Complete College Mathematics Masterclass

📚 Topics Covered

• Review of functions: domain, range, composition
• Inverse functions and their properties
• Exponential and logarithmic functions
• Trigonometric functions and identities
• Intuitive notion of limits
• Formal epsilon-delta definition of limits
• Limit laws and computation techniques
• One-sided limits and infinite limits
• Limits at infinity and horizontal asymptotes
• Continuity at a point and on intervals

• Intermediate Value Theorem
• Types of discontinuities

📚 Topics Covered

• Definition of derivative as a limit
• Geometric interpretation: tangent lines
• Physical interpretation: rates of change
• Differentiability and continuity relationship
• Basic differentiation rules: power, constant, sum
• Product rule and quotient rule
• Chain rule and implicit differentiation
• Derivatives of trigonometric functions
• Derivatives of exponential and logarithmic functions
• Inverse function derivatives

• Higher-order derivatives
• Logarithmic differentiation

📚 Topics Covered

• Linear approximation and differentials
• L'Hôpital's Rule for indeterminate forms
• Critical points and extrema
• First Derivative Test
• Second Derivative Test
• Concavity and inflection points
• Curve sketching comprehensive method
• Optimization problems
• Related rates problems
• Mean Value Theorem

• Rolle's Theorem
• Newton's Method for root finding

📚 Topics Covered

• Antiderivatives and indefinite integrals
• Definite integrals and Riemann sums
• Fundamental Theorem of Calculus Part I
• Fundamental Theorem of Calculus Part II
• Basic integration rules
• U-substitution method
• Integration of trigonometric functions
• Integration of exponential and logarithmic functions
• Area between curves
• Volumes by cross-sections

• Volumes by cylindrical shells
• Average value of a function

📚 Topics Covered

• Integration by parts
• Trigonometric integrals and substitutions
• Partial fraction decomposition
• Rational function integration
• Improper integrals Type I and II
• Comparison tests for improper integrals
• Numerical integration: Trapezoidal rule
• Simpson's rule and error bounds
• Arc length calculations
• Surface area of revolution

• Work, force, and energy applications
• Center of mass and moments

📚 Topics Covered

• Sequences: convergence and divergence
• Monotonic sequences and bounded sequences
• Series and partial sums
• Geometric series and telescoping series
• Divergence test and p-series
• Comparison tests and limit comparison test
• Ratio test and root test
• Alternating series test and error bounds
• Absolute vs conditional convergence
• Power series and radius of convergence

• Taylor and Maclaurin series
• Taylor polynomial approximations and error

📚 Topics Covered

• Vector spaces: R^n and axioms
• Vector operations: addition, scalar multiplication
• Dot product and cross product
• Vector projections and orthogonality
• Linear combinations and span
• Linear independence and dependence
• Matrix operations: addition, multiplication
• Matrix transpose and properties
• Special matrices: identity, diagonal, symmetric
• Elementary row operations

• Row echelon form and reduced row echelon form
• Gaussian elimination algorithm

📚 Topics Covered

• Consistent vs inconsistent systems
• Homogeneous systems and nontrivial solutions
• Matrix representation of systems
• Augmented matrices and RREF
• Parametric solutions and free variables
• Matrix inverses and invertibility
• Computing inverses using row operations
• Determinants: definition and properties
• Cofactor expansion and row operations
• Cramer's rule for solving systems

• Applications to network flow
• Applications to economics and engineering

📚 Topics Covered

• Logic and truth tables
• Quantifiers: universal and existential
• Direct proof technique
• Proof by contradiction
• Proof by contrapositive
• Mathematical induction
• Strong induction and well-ordering
• Proof by cases
• Existence and uniqueness proofs
• Counterexamples

• Writing clear mathematical arguments
• Common proof strategies and patterns

📚 Topics Covered

• Eigenvalue and eigenvector definition
• Characteristic polynomial
• Finding eigenvalues and eigenvectors
• Algebraic and geometric multiplicity
• Diagonalization of matrices
• Powers of diagonalizable matrices
• Orthogonal matrices and rotations
• Symmetric matrices and spectral theorem
• Quadratic forms
• Principal axes theorem

• Applications to differential equations
• Applications to Markov chains

📚 Topics Covered

• Abstract vector spaces and subspaces
• Basis and dimension
• Coordinate systems and change of basis
• Row space, column space, null space
• Rank-nullity theorem
• Linear transformations: definition and properties
• Kernel and image of linear transformation
• Matrix representation of linear transformations
• Composition of linear transformations
• Isomorphisms and invertible transformations

• Similarity of matrices
• Jordan canonical form introduction

📚 Topics Covered

• Functions of several variables
• Level curves and level surfaces
• Limits in multiple dimensions
• Continuity for multivariable functions
• Partial derivatives: definition and notation
• Higher-order partial derivatives
• Clairaut's theorem on mixed partials
• Tangent planes and linear approximation
• The gradient vector and directional derivatives
• The chain rule for multivariable functions

• Implicit differentiation in multiple variables
• Jacobian matrices

📚 Topics Covered

• Critical points in multiple dimensions
• Second derivative test for two variables
• Hessian matrix and definiteness
• Global vs local extrema
• Constrained optimization: Lagrange multipliers
• Multiple constraints
• Vector fields and flow lines
• Conservative vector fields
• Potential functions
• Divergence and curl

• Physical interpretations
• Applications to physics and engineering

📚 Topics Covered

• Integration of calculus and linear algebra
• Comprehensive problem solving
• Proof portfolio creation
• Mathematical writing
• Presentation skills

