دانلود کتاب A Comprehensive Textbook of Classical Mathematics: A Contemporary Interpretation

فهرست مطالب :

Introduction to the Springer-Verlag Edition
Introduction
1. Origins and Purpose of the Book
2. Mode of Presentation
3. The Spiral Approach versus Pedantry: Notational Difficulties
4. Unification in Mathematics
5. Organization of the Text
6. The Exercises
7. Acknowledgements
Some \'Packages\' of Directed Reading
Textual Conventions
Contents
Part I. The Language of Mathematics
Chapter 1. Descriptive Theory of Sets
1.1 Notion of Set
1.2 Inclusion
1.3 Venn Diagrams
1.4 Equality
1.5 The Power Set
1.6 Union and Intersection
1.7 The Complement
1.8 Quantifiers
Chapter 2. Functions: Descriptive Theory
2.1 The Notion of Function
2.2 Equality of Functions
2.3 The Image
2.4 Injections, Surjections, and Equivalences
2.5 Examples
2.6 Notation and Abuse of Language
2.7 Composition of Functions
2.8 Composition of Injections, etc.
2.9 The Inversion Theorem
2.10 Equivalent Sets
2.11 Counting
Chapter 3. The Cartesian Product
3.1 Pairs and Products
3.2 Algebraic Properties
3.3 The Graph of a Function
3.4 The Notion of Function, again
3.5 Ordered Pairs again
3.6 Multiplicative Systems
Chapter 4. Relations
4.1 What is a Relation?
4.2 The RST Conditions
4.3 Linear Graphs
4.4 Orderings
4.5 Equivalence Relations
4.6 Partitionings
4.7 The Quotient Map
Chapter 5. Mathematical Induction
5.1 Physical and Mathematical Induction
5.2 A Bad Custom
5.3 The Method of Inductive Definition
Part II. Further Set Theory
Chapter 6. Sets of Functions
6.1 The Set B^A
6.2 Mappings of B^A
6.3 The Case when #B = 2
6.4 Shuffles, Permutations, and the Set I(A, B)
6.5 Combinations
6.6 The Set S(A, B)
Chapter 7. Counting and Transfinite Arithmetic
7.1 Counting
7.2 Transfinite Arithmetic
7.3 The Order Relation in Transfinite Arithmetic
7.4 The Axiom of Choice
Chapter 8. Algebra of Sets and the Propositional Calculus
8.1 Algebra of Sets
8.2 B-Algebras
8.3 The Propositional Calculus
8.4 Extension to More General Formulae
8.5 Implication and Deduction
Part III. Arithmetic
Chapter 9. Commutative Rings and Fields
9.1 The Set of Integers as an Algebraic System
9.2 Rings
9.3 Consequences
9.4 Sub-rings
9.5 Commutative Groups
9.6 Fields
Chapter 10. Arithmetic mod m
10.1 Residue Classes, and the Ring Z_m
10.2 Theory of Z_m
10.3 Euler\'s Totient Function
10.4 Solution of Congruences
Chapter 11. Rings with Integral Norm
11.1 Integral Norms
11.2 Examples
11.3 Factorization in Euclidean Domains
11.4 Ideals
11.5 HCF\'s
11.6 Euclid\'s Algorithm
11.7 LCM\'s
Chapter 12. Factorization Into Primes
12.1 Prime Numbers
12.2 Irreducibility and Primes
12.3 Existence and Uniqueness of Prime Factorization
12.4 Factorization in Z[x]
Chapter 13. Applications of the Theory of HCF\'s
13.1 Partial Fractions
13.2 Continued Fractions
Part IV. Geometry of R^3
Chapter 14. Vector Geometry of R^3
14.1 The Vector Space R^3
14.2 Linear Dependence; Bases
14.3 The Equation of a Line
14.4 Lengths
14.5 Spheres
14.6 Projections
14.7 Vectors
14.8 The Scalar Product
14.9 Planes
14.10 The Vector Product
14.11 Volumes
Chapter 15. Linear Algebra and Measure in R^3
15.1 Matrices and Determinants
15.2 Three Linear Equations
15.3 Linear Transformations
Appendix: Length and Area
15.4 Paths
15.5 Rectifiability
15.6 Jordan Arcs and Curves
15.7 Area
15.8 Polygons
15.9 Properties of α
15.10 Curved Boundaries
15.11 Lattices
15.12 A_Λ Related to A
Chapter 16. The Logic of Geometry
16.1 Philosophies of the Greeks and Others
16.2 Hilbert
16.3 Pedagogy
16.4 An Algebraic Model of R^3
16.5 The Pay-off
16.6 Plan for a Proof
16.7 Verifications
16.8 Parallels and Perpendiculars
Chapter 17. Projective Geometry
17.1 A Commercial
17.2 Perspective
17.3 Plane Projective Geometries
17.4 Duality
17.5 The Geometry P(R)
17.6 Relevance to R^2
17.7 Conics
17.8 Models of RP^2
17.9 Embedding P(R) in P(C)
17.10 Projection in R^3
17.11 Invariants: The Erlanger Program
Part V. Algebra
Chapter 18. Groups
18.1 Notion of a Group
18.2 Definition of a Group
18.3 Indices; Subgroups
18.4 Generators of a Group
18.5 Subgroups
18.6 Homomorphisms of Groups
18.7 Isomorphisms
18.8 Kernels and Images
