All the Mathematics You Missed: But Need to Know for Graduate SchoolCambridge University Press, 2002 - 347 σελίδες 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. |
Περιεχόμενα
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 |
Άλλες εκδόσεις - Προβολή όλων
All the Mathematics You Missed: But Need to Know for Graduate School Thomas A. Garrity Περιορισμένη προεπισκόπηση - 2002 |
All the Mathematics You Missed: But Need to Know for Graduate School Thomas A. Garrity Περιορισμένη προεπισκόπηση - 2004 |
All the Mathematics You Missed: But Need to Know for Graduate School Thomas A. Garrity Περιορισμένη προεπισκόπηση - 2002 |
Συχνά εμφανιζόμενοι όροι και φράσεις
a₁ algorithm analytic functions Axiom of Choice basic basis boundary Cauchy-Riemann equations chapter circle coefficients complex analysis complex numbers compute continuous function converges uniformly coordinates countable curve define definition denoted derivative differentiable functions differential equations differential k-form Divergence Theorem eigenvalues elements elliptic geometry example exterior derivative f(zo finite Fourier series Fourier transform function f Fundamental Theorem given goal graph hence holomorphic infinite interval intuitions inverse Jacobian k-forms Lebesgue integral Lemma linear algebra linear transformation loop manifold mathematics matrix means measure normal vector open set orientation parametrization path integral permutation plane polynomial problem proof rational numbers real numbers real-valued function roots solution Stokes subgroup subset surface tangent line tangent vector Theorem of Calculus tion topological space topology variable vector field vector space zero მა მთ