Advanced Calculus

Cover......Page 1
Series......Page 2
Title page......Page 3
Date-line......Page 4
Contents......Page 5
PREFACE......Page 9
TO THE STUDENT......Page 11
1.1 Introduction......Page 15
1.2 $\\mathbb{R}$ and $\\mathbb{R}^n$......Page 16
1.3 Distance......Page 25
1.4 Functions......Page 33
1.5 Topological Terminology......Page 42
1.6 Sequences......Page 51
1.7 Consequences of the Monotonic-Sequence Property......Page 71
1.8 Compact Sets......Page 78
2.1 Preview......Page 85
2.2 Basic Definitions......Page 86
2.3 Uniform Continuity......Page 95
2.4 Implications of Continuity......Page 103
2.5 Limits of Functions......Page 111
2.6 Discontinuities......Page 121
2.7 Inverses for Functions of One Variable......Page 125
3.2 Mean Value Theorems and L\'Hospital\'s Rule......Page 130
3.3 Derivatives for Functions on $\\mathbb{R}^n$......Page 139
3.4 Differentiation of Composite Functions......Page 149
3.5 Taylor\'s Theorem......Page 161
3.6 Extremal Problems......Page 168
4.1 Preview......Page 181
4.2 The Definite Integral......Page 182
4.3 Evaluation of Definite Integrals......Page 193
4.4 Substitution in Multiple Integrals......Page 215
4.5 Improper Integrals......Page 219
5.1 Preview......Page 241
5.2 Infinite Series......Page 242
5.3 Conditionally Convergent Series......Page 250
5.4 Double Series......Page 259
5.5 Some Sums......Page 264
6.1 Preview......Page 274
6.2 Series and Sequences of Functions......Page 275
6.3 Power Series......Page 292
6.4 Improper Integrals with Parameters......Page 299
6.5 The Gamma Function......Page 310
6.6 Fourier Series......Page 318
7.1 Preview......Page 341
7.2 Transformations......Page 342
7.3 Linear Functions and Transformations......Page 348
7.4 Differentials of Transformations......Page 355
7.5 Inverses of Transformations......Page 366
7.6 The Implicit Function Theorems......Page 376
7.7 Functional Dependence......Page 381
8.1 Preview......Page 389
8.2 Set Functions......Page 390
8.3 Transformations of Multiple Integrals......Page 396
8.4 Curves and Arc Length......Page 413
8.5 Surfaces arid Surface Area......Page 431
8.6 Integrals over Curves and Surfaces......Page 453
9.1 Preview......Page 459
9.2 Differential Forms......Page 460
9.3 Vector Analysis......Page 478
9.4 The Theorems of Green, Gauss, and Stokes......Page 491
9.5 Exact Forms and Closed Forms......Page 511
9.6 Applications......Page 526
10.1 Preview......Page 534
10.2 Locating Zeros......Page 535
10.3 Fixed-Point Methods......Page 541
10.4 Extremal Problems......Page 549
10.5 Miscellaneous Approximation Methods......Page 555
1 Logic and Set Theory......Page 563
2 Foundations of the Real Number System......Page 568
3 Linear Algebra......Page 576
4 Applications of Mathematics......Page 581
5 Introduction to Complex Analysis......Page 587
6 Further Topics in Real Analysis......Page 596
SUGGESTED READING......Page 602
HINTS AND ANSWERS......Page 605
LIST OF SYMBOLS......Page 624
INDEX......Page 627