The Integral Calculus

Mathematical Analysis I......Page 3
Terms and Conditions......Page 4
Contents......Page 5
Preface......Page 9
About the Author......Page 11
1-3 Sets and Operations on Sets. Quantifiers......Page 13
Problems in Set Theory......Page 18
4-7 Relations. Mappings......Page 20
Problems on Relations and Mappings......Page 26
8 Sequences......Page 27
9 Some Theorems on Countable Sets......Page 30
Problems on Countable and Uncountable Sets......Page 33
1-4 Axioms and Basic Definitions......Page 35
5-6 Natural Numbers. Induction......Page 39
Problems on Natural Numbers and Induction......Page 44
7 Integers and Rationals......Page 46
8-9 Upper and Lower Bounds. Completeness Axiom......Page 48
Problems on Upper and Lower Bounds......Page 52
10 Some Consequences of the Completeness Axiom......Page 55
11-12 Powers with Arbitrary Real Exponents. Irrationals......Page 58
Problems on Roots, Powers, and Irrationals......Page 62
13 The Infinities. Upper and Lower Limits of Sequences......Page 65
Problems on Upper and Lower Limits of Sequences in E*......Page 72
1-3 The Euclidean n-Space, En......Page 75
Problems on Vectors in En......Page 81
4-6 Lines and Planes in En......Page 83
Problems on Lines and Planes in En......Page 87
7 Intervals in En......Page 88
Problems on Intervals in En......Page 91
8 Complex Numbers......Page 92
Problems on Complex Numbers......Page 95
9 Vector Spaces. The Space Cn. Euclidean Spaces......Page 97
Problems on Linear Spaces......Page 101
10 Normed Linear Spaces......Page 102
Problems on Normed Linear Spaces......Page 105
11 Metric Spaces......Page 107
Problems on Metric Spaces......Page 110
12 Open and Closed Sets. Neighborhoods......Page 113
Problems on Neighborhoods, Open and Closed Sets......Page 118
13 Bounded Sets. Diameters......Page 120
Problems on Boundedness and Diameters......Page 124
14 Cluster Points. Convergent Sequences......Page 126
Problems on Cluster Points and Convergence......Page 130
15 Operations on Convergent Sequences......Page 132
Problems on Limits of Sequences......Page 135
16 More on Cluster Points and Closed Sets. Density......Page 147
Problems on Cluster Points, Closed Sets, and Density......Page 151
17 Cauchy Sequences. Completeness......Page 153
Problems on Cauchy Sequences......Page 156
1 Basic Definitions......Page 161
Problems on Limits and Continuity......Page 169
2 Some General Theorems on Limits and Continuity......Page 173
More Problems on Limits and Continuity......Page 178
3 Operations on Limits. Rational Functions......Page 182
Problems on Continuity of Vector-Valued Functions......Page 186
4 Infinite Limits. Operations on E*......Page 189
Problems on Limits and Operations in E*......Page 192
5 Monotone Functions......Page 193
Problems on Monotone Functions......Page 197
6 Compact Sets......Page 198
Problems on Compact Sets......Page 201
7 More on Compactness......Page 204
8 Continuity on Compact Sets. Uniform Continuity......Page 206
Problems on Uniform Continuity; Continuity on Compact Sets......Page 212
9 The Intermediate Value Property......Page 215
Problems on the Darboux Property and Related Topics......Page 221
10 Arcs and Curves. Connected Sets......Page 223
Problems on Arcs, Curves, and Connected Sets......Page 227
11 Product Spaces. Double and Iterated Limits......Page 230
Problems on Double Limits and Product Spaces......Page 236
12 Sequences and Series of Functions......Page 239
Problems on Sequences and Series of Functions......Page 244
13 Absolutely Convergent Series. Power Series......Page 249
More Problems on Series of Functions......Page 257
1 Derivatives of Functions of One Real Variable......Page 263
Problems on Derived Functions in One Variable......Page 269
2 Derivatives of Extended-Real Functions......Page 271
Problems on Derivatives of Extended-Real Functions......Page 277
3 L\'Hopital\'s Rule......Page 278
Problems on L\'Hopital\'s Rule......Page 281
4 Complex and Vector-Valued Functions on E1......Page 283
Problems on Complex and Vector-Valued Functions on E1......Page 287
5 Antiderivatives (Primitives, Integrals)......Page 290
Problems on Antiderivatives......Page 297
6 Differentials. Taylor\'s Theorem and Taylor\'s Series......Page 300
Problems on Taylor\'s Theorem......Page 308
7 The Total Variation (Length) of a Function......Page 312
Problems on Total Variation and Graph Length......Page 318
8 Rectifiable Arcs. Absolute Continuity......Page 320
9 Convergence Theorems in Differentiation and Integration......Page 326
Problems on Convergence in Differentiation and Integration......Page 333
10 Sufficient Condition of Integrability. Regulated Functions......Page 334
Problems on Regulated Functions......Page 341
11 Integral Definitions of Some Functions......Page 343
Problems on Exponential and Trigonometric Functions......Page 350
Index......Page 353