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ОбложкаGarrity T.A. All the mathematics you missed: but need to know for graduate school / figures by L.Pedersen. - New York: Cambridge University Press, 2002. - xxvii, 347 p.: ill. - Bibliogr.: p.329-337. - Ind.: p.338-347. - ISBN 978-0-521-79707-8
Шифр: (И/В1-G22) 02
 

Место хранения: 02 | Отделение ГПНТБ СО РАН | Новосибирск

Оглавление / Contents
 
 
Preface ...................................................... xiii
On the Structure of Mathematics ............................... xix
Brief Summaries of Topics ................................... xxiii
   0.1  Linear Algebra ...................................... xxiii
   0.2  Real Analysis ....................................... xxiii
   0.3  Differentiating Vector-Valued Functions ............. xxiii
   0.4  Point Set Topology ................................... xxiv
   0.5  Classical Stokes' Theorems ........................... xxiv
   0.6  Differential Forms and Stokes' Theorem ............... xxiv
   0.7  Curvature for Curves and Surfaces .................... xxiv
   0.8  Geometry .............................................. xxv
   0.9  Complex Analysis ...................................... xxv
   0.10 Countability and the Axiom of Choice ................. xxvi
   0.11 Algebra .............................................. xxvi
   0.12 Lebesgue Integration ................................. xxvi
   0.13 Fourier Analysis ..................................... xxvi
   0.14 Differential Equations .............................. xxvii
   0.15 Combinatorics and Probability Theory ................ xxvii
   0.16 Algorithms .......................................... xxvii
1  Linear Algebra ............................................... 1
   1.1  Introduction ............................................ 1
   1.2  The Basic Vector Space fig.1n ............................... 2
   1.3  Vector Spaces and Linear Transformations ................ 4
   1.4  Bases and Dimension ..................................... 6
   1.5  The Determinant ......................................... 9
   1.6  The Key Theorem of Linear Algebra ...................... 12
   1.7  Similar Matrices ....................................... 14
   1.8  Eigenvalues and Eigenvectors ........................... 15
   1.9  Dual Vector Spaces ..................................... 20
   1.10 Books .................................................. 21
   1.11 Exercises .............................................. 21
2  ε and δ Real Analysis ....................................... 23
   2.1  Limits ................................................. 23
   2.2  Continuity ............................................. 25
   2.3  Differentiation ........................................ 26
   2.4  Integration ............................................ 28
   2.5  The Fundamental Theorem of Calculus .................... 31
   2.6  Pointwise Convergence of Functions ..................... 35
   2.7  Uniform Convergence .................................... 36
   2.8  The Weierstrass M-Test ................................. 38
   2.9  Weierstrass' Example ................................... 40
   2.10 Books .................................................. 43
   2.11 Exercises .............................................. 44
3  Calculus for Vector-Valued Functions ........................ 47
   3.1  Vector-Valued Functions ................................ 47
   3.2  Limits and Continuity .................................. 49
   3.3  Differentiation and Jacobians .......................... 50
   3.4  The Inverse Function Theorem ........................... 53
   3.5  Implicit Function Theorem .............................. 56
   3.6  Books .................................................. 60
   3.7  Exercises .............................................. 60
4  Point Set Topology .......................................... 63
   4.1  Basic Definitions ...................................... 63
   4.2  The Standard Topology on fig.1n ............................ 66
   4.3  Metric Spaces .......................................... 72
   4.4  Bases for Topologies ................................... 73
   4.5  Zariski Topology of Commutative Rings .................. 75
   4.6  Books .................................................. 77
   4.7  Exercises .............................................. 78
5  Classical Stokes' Theorems .................................. 81
   5.1  Preliminaries about Vector Calculus .................... 82
        5.1.1  Vector Fields ................................... 82
        5.1.2  Manifolds and Boundaries ........................ 84
        5.1.3  Path Integrals .................................. 87
        5.1.4  Surface Integrals ............................... 91
        5.1.5  The Gradient .................................... 93
        5.1.6  The Divergence .................................. 93
        5.1.7  The Curl ........................................ 94
        5.1.8  Orientability ................................... 94
   5.2  The Divergence Theorem and Stokes' Theorem ............. 95
   5.3  Physical Interpretation of Divergence Thm .............. 97
   5.4  A Physical Interpretation of Stokes' Theorem ........... 98
   5.5  Proof of the Divergence Theorem ........................ 99
   5.6  Sketch of a Proof for Stokes' Theorem ................. 104
   5.7  Books ................................................. 108
   5.8  Exercises ............................................. 108
6  Differential Forms and Stokes' Thm. ........................ 111
   6.1  Volumes of Parallelepipeds ............................ 112
   6.2  Diff. Forms and the Exterior Derivative ............... 115
        6.2.1  Elementary k-forms ............................. 115
        6.2.2  The Vector Space of k-forms .................... 118
        6.2.3  Rules for Manipulating k-forms ................. 119
        6.2.4  Differential k-forms and the Exterior
               Derivative ..................................... 122
   6.3  Differential Forms and Vector Fields .................. 124
   6.4  Manifolds ............................................. 126
   6.5  Tangent Spaces and Orientations ....................... 132
        6.5.1  Tangent Spaces for Implicit and Parametric
               Manifolds ...................................... 132
