Introductory Mathematics: Algebra and Analysis [electronic resource] /

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 Completeness Axiom -- 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.

Bibliographic Details

Main Authors:	Smith, Geoff. author., SpringerLink (Online service)
Format:	Texto biblioteca
Language:	eng
Published:	London : Springer London : Imprint: Springer, 1998
Subjects:	Mathematics., Algebra., Mathematical analysis., Analysis (Mathematics)., Analysis., Mathematics, general.,
Online Access:	http://dx.doi.org/10.1007/978-1-4471-0619-7
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