# All the Mathematics You Missed: But Need to Know for Graduate School

Cambridge University Press, 2002 - 347 Seiten
Few beginning graduate students in mathematics and other quantitative subjects possess the daunting breadth of mathematical knowledge expected of them when they begin their studies. This book will offer students a broad outline of essential mathematics and will help to fill in the gaps in their knowledge. The author explains the basic points and a few key results of all the most important undergraduate topics in mathematics, emphasizing the intuitions behind the subject. The topics include linear algebra, vector calculus, differential and analytical geometry, real analysis, point-set topology, probability, complex analysis, set theory, algorithms, and more. An annotated bibliography offers a guide to further reading and to more rigorous foundations.

### Was andere dazu sagen -Rezension schreiben

Es wurden keine Rezensionen gefunden.

### Inhalt

 Preface xiii On the Structure of Mathematics xix Brief Summaries of Topics xxiii 04 Point Set Topology xxiv 08 Geometry xxv 010 Countability and the Axiom of Choice xxvi 014 Differential Equations xxvii Linear Algebra 1
 Curvature for Curves and Surfaces 145 72 Space Curves 148 73 Surfaces 152 74 The GaussBonnet Theorem 157 75 Books 158 Geometry 161 81 Euclidean Geometry 162 82 Hyperbolic Geometry 163

 12 The Basic Vector Space Rⁿ 2 13 Vector Spaces and Linear Transformations 4 14 Bases Dimension and Linear Transformations as Matrices 6 15 The Determinant 9 16 The Key Theorem of Linear Algebra 12 17 Similar Matrices 14 18 Eigenvalues and Eigenvectors 15 19 Dual Vector Spaces 20 110 Books 21 ϵ and 𝛿 Real Analysis 23 22 Continuity 25 23 Differentiation 26 24 Integration 28 25 The Fundamental Theorem of Calculus 31 26 Pointwise Convergence of Functions 35 27 Uniform Convergence 36 28 The Weierstrass MTest 38 29 Weierstrass Example 40 210 Books 43 211 Exercises 44 Calculus for VectorValued Functions 47 32 Limits and Continuity of VectorValued Functions 49 33 Differentiation and Jacobians 50 34 The Inverse Function Theorem 53 35 Implicit Function Theorem 56 36 Books 60 Point Set Topology 63 42 The Standard Topology on Rⁿ 66 43 Metric Spaces 72 44 Bases for Topologies 73 45 Zariski Topology of Commutative Rings 75 46 Books 77 47 Exercises 78 Classical Stokes Theorems 81 51 Preliminaries about Vector Calculus 82 512 Manifolds and Boundaries 84 513 Path Integrals 87 514 Surface Integrals 91 515 The Gradient 93 517 The Curl 94 52 The Divergence Theorem and Stokes Theorem 95 53 Physical Interpretation of the Divergence Thm 97 54 A Physical Interpretation of Stokes Theorem 98 55 Proof of the Divergence Theorem 99 56 Sketch of a Proof for Stokes Theorem 104 57 Books 108 Differential Forms and Stokes Theorem 111 61 Volumes of Parallelepipeds 112 62 Diff Forms and the Exterior Derivative 115 622 The Vector Space of 𝓀forms 118 623 Rules for Manipulating 𝓀forms 119 624 Differential 𝓀forms and the Exterior Derivative 122 63 Differential Forms and Vector Fields 124 64 Manifolds 126 65 Tangent Spaces and Orientations 132 652 Tangent Spaces for Abstract Manifolds 133 653 Orientation of a Vector Space 135 654 Orientation of a Manifold and its Boundary 136 66 Integration on Manifolds 137 67 Stokes Theorem 139 68 Books 142 69 Exercises 143
 83 Elliptic Geometry 166 84 Curvature 167 85 Books 168 86 Exercises 169 Complex Analysis 171 91 Analyticity as a Limit 172 92 CauchyRiemann Equations 174 93 Integral Representations of Functions 179 94 Analytic Functions as Power Series 187 95 Conformal Maps 191 96 The Riemann Mapping Theorem 194 Hartogs Theorem 196 98 Books 197 99 Exercises 198 Countability and the Axiom of Choice 201 102 Naive Set Theory and Paradoxes 205 103 The Axiom of Choice 207 104 Nonmeasurable Sets 208 105 Gödel and Independence Proofs 210 106 Books 211 Algebra 213 112 Representation Theory 219 113 Rings 221 114 Fields and Galois Theory 223 115 Books 228 116 Exercises 229 Lebesgue Integration 231 122 The Cantor Set 234 123 Lebesgue Integration 236 124 Convergence Theorems 239 125 Books 241 Fourier Analysis 243 132 Fourier Series 244 133 Convergence Issues 250 134 Fourier Integrals and Transforms 252 135 Solving Differential Equations 256 136 Books 258 Differential Equations 261 142 Ordinary Differential Equations 262 1431 Mean Value Principle 266 1432 Separation of Variables 267 1433 Applications to Complex Analysis 270 1451 Derivation 273 1452 Change of Variables 277 Integrability Conditions 279 147 Lewys Example 281 148 Books 282 Combinatorics and Probability Theory 285 152 Basic Probability Theory 287 153 Independence 290 154 Expected Values and Variance 291 155 Central Limit Theorem 294 156 Stirlings Approximation for 𝑛 300 157 Books 305 Chapter 16 Algorithms 307 161 Algorithms and Complexity 308 163 Sorting and Trees 313 164 PNP? 316 Newtons Method 317 166 Books 324 Urheberrecht