        6.5.2  Tangent Spaces for Abstract Manifolds .......... 133
        6.5.3  Orientation of a Vector Space .................. 135
        6.5.4  Orientation of a Manifold and its Boundary ..... 136
   6.6  Integration on Manifolds .............................. 137
   6.7  Stokes' Theorem ....................................... 139
   6.8  Books ................................................. 142
   6.9  Exercises ............................................. 143
7  Curvature for Curves and Surfaces .......................... 145
   7.1  Plane Curves .......................................... 145
   7.2  Space Curves .......................................... 148
   7.3  Surfaces .............................................. 152
   7.4  The Gauss-Bonnet Theorem .............................. 157
   7.5  Books ................................................. 158
   7.6  Exercises ............................................. 158
8  Geometry ................................................... 161
   8.1  Euclidean Geometry .................................... 162
   8.2  Hyperbolic Geometry ................................... 163
   8.3  Elliptic Geometry ..................................... 166
   8.4  Curvature ............................................. 167
   8.5  Books ................................................. 168
   8.6  Exercises ............................................. 169
9  Complex Analysis ........................................... 171
   9.1  Analyticity as a Limit ................................ 172
   9.2  Cauchy-Riemann Equations .............................. 174
   9.3  Integral Representations of Functions ................. 179
   9.4  Analytic Functions as Power Series .................... 187
   9.5  Conformal Maps ........................................ 191
   9.6  The Riemann Mapping Theorem ........................... 194
   9.7  Several Complex Variables: Hartog's Theorem ........... 196
   9.8  Books ................................................. 197
   9.9  Exercises ............................................. 198
10 Countability and the Axiom of Choice ....................... 201
   10.1 Countability .......................................... 201
   10.2 Naive Set Theory and Paradoxes ........................ 205
   10.3 The Axiom of Choice ................................... 207
   10.4 Non-measurable Sets ................................... 208
   10.5 Gödel and Independence Proofs ......................... 210
   10.6 Books ................................................. 211
   10.7 Exercises ............................................. 211
11 Algebra .................................................... 213
   11.1 Groups ................................................ 213
   11.2 Representation Theory ................................. 219
   11.3 Rings ................................................. 221
   11.4 Fields and Galois Theory .............................. 223
   11.5 Books ................................................. 228
   11.6 Exercises ............................................. 229
12 Lebesgue Integration ....................................... 231
   12.1 Lebesgue Measure ...................................... 231
   12.2 The Cantor Set ........................................ 234
   12.3 Lebesgue Integration .................................. 236
   12.4 Convergence Theorems .................................. 239
   12.5 Books ................................................. 241
   12.6 Exercises ............................................. 241
13 Fourier Analysis ........................................... 243
   13.1 Waves, Periodic Functions and Trigonometry ............ 243
   13.2 Fourier Series ........................................ 244
   13.3 Convergence Issues .................................... 250
   13.4 Fourier Integrals ала Transforms ...................... 252
   13.5 Solving Differential Equations ........................ 256
   13.6 Books ................................................. 258
   13.7 Exercises ............................................. 258
14 Differential Equations ..................................... 261
   14.1 Basics ................................................ 261
   14.2 Ordinary Differential Equations ....................... 262
   14.3 The Laplacian ......................................... 266
        14.3.1  Mean Value Principle .......................... 266
        14.3.2  Separation of Variables ....................... 267
        14.3.3  Applications to Complex Analysis .............. 270
   14.4 The Heat Equation ..................................... 270
   14.5 The Wave Equation ..................................... 273
        14.5.1 Derivation ..................................... 273
        14.5.2 Change of Variables ............................ 277
   14.6 Integrability Conditions .............................. 279
   14.7 Lewy's Example ........................................ 281
   14.8 Books ................................................. 282
   14.9 Exercises ............................................. 282
15 Combinatorics and Probability .............................. 285
   15.1 Counting .............................................. 285
   15.2 Basic Probability Theory .............................. 287
   15.3 Independence .......................................... 290
   15.4 Expected Values and Variance .......................... 291
   15.5 Central Limit Theorem ................................. 294
   15.6 Stirling's Approximation for n ........................ 300
   15.7 Books ................................................. 305
   15.8 Exercises ............................................. 305
16 Algorithms ................................................. 307
   16.1 Algorithms and Complexity ............................. 308
   16.2 Graphs: Euler and Hamiltonian Circuits ................ 308
   16.3 Sorting and Trees ..................................... 313
   16.4 P=NP? ................................................. 316
   16.5 Numerical Analysis: Newton's Method ................... 317
   16.6 Books ................................................. 324
   16.7 Exercises ............................................. 324
A  Equivalence Relations ...................................... 327

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