Mathematical analysis 1

Cover......Page 1
Mathematical Analysis I......Page 4
Copyright - ISBN 3540403868......Page 5
Preface to the English Edition......Page 6
Preface to the Second Russian Edition......Page 8
Prom the Preface to the First Russian Edition......Page 10
Table of Contents......Page 14
1.1.1 Connectives and Brackets......Page 20
1.1.2 Remarks on Proofs......Page 21
1.1.4 Concluding Remarks......Page 22
1.1.5 Exercises......Page 23
1.2.1 The Concept of a Set......Page 24
1.2.2 The Inclusion Relation......Page 26
1.2.3 Elementary Operations on Sets......Page 27
1.2.4 Exercises......Page 29
1.3.1 The Concept of a Function (Mapping)......Page 30
1.3.2 Elementary Classification of Mappings......Page 34
1.3.3 Composition of Functions and Mutually Inverse Mappings......Page 35
1.3.4 Functions as Relations. The Graph of a Function......Page 38
1.3.5 Exercises......Page 41
1.4.1 The Cardinality of a Set (Cardinal Numbers)......Page 44
1.4.2 Axioms for Set Theory......Page 46
1.4.3 Remarks on the Structure of Mathematical Propositions and Their Expression in the Language of Set Theory......Page 48
1.4.4 Exercises......Page 50
2.1.1 Definition of the Set of Real Numbers......Page 54
2.1.2 Some General Algebraic Properties of Real Numbers......Page 58
2.1.3 The Completeness Axiom and the Existence of a Least Upper (or Greatest Lower) Bound of a Set of Numbers......Page 63
2.2.1 The Natural Numbers and the Principle of Mathematical Induction......Page 65
2.2.2 Rational and Irrational Numbers......Page 68
2.2.3 The Principle of Archimedes......Page 71
2.2.4 The Geometric Interpretation of the Set of Real Numbers and Computational Aspects of Operations with Real Numbers......Page 73
2.2.5 Problems and Exercises......Page 85
2.3 Basic Lemmas Connected with the Completeness of the Real Numbers......Page 89
2.3.2 The Finite Covering Lemma (Borel—Lebesgue Principle, or Heine—Borel Theorem)......Page 90
2.3.3 The Limit Point Lemma (Bolzano—Weierstrass Principle)......Page 91
2.3.4 Problems and Exercises......Page 92
2.4.1 Countable Sets......Page 93
2.4.3 Problems and Exercises......Page 95
3.1.1 Definitions and Examples......Page 98
3.1.2 Properties of the Limit of a Sequence......Page 100
3.1.3 Questions Involving the Existence of the Limit of a Sequence......Page 104
3.1.4 Elementary Facts about Series......Page 114
3.1.5 Problems and Exercises......Page 123
3.2.1 Definitions and Examples......Page 126
3.2.2 Properties of the Limit of a Function......Page 130
3.2.3 The General Definition of the Limit of a Function (Limit over a Base)......Page 146
3.2.4 Existence of the Limit of a Function......Page 150
3.2.5 Problems and Exercises......Page 166
4.1.1 Continuity of a Function at a Point......Page 170
4.1.2 Points of Discontinuity......Page 174
4.2.1 Local Properties......Page 177
4.2.2 Global Properties of Continuous Functions......Page 179
4.2.3 Problems and Exercises......Page 188
5.1.1 Statement of the Problem and Introductory Considerations......Page 192
5.1.2 Functions Differentiable at a Point......Page 197
5.1.3 The Tangent Line; Geometric Meaning of the Derivative and Differential......Page 200
5.1.4 The Role of the Coordinate System......Page 203
5.1.5 Some Examples......Page 204
5.1.6 Problems and Exercises......Page 210
5.2.1 Differentiation and the Arithmetic Operations......Page 212
5.2.2 Differentiation of a Composite Function (chain rule)......Page 215
5.2.3 Differentiation of an Inverse Function......Page 218
5.2.5 Differentiation of a Very Simple Implicit Function......Page 223
5.2.6 Higher-order Derivatives......Page 228
5.2.7 Problems and Exercises......Page 231
5.3.1 Fermat\'s Lemma and Rolle\'s Theorem......Page 233
5.3.2 The theorems of Lagrange and Cauchy on finite increments......Page 235
5.3.3 Taylor\'s Formula......Page 238
5.3.4 Problems and Exercises......Page 251
5.4.1 Conditions for a Function to be Monotonic......Page 255
5.4.2 Conditions for an Interior Extremum of a Function......Page 256
5.4.3 Conditions for a Function to be Convex......Page 262
5.4.4 L\'Hôpital\'s Rule......Page 269
5.4.5 Constructing the Graph of a Function......Page 271
5.4.6 Problems and Exercises......Page 280
5.5.1 Complex Numbers......Page 284
5.5.2 Convergence in C and Series with Complex Terms......Page 287
5.5.3 Euler\'s Formula and the Connections Among the Elementary Functions......Page 292
5.5.4 Power Series Representation of a Function. Analyticity......Page 295
5.5.5 Algebraic Closedness of the Field C of Complex Numbers......Page 301
5.5.6 Problems and Exercises......Page 306
5.6.1 Motion of a Body of Variable Mass......Page 308
5.6.2 The Barometric Formula......Page 310
5.6.3 Radioactive Decay, Chain Reactions, and Nuclear Reactors......Page 312
5.6.4 Falling Bodies in the Atmosphere......Page 314
5.6.5 The Number e and the Function exp x Revisited......Page 316
5.6.6 Oscillations......Page 319
