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Viser: Introductory Mathematics - Algebra and Analysis
Introductory Mathematics
Algebra and Analysis
Geoff Smith
(2000)
Sprog: Engelsk
Springer London, Limited
387,00 kr.
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Detaljer om varen
- 1. Udgave
- Paperback
- Udgiver: Springer London, Limited (Februar 2000)
- ISBN: 9783540761785
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.
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.- Further Reading.- Solutions.
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.- Further Reading.- Solutions.
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