Advanced Calculus of real valued functions of real variable and vectored valued functions of a vector 0

17. Index......Page 0
1.1 Set Language-A mathematical Shorthand......Page 2
 1.2 Generalized Unions and Intersections; DeMorgan's Law......Page 7
1.3 The Russell Paradox*......Page 8
1.4 Cartesian Products and Functions......Page 9
1.5 The Inverse Function......Page 15
1.6 The Field of Rational Numbers......Page 21
1.7 Ordered Fields......Page 24
1.8 The Real Numbers as a Complete Ordered Field......Page 29
1.9 Properties of the Least Upper Bound......Page 33
1.10 Representations of the Real Numbers by Decimals*......Page 37
1.11 Existence and Uniqueness of a Complete Ordered Field*......Page 42
1.12 The Hierarchy of Numbers......Page 47
1.13 Cardinal Numbers......Page 49
1.14 The Cantor Set*......Page 56
2.15 Limit of a Sequence......Page 68
2.16 Three Fundamental Properties of Convergent Sequences......Page 71
2.17 Manipulation of Sequences......Page 73
2.18 Monotonic Sequences......Page 78
2.19 The Geometric Series......Page 80
2.20 The Number Line......Page 83
2.21 The Nested-Interval Property of R......Page 85
2.22 The Location of the Rationals and Irrationals Relative to the Reals......Page 88
2.23  Accumulation Point......Page 90
2.24 Limit of a Function......Page 93
2.25 Limit of a Function-Neighborhood Definition......Page 97
2.26 Continuous Functions......Page 100
2.27 Properties of Continuous Functions......Page 105
2.28 Some Elementary Point Set Topology......Page 110
2.29 Unions and Intersections of Open Sets and Closed Sets......Page 113
2.30 The Bolzano-Weierstrass Theorem......Page 117
2.31 The Heine-Borel Property......Page 118
2.32 Compact Sets......Page 120
2.33 Uniformly Continuous Functions......Page 123
2.34 The Continuous Image of a Compact Set......Page 126
3.35 Definition of the Derivative......Page 130
3.36 Differentiation of a Composite Function......Page 134
3.37 Mean Value Theorems......Page 138
3.38 The Intermediate-Value Property of Derivatives......Page 146
3.39 The Differential*......Page 148
4.40 Area Measure......Page 151
4.41 Properties of Area Measure......Page 153
4,42 Area Measure of Rectangles*......Page 156
4.43 Approximation by Polygonal Regions......Page 158
4.44 Upper and Lower Sums......Page 161
4.45 The Riemann Integral......Page 164
4.46 The Riemann Criterion for Integrability......Page 167
4.47 Integration of Continuous Functions and Monotonic Functions......Page 171
4.48 Sets of Lebesgue Measure Zero......Page 175
4.49 Characterization of Integrable Functions......Page 180
4.50 The Linearity of the Integral......Page 186
4.51 Properties of the Riemann Integral......Page 188
4.52 The Riemann Integral with a Variable Upper Limit......Page 192
4.53 Improper Integrals*......Page 200
5.54 Introduction......Page 207
5.55 The Cartesian n-Space......Page 208
5.56 Dot Product......Page 211
5.57 Norm......Page 214
5.58 The Euclidean n-Space......Page 218
5.59 Open and Closed Sets in En......Page 221
5.60 The Nested-Interval Property in En and the Bolzano-Weierstrass Theorem......Page 225
5.61 Sequences in En......Page 228
5.62 Cauchy Sequences......Page 231
6.63 Notation......Page 236
6.64 Limit of a Function......Page 239
6.65 Continuity......Page 241
6.66 Direct and Inverse Images......Page 243
6.67 Composite Functions......Page 250
6.68 Global Characterization of Continuous Functions......Page 253
6.69 Open Maps......Page 258
6.70 Compact Sets and the Heine-Borel Theorem......Page 259
6.71 The Continuous Image of a Compact Set......Page 262
6.72 Connected Sets......Page 263
6.73 The Continuous Image of a Connected Set......Page 267
6.74 Uniformly Continuous Functions......Page 270
