Handbook of Mathematics.
Ebook published on September, 2023
Hardcover published on August, 2023
Softcover published on September, 2023
Format: 15.5x22 cm, 1132 pages, revised
ISBN 978-2-9551990-5-3 : Hardcover
ISBN 978-2-9551990-4-6 : Softcover
ISBN 978-2-9551990-3-9 : EBook
Author: Thierry Vialar
Publisher: HDBOM

table of contents (level 1)

Introduction

Part I. Foundations of Mathematics.

Chap.1. Mathematical Logic.

Section 1. Propositions, Connections

Section 2. Propositional and Predicate Calculus

Section 3. Extension of First-order Predicate Calculus

Section 4. Formal System

Section 5. Demonstrations and Definitions

Chap.2. Set Theory.

Section 1. Basic Concepts

Section 2. Set Algebra

Section 3. Lattice Theory

Section 4. Foundations of Sets

Section 5. Problems of Set Theory

Section 6. Zermelo-Fraenkel Set Theory (ZFC)

Section 7. Banach-Tarski paradox and ZF

Section 8. Hahn-Banach Theorems

Chap.3. Relations and Structures.

Section 1. Relations

Section 2. Maps, Functions

Section 3. Families

Section 4. Laws of Composition

Section 5. Power, Cardinal, Denumerability

Section 6. Cardinals

Section 7. Structures

Section 8. Algebraic Structures

Section 9. Order Structure

Section 10. Ordinals

Section 11. Topological Structures

Chap.4. Arithmetic.

Section 1. Set of Natural Numbers ℕ

Section 2. Denumerability (Counting)

Section 3. Divisibility

Section 4. Integers ℤ

Section 5. Rational Numbers ℚ

Section 6. A General Exercise

Chap.5. Construction of Number System.

Section 1. Semigroup of Natural Numbers

Section 2. Ring of Integers

Section 3. Field of Rational Numbers

Section 4. Real Numbers

Section 5. Complex Numbers

Section 6. Synthesis, Generalization

Part II. Algebra.

Chap.6. Algebra

Section 1. Introduction

Section 2. Group Theory

Section 3. Rings and Fields

Section 4. Modules, Vector Spaces

Section 5. Linear Maps and Matrices

Section 6. Equation and System of Equations

Section 7. Algebra of Polynomials

Section 8. Polynomials over ℝ and ℂ

Section 9. Fractions and Rational Functions

Section 10. Polynomial Rings

Section 11. Field Extensions

Section 12. Prime fields, Finite fields

Section 13. Galois Theory

Section 14. Galois Theory Application

Section 15. Lie Groups

Section 16. Linear Algebra

Section 17. Eigenvalue, Eigensubspace

Section 18. Hermitian Form, Pre-Hilbert Space

Section 19. Exterior Algebra of Vector Space

Part III. Number Theory.

Chap.7. Number Theory

Section 1. Divisibility in an integral domain

Section 2. Diophantine Equations and Residues

Section 3. Absolute Value, Valuation

Section 4. Prime Numbers

Part IV. Geometry.

Chap.8. Geometry

Section 1. Contruction, and Overview

Section 2. Fundamental Concepts

Section 3. Absolute geometry

Section 4. Euclidean and non Euclidean Metrics

Section 5. Affine and Projective Planes

Section 6. Collineations and Correlations

Section 7. Ideal Plane, Coordinates

Section 8. Projective Metric

Section 9. Order and Orientation

Section 10. Angles and Measurements

Section 11. Rigid Transformations

Section 12. Similarity in Geometry

Section 13. Affine Maps

Section 14. Projective maps

Section 15. Analytic Representations

Section 16. Descriptive Geometry

Section 18. Hyperbolic Geometry

Section 19. Elliptic Geometry

Part V. Analytic Geometry.

Chap.9. Analytic Geometry

Section 1. Vector spaces V³

Section 2. Scalar, Vector and Mixed products

Section 3. Line and Plane Equations

Section 4. Sphere and Conics

Section 5. Displacement, Affine map in R³

Section 6. Quadrics

Section 7. Affine Space on Rⁿ

Part VI. Topology.

