The way of analysis:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boston [u.a.]
Jones and Bartlett
2000
|
| Edition: | Rev. ed. |
| Series: | Jones and Bartlett books in mathematics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIX, 739 S. |
| ISBN: | 0763714976 |
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MARC
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Record in the Search Index
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|---|---|
| adam_text | Titel: The way of analysis
Autor: Strichartz, Robert S
Jahr: 2000
Contents
Preface xiii
1 Preliminaries 1
1.1 The Logic of Quantifiers....................................1
1.1.1 Rules of Quantifiers................................1
1.1.2 Examples............................................4
1.1.3 Exercises ..............................................7
1.2 Infinite Sets..................................................8
1.2.1 Countable Sets......................................8
1.2.2 Uncountable Sets....................................10
1.2.3 Exercises ............................................13
1.3 Proofs........................................................13
1.3.1 How to Discover Proofs............................13
1.3.2 How to Understand Proofs ........................17
1.4 The Rational Number System..............................18
1.5 The Axiom of Choice*......................................21
2 Construction of the Real Number System 25
2.1 Cauchy Sequences ..........................................25
2.1.1 Motivation ..........................................25
2.1.2 The Definition......................................30
2.1.3 Exercises ............................................37
2.2 The Reals as an Ordered Field............................38
2.2.1 Defining Arithmetic................................38
2.2.2 The Field Axioms..................................41
2.2.3 Order................................................45
2.2.4 Exercises ............................................4®
v
vi
Contents
2.3 Limits and Completeness .................
2.3.1 Proof of Completeness..............................50
2.3.2 Square Roots........................................52
_ . KA
2.3.3 Exercises......................
2.4 Other Versions and Visions................................56
2.4.1 Infinite Decimal Expansions........................56
2.4.2 Dedekind Cuts* ....................................59
2.4.3 Non-Standard Analysis*............................63
2.4.4 Constructive Analysis*..............................66
2.4.5 Exercises............................................68
2.5 Summary....................................................69
3 Topology of the Real Line 73
3.1 The Theory of Limits......................................73
3.1.1 Limits, Sups, and Infs..............................73
3.1.2 Limit Points ........................................78
3.1.3 Exercises............................................84
3.2 Open Sets and Closed Sets ................................86
3.2.1 Open Sets............................................86
3.2.2 Closed Sets..........................................91
3.2.3 Exercises............................................98
3.3 Compact Sets................................................99
3.3.1 Exercises......................106
3.4 Summary..........................107
4 Continuous Functions 111
4.1 Concepts of Continuity...................Ill
4.1.1 Definitions .....................Ill
4.1.2 Limits of Functions and Limits of Sequences . . . 119
4.1.3 Inverse Images of Open Sets............121
4.1.4 Related Definitions.................123
4.1.5 Exercises......................125
4.2 Properties of Continuous Functions............127
4.2.1 Basic Properties..................127
4.2.2 Continuous Functions on Compact Domains . . . 131
4.2.3 Monotone Functions................134
4.2.4 Exercises.........................
4.3 Summary..........................................j^q
Contents vii
5 Differential Calculus 143
5.1 Concepts of the Derivative.................143
5.1.1 Equivalent Definitions...............143
5.1.2 Continuity and Continuous Differentiability . . . 148
5.1.3 Exercises......................152
5.2 Properties of the Derivative................153
5.2.1 Local Properties..................153
5.2.2 Intermediate Value and Mean Value Theorems . 157
5.2.3 Global Properties..................162
5.2.4 Exercises......................163
5.3 The Calculus of Derivatives................165
5.3.1 Product and Quotient Rules............165
5.3.2 The Chain Rule ..................168
5.3.3 Inverse Function Theorem.............171
5.3.4 Exercises......................176
5.4 Higher Derivatives and Taylor s Theorem ........177
5.4.1 Interpretations of the Second Derivative.....177
5.4.2 Taylor s Theorem..................181
5.4.3 L Hopital s Rule*..................185