18.9 Subgroups, Quotient Spaces, and Quotient Groups
18.10 Rings
Chapter 19. Vector Spaces and Linear Equations
19.1 Preliminary Definitions
19.2 Bases
19.3 Subspaces
19.4 Homomorphisms: Matrices
19.5 Rank of a Linear Transformation
19.6 Linear Equations
Chapter 20. Inner Product Spaces and Duality
20.1 Scalar Products; Distance
20.2 Geometry in V
20.3 Orthogonality
20.4 Duality
20.5 Orthogonal Transformations
Chapter 21. Inequalities and Boolean Algebra
21.1 Inequalities
21.2 Some Applications
21.3 Dedekind\'s Completion of the Rationals
21.4 Boolean Algebra
21.5 Ordering a Boolean Algebra
21.6 Homomorphisms
Chapter 22. Polynomials and Equations of Degree n
22.1 Polynomial Forms
22.2 Substitution
22.3 The Remainder Theorem
22.4 Polynomial Functions
22.5 Real and Complex Polynomials
22.6 Derivation
22.7 Solution of Polynomial Equations
22.8 Application to Finite Fields
Part VI. Number Systems and Topology
Chapter 23. The Rational Numbers
23.1 The Peano Axioms
23.2 The System Z
23.3 The system Q
Chapter 24. The Real and Complex Numbers
24.1 The Inadequacy of Q
24.2 Sequences
24.3 Structure of R
24.4 The Order Relation in R
24.5 Decimals
24.6 The Completeness of R
24.7 The Complex Numbers
24.8 Completeness of C
24.9 Quaternions and Hypercomplex Numbers
Chapter 25. Topology of R^n
25.1 Introduction
25.2 Topology within the Erlanger Program
25.3 Some Homeomorphisms
25.4 The Cartesian Product
25.5 Metric Spaces
25.6 Closed and Open Sets
25.7 Dimension
25.8 Compact Spaces
25.9 Quotient Spaces
25.10 Simply Connected Spaces: Homotopy
25.11 The Algebraic Approach
25.12 Manifolds
25.13 Applications and Further Outlook
25.14 Some Books
Part VII. Calculus
Chapter 26. The Algebra R^I
26.1 Intervals
26.2 Algebraic Operations
26.3 Polynomials
26.4 The Reciprocal
26.5 The Order Relation
Chapter 27. Limiting Processes
27.1 Limits
27.2 The Algebra of Limits
27.3 Infinite Limits
27.4 Sequences
Chapter 28. Continuous Functions
28.1 The Algebra C(I)
28.2 Composition
28.3 The Principle of Preservation of Inequalities
28.4 Max and Min
28.5 Two Deeper Theorems
28.6 The Laws of Indices
Chapter 29. Differentiable Functions
29.1 The Differential Coefficient
29.2 The Derivative
29.3 The Algebra D(I)
29.4 Composition
29.5 The Differential d_c f
29.6 Higher Derivatives
29.7 The Rolle Conditions
29.8 Example (The Trigonometric Functions)
29.9 Inverse Functions
Chapter 30. Integration
30.1 The Problem
30.2 Rules for Integration
30.3 Integration by Substitution
30.4 Convergence of Integrals
Part VII continued. Additional Topics in the Calculus
Chapter 31. The Logarithm and the Exponential Function
31.1 The Logarithm
31.2 The Function exp
31.3 The Laws of Indices
Chapter 32. Differential Equations
32.1 Linear First-Order Equations
32.2 Second-Order Equations
Chapter 33. Complex-valued Functions
33.1 Differentiation
33.2 The Function cis
33.3 Algebra of e^z
Chapter 34. Approximation and Iteration
34.1 Taylor\'s Expansion
34.2 Maxima and Minima
34.3 Newton\'s Method of Approximation
34.4 Approximate Integration
34.5 Series
34.6 Further Outlook
Chapter 35. Functions of Several Real Variables
35.1 The Problem
35.2 Continuity
35.3 The Differential
35.4 The Formula for Small Errors
35.5 Differentiability and Derivatives
Chapter 36. Vector-valued Functions
36.1 Differentiability
36.2 Composition
36.3 Co-ordinate Systems
36.4 The Chain Rule of Differentiation
36.5 Summary of Principal Formulae
Chapter 37. C^r-functions
37.1 The Problem
37.2 Taylor\'s Expansion
37.3 Critical Points
37.4 Implicit Functions
37.5 A Clarification
Part VIII. Foundations
Chapter 38. Categories and Functors
38.1 Categories
38.2 Initial, Terminal, Zero Objects
38.3 Functors
38.4 Standard Notions in the Theory of Categories
Chapter 39. Mathematical Logic
39.1 Axioms
39.2 Sets
39.3 Consistency
39.4 Formal Systems
39.5 Examples of the \'Proof Game\'
39.6 Gödel\'s Theorems
39.7 Gödel\'s Proofs
39.8 The Axiom of Choice, and the Continuum Hypothesis
Bibliography
[1]-[14]
[15]-[40]
[41]-[67]
[68]-[94]
[95]-[124]
[125]-[139]
Index
A, B, C
D
E, F
G, H, I
J, K, L
M, N
O, P
Q, R, S
T
U, V, W, Z
Index of Special Symbols