5.6.7 Problems and Exercises......Page 322
5.7.1 The Primitive and the Indefinite Integral......Page 326
5.7.2 The Basic General Methods of Finding a Primitive......Page 328
5.7.3 Primitives of Rational Functions......Page 334
5.7.4 Primitives of the Form / R(cos x, sin x) dx......Page 338
5.7.5 Primitives of the Form / R(x, y(x)) dx......Page 340
5.7.6 Problems and Exercises......Page 343
6.1.1 The Problem and Introductory Considerations......Page 348
6.1.2 Definition of the Riemann Integral......Page 350
6.1.3 The Set of Integrable Functions......Page 352
6.1.4 Problems and Exercises......Page 364
6.2.2 The Integral as an Additive Function of the Interval of Integration......Page 366
6.2.3 Estimation of the Integral, Monotonicity of the Integral, and the Mean-value Theorem......Page 369
6.2.4 Problems and Exercises......Page 377
6.3.1 The Integral and the Primitive......Page 378
6.3.2 The Newton-Leibniz Formula......Page 380
6.3.3 Integration by Parts in the Definite Integral and Taylor\'s Formula......Page 381
6.3.4 Change of Variable in an Integral......Page 383
6.3.5 Some Examples......Page 386
6.3.6 Problems and Exercises......Page 390
6.4.1 Additive Interval Functions and the Integral......Page 393
6.4.2 Arc Length......Page 396
6.4.3 The Area of a Curvilinear Trapezoid......Page 402
6.4.4 Volume of a Solid of Revolution......Page 403
6.4.5 Work and Energy......Page 404
6.4.6 Problems and Exercises......Page 410
6.5.1 Definition, Examples, and Basic Properties of Improper Integrals......Page 412
6.5.2 Convergence of an Improper Integral......Page 417
6.5.3 Improper Integrals with More than one Singularity......Page 424
6.5.4 Problems and Exercises......Page 427
7.1.1 The Set IR^m and the Distance in it......Page 430
7.1.2 Open and Closed Sets in IR^m......Page 432
7.1.3 Compact Sets in IR^m......Page 434
7.1.4 Problems and Exercises......Page 436
7.2.1 The Limit of a Function......Page 437
7.2.2 Continuity of a Function of Several Variables and Properties of Continuous Functions......Page 442
7.2.3 Problems and Exercises......Page 447
8.1.1 IR^m as a Vector Space......Page 448
8.1.2 Linear Transformations L : IR^m -> IR^n......Page 449
8.1.3 The Norm in IR^m......Page 450
8.1.4 The Euclidean Structure on IR^m......Page 452
8.2.1 Differentiability and the Differential of a Function at a Point......Page 453
8.2.2 The Differential and Partial Derivatives of a Real-valued Function......Page 454
8.2.3 Coordinate Representation of the Differential of a Mapping. The Jacobi Matrix......Page 457
8.2.4 Continuity, Partial Derivatives, and Differentiability of a Function at a Point......Page 458
8.3.1 Linearity of the Operation of Differentiation......Page 459
8.3.2 Differentiation of a Composition of Mappings (Chain Rule)......Page 461
8.3.3 Differentiation of an Inverse Mapping......Page 467
8.3.4 Problems and Exercises......Page 468
8.4.1 The Mean-value Theorem......Page 474
8.4.2 A Sufficient Condition for Differentiability of a Function of Several Variables......Page 476
8.4.3 Higher-order Partial Derivatives......Page 477
8.4.4 Taylor\'s Formula......Page 480
8.4.5 Extrema of Functions of Several Variables......Page 482
8.4.6 Some Geometric Images Connected with Functions of Several Variables......Page 489
8.4.7 Problems and Exercises......Page 493
8.5.1 Statement of the Problem and Preliminary Considerations......Page 499
8.5.2 An Elementary Version of the Implicit Function Theorem......Page 501
8.5.3 Transition to the Case of a Relation F(x^1, ..., x^m, y) = 0......Page 505
8.5.4 The Implicit Function Theorem......Page 508
8.5.5 Problems and Exercises......Page 513
8.6.1 The Inverse Function Theorem......Page 517
8.6.2 Local Reduction of a Smooth Mapping to Canonical Form......Page 522
8.6.3 Functional Dependence......Page 527
8.6.4 Local Resolution of a Diffeomorphism into a Composition of Elementary Ones......Page 528
8.6.5 Morse\'s Lemma......Page 531
8.6.6 Problems and Exercises......Page 534
8.7.1 κ-Dimensional Surfaces in IR^n......Page 536
8.7.2 The Tangent Space......Page 541
8.7.3 Extrema with Constraint......Page 546
8.7.4 Problems and Exercises......Page 559
1. Introduction to Analysis (Numbers, Functions, Limits)......Page 564
2. One-variable Differential Calculus......Page 565
3. Integration and Introduction to Several Variables......Page 566
4. Differential Calculus of Several Variables......Page 568
1.1. Introduction to Analysis and One-variable Differential Calculus......Page 570
2.1. Integration. Multivariable Differential Calculus......Page 572
1.3. Classical courses of analysis from the first half of the twentieth century......Page 576
3. Classroom Materials......Page 577
4. Further Reading......Page 578
B......Page 580
C......Page 581
E......Page 582
F......Page 583
I......Page 584
L......Page 585
O......Page 586
P......Page 587
S......Page 588
Z......Page 590
Name Index......Page 592