5.4.4 Lagrange Remainder Formula*..........188
5.4.5 Orders of Zeros*..................190
5.4.6 Exercises ......................192
5.5 Summary..........................195
6 Integral Calculus 201
6.1 Integrals of Continuous Functions.............201
6.1.1 Existence of the Integral..............201
6.1.2 Fundamental Theorems of Calculus........207
6.1.3 Useful Integration Formulas............212
6.1.4 Numerical Integration...............214
6.1.5 Exercises ......................217
6.2 The Riemann Integral...................219
6.2.1 Definition of the Integral .............219
6.2.2 Elementary Properties of the Integral......224
6.2.3 Functions with a Countable Number of Discon-
tinuities* ......................227
6.2.4 Exercises ......................231
6.3 Improper Integrals* ....................232
viii
Contents
6.3.1 Definitions and Examples.............232
6.3.2 Exercises ......................235
6.4 Summary..........................236
7 Sequences and Series of Functions 241
7.1 Complex Numbers.....................241
7.1.1 Basic Properties of C................241
7.1.2 Complex-Valued Functions............247
7.1.3 Exercises......................249
7.2 Numerical Series and Sequences..............250
7.2.1 Convergence and Absolute Convergence.....250
7.2.2 Rearrangements ..................256
7.2.3 Summation by Parts*...............260
7.2.4 Exercises ......................262
7.3 Uniform Convergence ...................263
7.3.1 Uniform Limits and Continuity..........263
7.3.2 Integration and Differentiation of Limits.....268
7.3.3 Unrestricted Convergence* ............272
7.3.4 Exercises ......................274
7.4 Power Series.........................276
7.4.1 The Radius of Convergence............276
7.4.2 Analytic Continuation...............281
7.4.3 Analytic Functions on Complex Domains* .... 286
7.4.4 Closure Properties of Analytic Functions* .... 288
7.4.5 Exercises ......................294
7.5 Approximation by Polynomials..............296
7.5.1 Lagrange Interpolation...............296
7.5.2 Convolutions and Approximate Identities .... 297
7.5.3 The Weierstrass Approximation Theorem .... 301
7.5.4 Approximating Derivatives............305
7.5.5 Exercises.........................
7.6 Equicontinuity..........................................3qq
7.6.1 The Definition of Equicontinuity.........309
7.6.2 The Arzela-Ascoli Theorem..................312
7.6.3 Exercises ................................3^
7.7 Summary......................................2^
Contents ix
8 Transcendental Functions 323
8.1 The Exponential and Logarithm .............323
8.1.1 Five Equivalent Definitions............323
8.1.2 Exponential Glue and Blip Functions.......329
8.1.3 Functions with Prescribed Taylor Expansions* . 332
8.1.4 Exercises ......................335
8.2 Trigonometric Functions..................337
8.2.1 Definition of Sine and Cosine...........337
8.2.2 Relationship Between Sines, Cosines, and Com-
plex Exponentials .................344
8.2.3 Exercises ......................349
8.3 Summary..........................350
9 Euclidean Space and Metric Spaces 355
9.1 Structures on Euclidean Space ..............355
9.1.1 Vector Space and Metric Space..........355
9.1.2 Norm and Inner Product .............358
9.1.3 The Complex Case.................364
9.1.4 Exercises......................366
9.2 Topology of Metric Spaces.................368
9.2.1 Open Sets......................368
9.2.2 Limits and Closed Sets...............373
9.2.3 Completeness....................374
9.2.4 Compactness....................377
9.2.5 Exercises ......................384
9.3 Continuous Functions on Metric Spaces.........386
9.3.1 Three Equivalent Definitions...........386
9.3.2 Continuous Functions on Compact Domains . . . 391
9.3.3 Connectedness...................393
9.3.4 The Contractive Mapping Principle .......397
9.3.5 The Stone-Weierstrass Theorem*.........399
9.3.6 Nowhere Differentiable Functions, and Worse* . 403
9.3.7 Exercises ......................409
9.4 Summary..........................412
10 Differential Calculus in Euclidean Space 419
10.1 The Differential.......................419
10.1.1 Definition of Differentiability...........419
X
Contents
10.1.2 Partial Derivatives.................423
10.1.3 The Chain Rule ..................428
10.1.4 Differentiation of Integrals.............432
10.1.5 Exercises......................435
10.2 Higher Derivatives.....................437
10.2.1 Equality of Mixed Partials.............437
10.2.2 Local Extrema...................441
10.2.3 Taylor Expansions.................448
10.2.4 Exercises......................452
10.3 Summary..........................454
11 Ordinary Differential Equations 459
11.1 Existence and Uniqueness.................459
11.1.1 Motivation.....................459
11.1.2 Picard Iteration...................467
