Mathematical analysis 2

Mathematical Analysis II......Page 3
Terms and Conditions......Page 4
Contents......Page 5
Preface......Page 9
About the Author......Page 11
1 Directional and Partial Derivatives......Page 13
Problems on Directional and Partial Derivatives......Page 18
2 Linear Maps and Functionals. Matrices......Page 19
Problems on Linear Maps and Matrices......Page 26
3 Differentiable Functions......Page 28
Problems on Differentiable Functions......Page 37
4 The Chain Rule. The Cauchy Invariant Rule......Page 40
Further Problems on Differentiable Functions......Page 45
5 Repeated Differentiation. Taylor's Theorem......Page 47
Problems on Repeated Differentiation and Taylor Expansions......Page 56
6 Determinants. Jacobians. Bijective Linear Operators......Page 59
Problems on Bijective Linear Maps and Jacobians......Page 67
7 Inverse and Implicit Functions. Opens and Closed Maps......Page 69
Problems on Inverse and Implicit Functions, Open and Closed Maps......Page 79
8 Baire Categories. More on Linear Maps......Page 82
Problems on Baire Categories and Linear Maps......Page 88
9 Local Extrema. Maxima and Minima......Page 91
Problems on Maxima and Minima......Page 96
10 More on Implicit Differentiation. Conditional Extrema......Page 99
Further Problems on Maxima and Minima......Page 106
1 More on Intervals. Semirings of Sets......Page 109
2 C_sigma-Sets, Countable Additivity. Permutable Series......Page 116
Problems on C_sigma-Sets, Sigma-Additivity, and Permutable Series......Page 124
3 More on Set Families......Page 127
Problems on Set Families......Page 133
4 Set Functions. Additivity. Continuity......Page 136
Problems on Set Functions......Page 145
5 Nonnegative Set Functions. Premeasures. Outer Measures......Page 148
Problems on Premeasures and Related Topics......Page 153
6 Measure Spaces. More on Outer Measures......Page 159
Problems on Measures and Outer Measures......Page 167
7 Topologies. Borel Sets. Borel Measures......Page 172
Problems on Topologies, Borel Sets, and Regular Measures......Page 177
8 Lebesgue Measure......Page 180
Problems on Lebesgue Measure......Page 184
9 Lebesgue-Stieltjes Measures in E1......Page 188
Problems on Lebesgue Measures......Page 190
10 Vitali Coverings on En......Page 192
Problems on Vitali Coverings......Page 202
11 Generalized Measures. Absolute Continuity......Page 206
Problems on Generalized Measures......Page 218
12 Differentiation of Set Functions......Page 222
Problems on Differentiations of Set Functions......Page 227
1 Elementary and Measurable Functions......Page 229
Problems on Measurable and Elementary Functions in (S,M)
......Page 234
2 Measurability of Extended-Real Functions......Page 236
Further Problems on Measurable Functions in (S,M)
......Page 241
3 Measurable Functions in (S,M,m)......Page 243
Problems on Measurable Functions in (S,M,m)
......Page 250
4 Integration of Elementary Functions......Page 253
Problems on Integration of Elementary Functions......Page 260
5 Integration of Extended-Real Functions......Page 263
Problems on Integration of Extended-Real Functions......Page 277
6 Integrable Functions. Convergence Theorems......Page 279
Problems on Integrability and Convergence Theorems......Page 289
7 Integration of Complex and Vector-Valued Functions......Page 295
Problems on Integration of Complex and Vector-Valued Functions......Page 303
8 Product Measures. Iterated Integrals......Page 305
Problems on Product Measures and Fubini Theorems......Page 314
9 Riemann Integration. Stieltjes Integrals......Page 318
Problems on Riemann and Stieltjes Integrals......Page 331
10 Integration in Generalized Measure Spaces......Page 335
Problems on Generalized Integration......Page 345
11 The Radon-Nikodym Theorem. Lebesgue Decomposition......Page 348
Problems on Radon-Nikodym Derivatives and Lebesgue Decomposition......Page 356
12 Integration and Differentiation......Page 357
Problems on Differentiation and Related Topics......Page 366
1 L-Integrals and Antiderivatives......Page 369
Problems on L-Integrals and Antiderivatives......Page 379
2 More on L-Integrals and Absolute Continuity......Page 384
Problems on L-Integrals and Absolute Continuity......Page 396
3 Improper (Cauchy) Integrals......Page 399
Problems on Cauchy Integrals......Page 409
4 Convergence of Parametrized Integrals and Functions......Page 414
Problems on Uniform Convergence of Functions and C-Integrals......Page 424
Index......Page 429