Mathematics Foundations

Table of Contents

Introductory Material

1. About This Book
2. Mathematics Foundations/How to Use This Textbook

Foundations

1. Set Theory
   • 1.1 Basic Concepts and Notation
   • 1.2 Set Operations
   • 1.3 Relations and Functions
   • 1.4 Cardinality
   • 1.5 Axioms of Set Theory
2. Logic and Proof
   • 2.1 Propositional Logic
   • 2.2 Predicate Logic
   • 2.3 Methods of Proof
   • 2.4 Mathematical Induction
   • 2.5 Contradiction and Contraposition

Number Systems

1. Number Theory
   • 3.1 Natural Numbers and Integers
   • 3.2 Divisibility and Prime Numbers
   • 3.3 Greatest Common Divisors and Least Common Multiples
   • 3.4 Modular Arithmetic
   • 3.5 Diophantine Equations
2. Real Numbers
   • 4.1 Rational and Irrational Numbers
   • 4.2 Completeness of the Real Number Line
   • 4.3 Sequences and Series
   • 4.4 Limits and Convergence
   • 4.5 Decimal Representations
3. Complex Numbers
   • 5.1 Definitions and Basic Properties
   • 5.2 Geometric Interpretation
   • 5.3 Polar Form and Euler's Formula
   • 5.4 Complex Functions
   • 5.5 Applications in Physics and Engineering

Algebra

1. Linear Algebra
   • 6.1 Vectors and Vector Spaces
   • 6.2 Matrices and Matrix Operations
   • 6.3 Systems of Linear Equations
   • 6.4 Determinants
   • 6.5 Eigenvalues and Eigenvectors
   • 6.6 Linear Transformations
2. Abstract Algebra
   • 7.1 Groups
   • 7.2 Rings and Fields
   • 7.3 Homomorphisms and Isomorphisms
   • 7.4 Polynomial Rings
   • 7.5 Galois Theory
3. Polynomials and Equations
   • 8.1 Polynomial Functions
   • 8.2 Roots and Factorization
   • 8.3 Fundamental Theorem of Algebra
   • 8.4 Solving Equations
   • 8.5 Numerical Methods

Calculus

1. Differential Calculus
   • 9.1 Limits and Continuity
   • 9.2 Derivatives and Differentiation Rules
   • 9.3 Applications of Derivatives
   • 9.4 Optimization Problems
   • 9.5 Taylor Series
2. Integral Calculus
   • 10.1 Definite and Indefinite Integrals
   • 10.2 Techniques of Integration
   • 10.3 Applications of Integration
   • 10.4 Improper Integrals
   • 10.5 Numerical Integration
3. Multivariable Calculus
   • 11.1 Functions of Several Variables
   • 11.2 Partial Derivatives
   • 11.3 Multiple Integrals
   • 11.4 Vector Calculus
   • 11.5 Line and Surface Integrals
   • 11.6 Theorems of Green, Stokes, and Gauss
4. Differential Equations
   • 12.1 First-Order Differential Equations
   • 12.2 Second-Order Linear Equations
   • 12.3 Systems of Differential Equations
   • 12.4 Series Solutions
   • 12.5 Partial Differential Equations
   • 12.6 Applications in Science and Engineering

Geometry and Topology

1. Euclidean Geometry
   • 13.1 Points, Lines, and Planes
   • 13.2 Triangles and Polygons
   • 13.3 Circles and Spheres
   • 13.4 Coordinate Geometry
   • 13.5 Transformations
2. Non-Euclidean Geometry
   • 14.1 Hyperbolic Geometry
   • 14.2 Elliptic Geometry
   • 14.3 Projective Geometry
   • 14.4 Differential Geometry
   • 14.5 Riemannian Geometry
3. Topology
   • 15.1 Topological Spaces
   • 15.2 Continuity and Homeomorphisms
   • 15.3 Connectedness and Compactness
   • 15.4 Metric Spaces
   • 15.5 Manifolds

Applied Mathematics

1. Probability Theory
   • 16.1 Sample Spaces and Events
   • 16.2 Probability Axioms
   • 16.3 Conditional Probability
   • 16.4 Random Variables
   • 16.5 Probability Distributions
   • 16.6 Expectation and Variance
2. Statistics
   • 17.1 Descriptive Statistics
   • 17.2 Statistical Inference
   • 17.3 Hypothesis Testing
   • 17.4 Regression Analysis
   • 17.5 Bayesian Statistics
3. Numerical Analysis
   • 18.1 Error Analysis
   • 18.2 Interpolation and Approximation
   • 18.3 Numerical Integration and Differentiation
   • 18.4 Numerical Linear Algebra
   • 18.5 Numerical Solutions to Differential Equations
4. Optimization
   • 19.1 Linear Programming
   • 19.2 Nonlinear Programming
   • 19.3 Dynamic Programming
   • 19.4 Convex Optimization
   • 19.5 Game Theory

Advanced Topics

1. Functional Analysis
   • 20.1 Normed Spaces
   • 20.2 Banach and Hilbert Spaces
   • 20.3 Linear Operators
   • 20.4 Spectral Theory
   • 20.5 Applications to Differential Equations
2. Measure Theory
   • 21.1 Measures and Measurable Functions
   • 21.2 Lebesgue Integration
   • 21.3 Convergence Theorems
   • 21.4 Product Measures
   • 21.5 Radon-Nikodym Theorem
3. Number Theory (Advanced)
   • 22.1 Analytic Number Theory
   • 22.2 Algebraic Number Theory
   • 22.3 Elliptic Curves
   • 22.4 Diophantine Approximation
   • 22.5 Transcendental Numbers