem 276
10.4 Global inverses 284
10.5 Curvilinear coordinates 285
10.6 Implicit functions 289
10.7 Extreme values 294
10.8 Extreme values under constraints 297
Chapter 11 Multiple Integrals 303
11.1 Integrals over rectangles 303
11.2 Properties of the integral. Classes of integrable functions 310
11.3 Iterated integrals 311
11.4 Integration over regions. Area and volume 316
xiv Contents
Chapter 12 Line and Surface Integrals 325
12.1 Line integrals. Potentials 325
12.2 Green s theorem 336
12.3 Surfaces. Area 345
12A Surface integrals. The divergence theorem 351
12.5 Stokes theorem. Orientable surfaces 357
12.6 Some physical heuristics 363
12.7 Change of variables in multiple integrals 365
PART III THEORY OF CONVERGENCE 373
Chapter 13 Infinite Series 375
13.1 Convergence, absolute and conditional 375
13.2 Series with non negative terms: Comparison tests 379
13.3 Series with non negative terms. Ratio and root tests. 386
Remainders
13.4 Series with variable signs 389
13.5 More delicate tests 392
13.6 Rearrangements 394
13.7 Improvement of convergence 400
Chapter 14 Sequence and Series of Functions. Uniform Convergence 408
14.1 Introduction 408
14.2 Uniform convergence 409
14.3 Consequences of uniform convergence 415
14.4 Abel s and Dirichlet s tests 423
14.5 A theorem of Dini 427
Chapter 15 The Taylor Series 430
15.1 Power series. Interval of convergence 430
15.2 Properties of power series 436
15.3 The Taylor and Maclaurin series 442
15.4 The arithmetic of power series 447
15.5 Substitution and inversion 455
15.6 Complex series 457
15.7 Real analytic functions 460
Contents xv
Chapter 16 Improper Integrals 463
16.1 Improper integrals. Conditional and absolute
convergence 463
16.2 Improper integrals with non negative integrands 471
16.3 The Cauchy principal value 474
16.4 An alternation test 475
16.5 Improper multiple integrals 477
Chapter 17 Integral Representations of Functions 483
17.1 Introduction. Proper integrals 483
17.2 Uniform convergence 487
17.3 Consequences of uniform convergence 493
Chapter 18 Gamma and Beta Functions. Laplace s Method and
Stirling s Formula 509
18.1 The gamma function 509
18.2 The beta function 512
18.3 Laplace s method 575
18.4 Stirling s formula 520
Chapter 19 Fourier Series 523
19.1 Introduction 523
19.2 The class 3$2. Approximation in the mean. Bessel s
inequality 529
19.3 Some useful lemmas 532
19.4 Convergence theorems 536
19.5 Differentiation and integration. Uniform convergence 544
19.6 Sine and cosine series. Change of scale 548
19.7 Improvement of convergence 552
19.8 The Fourier integral 554
19.9 Function spaces. Complete orthonormal sets 560
Elementary Differentiation and Integration Formulas 567
Answers, Hints, and Solutions 569
Index 593
|
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| institution | BVB |
| language | English |
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| physical | XV, 597 S. graph. Darst. |
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| spelling | Fulks, Watson Verfasser aut Advanced calculus an introduction to analysis 2. ed. New York u.a. Wiley 1969 XV, 597 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Analysis swd Analysis (DE-588)4001865-9 gnd rswk-swf Vektoranalysis (DE-588)4191992-0 gnd rswk-swf Konvergenz (DE-588)4032326-2 gnd rswk-swf (DE-588)4151278-9 Einführung gnd-content Analysis (DE-588)4001865-9 s DE-604 Konvergenz (DE-588)4032326-2 s Vektoranalysis (DE-588)4191992-0 s HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=002583100&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Fulks, Watson Advanced calculus an introduction to analysis Analysis swd Analysis (DE-588)4001865-9 gnd Vektoranalysis (DE-588)4191992-0 gnd Konvergenz (DE-588)4032326-2 gnd |
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| title | Advanced calculus an introduction to analysis |
| title_auth | Advanced calculus an introduction to analysis |
| title_exact_search | Advanced calculus an introduction to analysis |
| title_full | Advanced calculus an introduction to analysis |
| title_fullStr | Advanced calculus an introduction to analysis |
| title_full_unstemmed | Advanced calculus an introduction to analysis |
| title_short | Advanced calculus |
| title_sort | advanced calculus an introduction to analysis |
| title_sub | an introduction to analysis |
| topic | Analysis swd Analysis (DE-588)4001865-9 gnd Vektoranalysis (DE-588)4191992-0 gnd Konvergenz (DE-588)4032326-2 gnd |
| topic_facet | Analysis Vektoranalysis Konvergenz Einführung |
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