6.75 Contraction Mappings......Page 272
7.76 Pointwise Convergence......Page 275
7.77 Uniform Convergence......Page 281
7.78 Uniformly Convergent Sequences of Continuous Functions and of Bounded Functions......Page 285
7.79 Weierstrass Approximation Theorem*......Page 288
7.80 A Continuous Function that is Nowhere Differentiable*......Page 291
7.80 Termwise Integration of Sequences......Page 295
7.82 Termwise Differentiation of Sequences......Page 300
8.83 Definition and Representation......Page 306
8.84 Linear Onto Functions and Linear One-to-One Functions......Page 310
8.85 Properties of Linear Functions......Page 316
9.86 Definition of the Derivative......Page 322
9.87 The Directional Derivative......Page 326
9.88 Partial Derivatives......Page 331
9.89 Representation of the Derivative......Page 333
9.90 Existence of the Derivative......Page 337
9.91 Differentiation Rules......Page 344
9.92 Differentiation of Composite Functions......Page 348
9.93 Mean-Value Theorems......Page 354
9.94 Partial Derivatives of Higher Order......Page 356
9.95 The Inverse-Function Theorem and the Implicit-Function Theorem*......Page 362
10.96 One-to-One Functions......Page 369
10.97 Onto Functions......Page 376
10.98 The Inverse-Function Theorem......Page 381
10.99 The Implicit-Function Theorem......Page 385
10.100 Implicit Differentiation......Page 390
10.101 Extreme Values*......Page 394
11.102 The Riemann Integral in En......Page 402
11.103 Existence of the Integral......Page 406
11.104 The Riemann Integral over Point Sets Other Than Intervals......Page 409
11.105 Jordan Content......Page 414
11.106 Properties of Jordan-Measurable Sets......Page 419
11.107 Integrals over Jordan-Measurable Sets......Page 421
11.108 Integration by Iterarion-Fubini's Theorem......Page 425
11.109 Applications of Fubini's Theorem......Page 430
11.110 Differentiation of an Integral with Respect to a Parameter*......Page 434
11.111 Transformation of Double Integrals*......Page 439
12.112 Images of Jordan-Measurable Sets......Page 448
12.113 Jordan Content of Linear Images of Jordan-Measurable Sets......Page 455
12.114 Jordan Content of General Images of Intervals......Page 460
12.115 Transformation of Multiple Integrals: Jacobi's Theorem......Page 467
12.116 Jordan Content as Length, Area, and Volume Measure*......Page 474
13.117 Introduction......Page 480
13.118 Curves......Page 483
13.119 Surfaces......Page 491
13.120 Smooth Manifolds......Page 495
13.121 Diffeomorphisms and Smooth Equivalence......Page 502
13.122 Tangent Lines and Tangent Vectors......Page 508
13.123 Tangent Planes and Normal Vectors......Page 513
13.124 Patches and Quilts......Page 520
13.125 Arc Length......Page 524
13.126 Surface Area......Page 530
13.127 Differential Forms......Page 534
13.128 Work and Steady Flow......Page 544
13.129 Differentials of k-forms......Page 551
13.130 Green's Theorem......Page 554
13.131 Stoke's Theorem......Page 562
13.132 The Theorem of Gauss......Page 570
14.133 Convergence of Infinite Series......Page 576
14.134 The Integral Test......Page 580
14.135 Absolute Convergence and Conditional Convergence......Page 584
14.136 Tests for Absolute Convergence......Page 590
14.137 The CBS Inequality and the Triangle Inequality for Infinite Series*......Page 598
14.138 Double Series......Page 600
14.139 Cauchy Products......Page 605
14.140 Infinite Series of Functions......Page 608
14.141 Power Series......Page 613
14.142 The Exponential Function and its Inverse Function*......Page 617
14.143 The Trigonometric Functions and Their Inverse Functions*......Page 622
14.144 Manipulations with Power Series......Page 628
14.145 Taylor Series......Page 632