A Comprehensive Textbook of Classical Mathematics: A Contemporary Interpretation

Description

arithmetic of the integers, linear algebra, an introduction to group theory, the theory of polynomial functions and polynomial equations, and some Boolean algebra. It could be supplemented, of course, by material from other chapters. Again, Course 5 (Calculus) aiscusses the differential and integral calculus more or less from the beginnings of these theories, and proceeds through functions of several real variables, functions of a complex variable, and topics of real analysis such as the implicit function theorem. We would, however, like to make a further point with regard to the appropriateness of our text in course work. We emphasized in the Introduction to the original edition that, in the main, we had in mind the reader who had already met the topics once and wished to review them in the light of his (or her) increased knowledge and mathematical maturity. We therefore believe that our book could form a suitable basis for American graduate courses in the mathematical sciences, especially those prerequisites for a Master's degree. 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