Introductory Mathematics Algebra And Analysis Springer Undergraduate Mathematics Series 2nd Edition by Geoffrey C Smith ISBN 3540761780 978-3540761785
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Product details:
ISBN 10: 3540761780
ISBN 13: 978-3540761785
Author: Geoffrey C. Smith
This text provides a lively introduction to pure mathematics. It begins with sets, functions and relations, proof by induction and contradiction, complex numbers, vectors and matrices, and provides a brief introduction to group theory. It moves onto analysis, providing a gentle introduction to epsilon-delta technology and finishes with continuity and functions. The book features numerous exercises of varying difficulty throughout the text.
Table of contents:
1. Sets, Functions and Relations.
1.1 Sets.
1.2 Subsets.
1.3 Well-known Sets.
1.4 Rationals, Reals and Pictures.
1.5 Set Operations.
1.6 Sets of Sets.
1.7 Paradox.
1.8 Set-theoretic Constructions.
1.9 Notation.
1.10 Venn Diagrams.
1.11 Quantifiers and Negation.
1.12 Informal Description of Maps.
1.13 Injective, Surjective and Bijective Maps.
1.14 Composition of Maps.
1.15 Graphs and Respectability Reclaimed.
1.16 Characterizing Bijections.
1.17 Sets of Maps.
1.18 Relations.
1.19 Intervals.
2. Proof.
2.1 Induction.
2.2 Complete Induction.
2.3 Counter-examples and Contradictions.
2.4 Method of Descent.
2.5 Style.
2.6 Implication.
2.7 Double Implication.
2.8 The Master Plan.
3. Complex Numbers and Related Functions.
3.1 Motivation.
3.2 Creating the Complex Numbers.
3.3 A Geometric Interpretation.
3.4 Sine, Cosine and Polar Form.
3.5 e.
3.6 Hyperbolic Sine and Hyperbolic Cosine.
3.7 Integration Tricks.
3.8 Extracting Roots and Raising to Powers.
3.9 Logarithm.
3.10 Power Series.
4. Vectors and Matrices.
4.1 Row Vectors.
4.2 Higher Dimensions.
4.3 Vector Laws.
4.4 Lengths and Angles.
4.5 Position Vectors.
4.6 Matrix Operations.
4.7 Laws of Matrix Algebra.
4.8 Identity Matrices and Inverses.
4.9 Determinants.
4.10 Geometry of Determinants.
4.11 Linear Independence.
4.12 Vector Spaces.
4.13 Transposition.
5. Group Theory.
5.1 Permutations.
5.2 Inverse Permutations.
5.3 The Algebra of Permutations.
5.4 The Order of a Permutation.
5.5 Permutation Groups.
5.6 Abstract Groups.-
5.7 Subgroups.
5.8 Cosets.
5.9 Cyclic Groups.
5.10 Isomorphism.
5.11 Homomorphism.
6. Sequences and Series.
6.1 Denary and Decimal Sequences.
6.2 The Real Numbers.
6.3 Notation for Sequences.
6.4 Limits of Sequences.
6.5 The CompletenessAxiom.
6.6 Limits of Sequences Revisited.
6.7 Series.
7. Mathematical Analysis.
7.1 Continuity.
7.2 Limits.
8. Creating the Real Numbers.
8.1 Dedekind’s Construction.
8.2 Construction via Cauchy Sequences.
8.3 A Sting in the Tail: p-adic numbers.
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