Chap.10. Topology

Section 1. Overview

Section 3. Basic Topological Notions of ℝ^p

Section 4. Definition of a Topological Space

Section 5. Metric Space, Basis, Neighborhood Basis

Section 6. Topological Map, Topological Subspace

Section 7. Quotient, Product and Sum Spaces

Section 8. Connectedness, Arc-connected space

Section 9. Sequence Convergence and Filter Base

Section 10. Separation Axioms

Section 11. Compactness

Section 12. Metrization

Section 13. Dimension Theory

Section 14. Theory of Curves

Section 15. Completion

Chap.11. Topology (Advanced)

Section 1. Introduction

Section 2. Prerequisites, Reminders

Section 3. Distances and Metric Spaces

Section 4. Limits and Cluster Points

Section 5. Compact and locally compact spaces

Section 6. Connected spaces

Section 7. Metric and Semi-metric Spaces

Section 8. Baire Spaces

Section 9. Mapping Spaces (or Function Spaces)

Section 10. Normed Vector Spaces

Section 11. Hilbert Spaces

Part VII. Algebraic Topology.

Chap.12. Algebraic Topology

Section 1. Homotopy

Section 2. Polyhedra

Section 3. Fundamental Group of a Connected Polyhedron

Section 4. Surfaces

Section 5. Homology Theory

Section 6. Graph Theory

Section 7. Bundles

Chap.13. Algebraic Topology (Advanced)

Section 1. Introduction

Section 2. Some Topological Notions

Section 3. Simplexes

Section 4. The Fundamental Group

Section 5. Singular Homology

Section 6. Long Exact Sequences

Section 7. Excision

Section 8. Simplicial Complexes

Section 9. CW Complexes

Section 10. Natural Transformations

Section 11. Covering Spaces

Section 12. Homotopy Groups

Section 13. Cohomology

Part VIII. Analysis.

Chap.14. Real Analysis

Section 1. Structures on ℝ

Section 2. Sequences, Series

Section 3. Real Functions

Chap.15. Differential Calculus

Section 1. Overview

Section 2. Functions of Differentiable Real Variable

Section 3. Mean Value Theorems

Section 4. Expansion In Series

Section 5. Rational Functions

Section 6. Algebraic Functions

Section 7. Non-algebraic Functions

Section 8. Approximation Theory

Section 9. Interpolation Theory

Section 10. Numerical Resolution of Equations

Section 11. Differential Calculus in ℝⁿ

Chap.16. Integral Calculus

Section 1. Overview

Section 2. Riemann Integral

Section 3. Integration Rules, R-integrable Functions

Section 4. Primitive Functions, Indefinite Integrals

Section 5. Integration Methods, Series Integration

Section 6. Approximation Methods, Generalized Integrals

Section 7. Riemann Integral of Functions of Several Variables

Section 8. Successive Integrations, Change of Variables

Section 9. Riemann Sums and Applications

Section 10. Curvilinear Integral, Surface Integral

Section 11. Field, force, work-done

Section 12. Integration Theorems

Section 13. Jordan Areolar Measure, Lebesgue Measure

Section 14. Measurable Functions, Lebesgue Integral

Chap.17. Functional Analysis

Section 1. Abstract Spaces

Section 2. Differentiable Operators

Section 3. Calculus of Variations

Section 4. Integral Equations

Section 5. Compact Operators

Chap.18. Differential Equations

Section 1. Classic Differential Equations

Section 2. First-order Differential Equations

Section 3. Second-order Differential Equations

Section 4. Linear n-order Differential Equations

Section 5. Systems of Differential Equations

Section 6. Existence and Uniqueness Theorems

Section 7. Numerical Methods

Section 8. Partial Differential Equations

Chap.19. Differential Geometry

Section 1. Curves in R³

Section 2. Plane Curves

Section 3. Regular Patches, Surfaces

Section 4. First Fundamental Form

Section 5. Second Fundamental Form, Curvature

Section 6. Fundamental Theorem

Section 7. Tensors

Section 8. Differential Forms

Section 9. Manifolds, Riemannian Geometry

Section 10. Complements of Differential Geometry

Chap.20. Function Theory (Complex Analysis)

