Management of The Object-Oriented Development Process [UML]

Cover......Page 1
Title......Page 4
Copyright Page......Page 5
Foreword to the First Edition......Page 6
Foreword to the Second Edition......Page 10
Contents......Page 14
Part One: Review of Calculus......Page 20
1.  Sets......Page 22
2.  Mappings......Page 23
3. Natural Numbers  and  Induction......Page 27
4.  Denumerable  Sets......Page 30
5.  Equivalence Relations......Page 34
1. Algebraic Axioms......Page 36
2.  Ordering Axioms......Page 40
3.  Integers  and Rational Numbers......Page 44
4. The Completeness Axiom......Page 48
1.  Sequences  of Numbers......Page 53
2.  Functions  and Limits......Page 60
3.  Limits with  Infinity......Page 69
4.  Continuous Functions......Page 78
1.  Properties  of  the Derivative......Page 85
2.  Mean Value Theorem......Page 89
3.  Inverse Functions......Page 93
1.  Exponential......Page 97
2.  Logarithm......Page 102
3.  Sine  and Cosine......Page 109
4.  Complex Numbers......Page 114
1.  Characterization of the  Integral......Page 120
2.  Properties  of  the  Integral......Page 123
3.  Taylor\'s Formula......Page 128
4.  Asymptotic Estimates and  Stirling\'s Formula......Page 135
Part  Two: Convergence......Page 146
1. Vector Spaces......Page 148
2.  Normed Vector Spaces......Page 150
3.  n-Space  and Function  Spaces......Page 156
4.  Completeness......Page 162
5.  Open  and Closed  Sets......Page 170
1.  Basic Properties......Page 179
2.  Continuous Maps......Page 189
3.  Limits  in  Function  Spaces......Page 198
4.  Completion  of a Normed Vector Space......Page 207
1.  Basic Properties  of Compact  Sets......Page 212
2.  Continuous Maps  on Compact  Sets......Page 216
3.  Algebraic  Closure  of the Complex Numbers......Page 220
4.  Relation with Open  Coverings......Page 222
1. Basic Definitions......Page 225
2.  Series  of Positive Numbers......Page 227
3.  Non-Absolute Convergence......Page 236
4.  Absolute Convergence  in Vector Spaces......Page 244
5.  Absolute  and Uniform Convergence......Page 248
6.  Power Series......Page 253
7.  Differentiation  and  Integration  of Series......Page 258
1.  Extension Theorem  for Linear Maps......Page 265
2.  Integral  of Step Maps......Page 267
3.  Approximation  by  Step Maps......Page 271
4.  Properties  of the  Integral......Page 274
Appendix.  The Lebesgue  Integral......Page 281
5.  The Derivative......Page 286
6.  Relation  Between  the  Integral  and  the Derivative......Page 291
7.  Interchanging Derivatives  and  Integrals......Page 294
Part Three: Applications  of  the  Integral......Page 300
1.  Dirac  Sequences......Page 302
2.  The Weierstrass Theorem......Page 306
1. Hermitian  Products  and Orthogonality......Page 310
2.  Trigonometric Polynomials  as a  Total  Family......Page 325
3.  Explicit Uniform  Approximation......Page 330
4.  Pointwise Convergence......Page 336
1. Definition......Page 345
2.  Criteria  for Convergence......Page 349
3.  Interchanging Derivatives  and  Integrals......Page 355
4.  The Heat Kernel......Page 366
1. The  Schwartz  Space......Page 372
2.  The  Fourier Inversion  Formula......Page 378
3.  An Example  of Fourier Transform  not  in  the  Schwartz  Space......Page 382
Part Four: Calculus  in Vector  Spaces......Page 388
1.  Partial Derivatives......Page 390
2.  Differentiability  and  the Chain Rule......Page 398
3.  Potential  Functions......Page 407
4.  Curve  Integrals......Page 414
5.  Taylor\'s Formula......Page 424
6.  Maxima  and  the Derivative......Page 430
CHAPTER  XVI: The Winding  Number  and  Global  Potential  Functions......Page 436
1. Another Description  of  the  Integral Along a  Path......Page 437
2.  The Winding Number and Homology......Page 439
3.  Proof of the Global  Integrability Theorem......Page 451
4.  The  Integral Over Continuous Paths......Page 457
5.  The Homotopy Form  of the  Integrability  Theorem......Page 463
6. More on Homotopies......Page 469
1. The  Space  of Continuous Linear Maps......Page 474
2.  The Derivative as a  Linear Map......Page 482
3.  Properties of the Derivative......Page 487
4. Mean Value Theorem......Page 492
5.  The  Second Derivative......Page 496
6.  Higher Derivatives  and Taylor\'s Formula......Page 506
7.  Partial Derivatives......Page 514
8.  Differentiating Under  the  Integral  Sign......Page 518
1. The Shrinking  Lemma......Page 521
2.  Inverse Mappings,  Linear Case......Page 525
3.  The  Inverse Mapping Theorem......Page 531
4.  Implicit  Functions  and Charts......Page 539
5.  Product Decompositions......Page 545
1.  Local Existence  and Uniqueness......Page 557
2.  Approximate  Solutions......Page 567
3.  Linear Differential  Equations......Page 571
4.  Dependence  on  Initial  Conditions......Page 576
Part Five: Multiple Integration......Page 582
1.  Elementary Multiple  Integration......Page 584
2.  Criteria  for Admissibility......Page 597
3.  Repeated  Integrals......Page 600
4.  Change of Variables......Page 603
5.  Vector Fields  on  Spheres......Page 621
1. Definitions......Page 626
2.  Stokes\' Theorem  for a Rectangle......Page 632
3.  Inverse  Image  of a  Form......Page 635
4.  Stokes\' Formula  for Simplices......Page 639
Appendix......Page 646
Index......Page 654