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.
Table Of Contents
Preface
On the Structure of Mathematics
Brief Summaries of Topics
1 Linear Algebra 1
2 [epsilon] and [delta] Real Analysis 23
3 Calculus for Vector-Valued Functions 47
4 Point Set Topology 63
5 Classical Stokes' Theorems 81
6 Differential Forms and Stokes' Thm 111
7 Curvature for Curves and Surfaces 145
8 Geometry 161
9 Complex Analysis 171
10 Countability and the Axiom of Choice 201
11 Algebra 213
12 Lebesgue Integration 231
13 Fourier Analysis 243
14 Differential Equations 261
15 Combinatorics and Probability 285
16 Algorithms 307
A: Equivalence Relations 327
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