Section 1. Introduction

Section 2. Complex Numbers, Compactification

Section 3. Sequences and Complex Functions

Section 4. Holomorphy

Section 5. Cauchy Integral Theorem and Formulas

Section 6. Power Series Expansion

Section 7. Analytic Continuation

Section 8. Singularities, Laurent Series

Section 9. Meromorphy, Residues

Section 10. Riemann Surfaces

Section 11. Entire Functions

Section 12. Meromorphic functions on C

Section 13. Periodic functions

Section 14. Algebraic functions

Section 15. Conformal transformations

Section 16. Functions of several variables

Chap.21. Complex Analysis (Advanced)

Section 1. Introduction

Section 2. Prerequisites

Section 3. Recall on Functions of Complex Variables

Section 4. Curvilinear Integral

Section 5. Differential Forms in the Plane

Section 6. Holomorphic Functions I

Section 7. Holomorphic Functions II

Section 8. Homotopy

Section 9. Topology of the Complex Plane

Section 10. Cauchy Theorem (Homology Version)

Section 11. Residues

Section 12. Runge Theorem, Applications

Section 13. Conformal Mapping

Section 14. Harmonic Functions

Section 15. Subharmonic Functions

Section 16. (A1) Convolution, Partition of Unity

Section 17. (A2) Distributions

Section 18. (A3) Topology of the Complex Plane II

Part IX. Category Theory.

Chap.22. Areas Involved in Category Theory

Section 1. Areas Involved

Section 2. Algebraic System, Universal Algebra

Section 3. Signatures

Section 4. Structures

Section 5. Interpretations

Section 6. Varieties

Section 7. Algebraic Structures

Section 8. Maps and Structure Homomorphisms

Section 9. Algebraic Structure Morphisms and Kernels

Section 10. Ring Theory

Section 11. Complexes

Section 12. Direct limits

Section 13. Topological spaces

Section 14. Manifolds

Section 15. Homology, Cohomology

Section 16. Sheaf Theory

Section 17. Bundles

Section 18. Scheme Theory

Section 19. Homotopy

Chap.23. Category Theory

Section 1. Introduction

Section 2. Universe

Section 3. Objects

Section 4. Categories

Section 5. Functors

Section 6. Morphisms

Section 7. Graphs and Diagrams

Section 8. Limits and Products

Section 9. Pullbacks and Pushouts

Section 10. Kernels and Cokernels

Section 11. Exact Sequences

Section 12. Sheaves

Section 13. Grothendieck topology, site, sieve

Section 14. Topos

Section 15. Étale

Section 16. Yoneda Lemma, Yoneda Embedding

Part X. Probability and Statistics.

Chap.24. Probability

Section 1. Combinatorial Analysis

Chap.25. Probability Calculation, Statistics

Section 1. Introduction

Section 2. Event, Probability

Section 3. Statistical Distribution, Cumulative Distribution

Section 4. Statistical Methods

Section 5. Statistical Models

Part XI. Applied Mathematics.

Chap.26. Miscellaneous

Section 1. Fourier Series and Fourier Transform

Section 2. Integral transformations

Section 3. Distribution Theory

Chap.27. Optimization

Section 1. Introduction

Section 2. Linear Optimization

Section 3. Simplex Method

Section 4. Other Optimizations

Section 5. Convex sets, concave and convex functions

Section 6. Marginal Optimality Conditions

Chap.28. Dynamical Systems

Section 1. Dynamical Systems

Section 2. Chaos Theory

Section 3. Example in Physics

Section 4. Examples in Economics

Appendices

Symbols, Tables

Mathematical Symbols

Transformation Tables

Statistical Tables

Bibliography

Index