11.1.3 Linear Equations..................473
11.1.4 Local Existence and Uniqueness*.........476
11.1.5 Higher Order Equations* .............481
11.1.6 Exercises......................483
11.2 Other Methods of Solution*................485
11.2.1 Difference Equation Approximation .......485
11.2.2 Peano Existence Theorem.............490
11.2.3 Power-Series Solutions...............494
11.2.4 Exercises......................500
11.3 Vector Fields and Flows*.................501
11.3.1 Integral Curves...................501
11.3.2 Hamiltonian Mechanics..............505
11.3.3 First-Order Linear P.D.E. s............506
11.3.4 Exercises......................507
11.4 Summary.............................
12 Fourier Series 5^5
12.1 Origins of Fourier Series..............................5^5
12.1.1 Fourier Series Solutions of P.D.E. s..........515
12.1.2 Spectral Theory ......................^2q
12.1.3 Harmonic Analysis................535
12.1.4 Exercises.............
12.2 Convergence of Fourier Series.....
Contents xi
12.2.1 Uniform Convergence for C1 Functions......531
12.2.2 Summability of Fourier Series...........537
12.2.3 Convergence in the Mean.............543
12.2.4 Divergence and Gibb s Phenomenon* ......550
12.2.5 Solution of the Heat Equation*..........555
12.2.6 Exercises ......................559
12.3 Summary..........................562
13 Implicit Functions, Curves, and Surfaces 567
13.1 The Implicit Function Theorem..............567
13.1.1 Statement of the Theorem.............567
13.1.2 The Proof*.....................573
13.1.3 Exercises......................580
13.2 Curves and Surfaces....................581
13.2.1 Motivation and Examples.............581
13.2.2 Immersions and Embeddings...........585
13.2.3 Parametric Description of Surfaces........591
13.2.4 Implicit Description of Surfaces..........597
13.2.5 Exercises ......................600
13.3 Maxima and Minima on Surfaces.............602
13.3.1 Lagrange Multipliers................602
13.3.2 A Second Derivative Test*.............605
13.3.3 Exercises ......................609
13.4 Arc Length.........................610
13.4.1 Rectifiable Curves.................610
13.4.2 The Integral Formula for Arc Length.......614
13.4.3 Arc Length Parameterization* ..........616
13.4.4 Exercises ......................617
13.5 Summary..........................618
14 The Lebesgue Integral 623
14.1 The Concept of Measure..................623
14.1.1 Motivation.....................623
14.1.2 Properties of Length................627
14.1.3 Measurable Sets..................631
14.1.4 Basic Properties of Measures...........634
14.1.5 A Formula for Lebesgue Measure.........636
14.1.6 Other Examples of Measures...........639
xy Contents
14.1.7 Exercises......................641
14.2 Proof of Existence of Measures*..............643
14.2.1 Outer Measures...................643
14.2.2 Metric Outer Measure...............647
14.2.3 Hausdorff Measures*................650
14.2.4 Exercises......................654
14.3 The Integral.........................655
14.3.1 Non-negative Measurable Functions .......655
14.3.2 The Monotone Convergence Theorem......660
14.3.3 Integrable Functions................664
14.3.4 Almost Everywhere ................667
14.3.5 Exercises......................668
14.4 The Lebesgue Spaces L1 and L2 .............670
14.4.1 L1 as a Banach Space...............670
14.4.2 L2 as a Hilbert Space...............673
14.4.3 Fourier Series for L2 Functions..........676
14.4.4 Exercises......................681
14.5 Summary..........................682
15 Multiple Integrals 691
15.1 Interchange of Integrals..................691
15.1.1 Integrals of Continuous Functions ........691
15.1.2 Fubini s Theorem..................694
15.1.3 The Monotone Class Lemma*...........700
15.1.4 Exercises......................703
15.2 Change of Variable in Multiple Integrals.........705
15.2.1 Determinants and Volume.............705
15.2.2 The Jacobian Factor*..................
15.2.3 Polar Coordinates....................
15.2.4 Change of Variable for Lebesgue Integrals* . 717
15.2.5 Exercises..........................................72q
15.3 Summary............................................^2
Index
727
|
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| spelling | Strichartz, Robert S. Verfasser aut The way of analysis Robert S. Strichartz Rev. ed. Boston [u.a.] Jones and Bartlett 2000 XIX, 739 S. txt rdacontent n rdamedia nc rdacarrier Jones and Bartlett books in mathematics Analyse (wiskunde) gtt Analyse mathématique Mathematical analysis Analysis (DE-588)4001865-9 gnd rswk-swf Analysis (DE-588)4001865-9 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009292927&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
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| title | The way of analysis |
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