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

Front Cover
Cambridge University Press, 2002 - Mathematics - 347 pages
3 Reviews
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.
 

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This is a fabulous book. It's chapters are logically organized, extremely accessible, actually enjoyable, and provide just what TG intends. Sure, for each topic there's much more one could describe, and for each one could take a different tack, but in any intro graduate course the prof will take their own trajectory, anyway.
Lovely.
 

User Review - Flag as inappropriate

奇書一本。當初於中大Swindon見此書,研讀月餘,然及後閒來再讀總覺有新得著。此書非為深入了解,但勝在精博,各方數學科目總有涉及。尤適合我輩在大學中未有下苦功讀數,後有悔忱之人。

Contents

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
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