Mathematics Foundations

__notoc__ Table of Contents Introductory Material About This Book Mathematics Foundations/How to Use This Textbook Foundations 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/ 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 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/ 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/ 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 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/ Abstract Algebra /7.1 Groups/ /7.2 Rings and Fields/ /7.3 Homomorphisms and Isomorphisms/ /7.4 Polynomial Rings/ /7.5 Galois Theory/ 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 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/ 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/ 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/ 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 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/ Non-Euclidean Geometry /14.1 Hyperbolic Geometry/ /14.2 Elliptic Geometry/ /14.3 Projective Geometry/ /14.4 Differential Geometry/ /14.5 Riemannian Geometry/ Topology /15.1 Topological Spaces/ /15.2 Continuity and Homeomorphisms/ /15.3 Connectedness and Compactness/ /15.4 Metric Spaces/ /15.5 Manifolds/ Applied Mathematics 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/ Statistics /17.1 Descriptive Statistics/ /17.2 Statistical Inference/ /17.3 Hypothesis Testing/ /17.4 Regression Analysis/ /17.5 Bayesian Statistics/ 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/ Optimization /19.1 Linear Programming/ /19.2 Nonlinear Programming/ /19.3 Dynamic Programming/ /19.4 Convex Optimization/ /19.5 Game Theory/ Advanced Topics 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/ 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/ 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/