Tools of the Trade: Introduction to Advanced Mathematics

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American Mathematical Soc., 2008 - Mathematics - 193 pages
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This book provides a transition from the formula-full aspects of the beginning study of college level mathematics to the rich and creative world of more advanced topics. It is designed to assist the student in mastering the techniques of analysis and proof that are required to do mathematics. Along with the standard material such as linear algebra, construction of the real numbers via Cauchy sequences, metric spaces and complete metric spaces, there are three projects at the end of each chapter that form an integral part of the text. These projects include a detailed discussion of topics such as group theory, convergence of infinite series, decimal expansions of real numbers, point set topology and topological groups. They are carefully designed to guide the student through the subject matter. Together with numerous exercises included in the book, these projects may be used as part of the regular classroom presentation, as self-study projects for students, or for Inquiry Based Learning activities presented by the students.
 

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User Review  - Brian - Goodreads

I am almost certainly biased, being a big fan of Mr. Sally and his brilliant pedagogy. Nevertheless, this book is a very clear, flowing primer to a great number of the basic mathematical entities and ... Read full review

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Contents

Sets Functions and Other Basic Ideas
1
1 Sets and Elements
2
3 The Algebra of Sets
4
4 Cartesian Products Counting and Power Sets
8
5 Some Sets of Numbers
10
6 Equivalence Relations and the Construction of Q
15
7 Functions
22
8 Countability and Other Basic Ideas
30
4 Intervals
97
5 The Construction of the Real Numbers
98
6 Convergence in R
102
7 Automorphisms of Fields
107
8 Construction of the Complex Numbers
108
9 Convergence in C
110
10 Independent Projects
115
Metric and Euclidean Spaces
125

9 Axiom of Choice
38
10 Independent Projects
41
Linear Algebra
47
1 Fundamentals of Linear Algebra
48
2 Linear Transformations
54
3 Linear Transformations and Matrices
56
4 Determinants
59
5 Geometric Linear Algebra
67
6 Independent Projects
76
The Construction of the Real and Complex Numbers
89
1 The Least Upper Bound Property and the Real Numbers
90
2 Consequences of the Least Upper Bound Property
92
3 Rational Approximation
94
2 Definition and Basic Properties of Metric Spaces
126
3 Topology of Metric Spaces
129
4 Limits and Continuous Functions
137
5 Compactness Completeness and Connectedness
145
6 Independent Projects
155
Complete Metric Spaces and the padic Completion of Q
167
1 The Contraction Mapping Theorem and Its Applications
168
2 The Baire Category Theorem and Its Applications
170
3 The StoneWeierstrass Theorem
172
4 The p adic Completion of Q
176
5 Challenge Problems
184
Index
189
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