📚 Topics Covered

• Double integrals over rectangles
• Iterated integrals and Fubini's theorem
• Double integrals over general regions
• Reversing order of integration
• Double integrals in polar coordinates
• Applications: area, volume, mass
• Center of mass and moments of inertia
• Triple integrals in rectangular coordinates
• Triple integrals in cylindrical coordinates
• Triple integrals in spherical coordinates

• Change of variables: Jacobians
• Applications to probability and physics

📚 Topics Covered

• Parametric curves and arc length
• Line integrals of scalar functions
• Line integrals of vector fields
• Work and circulation
• Fundamental theorem for line integrals
• Green's theorem and applications
• Parametric surfaces
• Surface area calculations
• Surface integrals of scalar functions
• Surface integrals of vector fields (flux)

• Stokes' theorem
• Divergence theorem (Gauss's theorem)

📚 Topics Covered

• Classification of differential equations
• Direction fields and solution curves
• Separable equations
• Linear first-order equations
• Integrating factors method
• Exact equations and exactness test
• Homogeneous equations
• Bernoulli equations
• Existence and uniqueness theorems
• Numerical methods: Euler's method

• Improved Euler and Runge-Kutta
• Applications to physics, biology, economics

📚 Topics Covered

• Second-order linear equations
• Homogeneous equations with constant coefficients
• Characteristic equation method
• Complex roots and oscillations
• Repeated roots and reduction of order
• Nonhomogeneous equations
• Method of undetermined coefficients
• Variation of parameters
• Higher-order linear equations
• Systems of first-order linear equations

• Matrix exponentials
• Phase plane analysis

📚 Topics Covered

• Definition and existence of Laplace transform
• Transforms of basic functions
• Linearity and shifting theorems
• Transforms of derivatives and integrals
• Inverse Laplace transforms
• Partial fraction decomposition for inverse transforms
• Convolution theorem
• Solving ODEs with Laplace transforms
• Systems of ODEs via Laplace
• Transfer functions

• Applications to control theory
• Discontinuous forcing functions

📚 Topics Covered

• Naive set theory and paradoxes
• Set operations: union, intersection, complement
• Power sets and Cartesian products
• Functions as relations
• Injective, surjective, bijective functions
• Cardinality and countability
• Cantor's diagonal argument
• Relations: reflexive, symmetric, transitive
• Equivalence relations and partitions
• Partial orders and Hasse diagrams

• Well-ordering principle
• Axiom of choice introduction

📚 Topics Covered

• Basic counting principles
• Permutations and combinations
• Binomial theorem and Pascal's triangle
• Multinomial coefficients
• Inclusion-exclusion principle
• Pigeonhole principle
• Generating functions
• Recurrence relations
• Solving linear recurrences
• Catalan numbers

• Stirling numbers
• Partitions of integers

📚 Topics Covered

• Graphs: vertices, edges, degree
• Types of graphs: simple, directed, weighted
• Graph representations: adjacency matrix, list
• Paths, cycles, and connectivity
• Trees and spanning trees
• Minimum spanning trees: Kruskal, Prim
• Shortest paths: Dijkstra, Bellman-Ford
• Eulerian and Hamiltonian paths
• Graph coloring and chromatic number
• Planar graphs and Euler's formula

• Bipartite graphs and matching
• Network flows and max-flow min-cut

📚 Topics Covered

• Divisibility and greatest common divisors
• Euclidean algorithm and extended version
• Prime numbers and fundamental theorem
• Modular arithmetic
• Chinese Remainder Theorem
• Fermat's Little Theorem
• Euler's theorem and totient function
• Wilson's theorem
• Quadratic residues
• Primitive roots

• RSA cryptography basics
• Diophantine equations

📚 Topics Covered

• Boolean operations and laws
• Truth tables and logical equivalence
• Normal forms: DNF and CNF
• Karnaugh maps
• Logic gates and circuits
• Minimization of Boolean functions
• Propositional logic and inference rules
• Predicate logic and quantifiers
• Formal proofs in logic
• Completeness and soundness

• Introduction to model theory
• Applications to computer science

📚 Topics Covered

• Sample spaces and events
• Probability axioms and properties
• Conditional probability
• Bayes' theorem and applications
• Independence of events
• Random variables: discrete and continuous
• Probability mass and density functions
• Cumulative distribution functions
• Expected value and variance
• Moment generating functions

• Common discrete distributions
• Common continuous distributions

📚 Topics Covered

• Joint probability distributions
• Marginal and conditional distributions
• Independence of random variables
• Covariance and correlation
• Conditional expectation
• Transformations of random variables
• Order statistics
• Law of large numbers
• Central limit theorem
• Normal approximations

• Characteristic functions
• Convergence concepts

📚 Topics Covered

• Point estimation: MLE and method of moments
• Properties of estimators: bias, consistency
• Confidence intervals
• Hypothesis testing framework
• Type I and Type II errors
• Power of tests
• t-tests and chi-square tests
• ANOVA basics
• Nonparametric tests
• Linear regression

• Multiple regression
• Model diagnostics

📚 Topics Covered

• Introduction to stochastic processes
• Markov chains: discrete time
• Transition matrices and steady states
• Classification of states
• Absorbing Markov chains
• Continuous-time Markov chains
• Poisson processes
• Birth-death processes
• Queueing theory basics
• Brownian motion introduction

• Random walks
• Applications to finance and biology

📚 Topics Covered

• Integration of discrete and continuous mathematics
• Statistical modeling project
• Algorithm implementation
• Research paper reading
• Technical presentation

📚 Topics Covered

• Metric spaces: definition and examples
• Open and closed sets
• Interior, closure, and boundary
• Convergent sequences in metric spaces
• Cauchy sequences and completeness
• Compactness: sequential and open cover
• Heine-Borel theorem
• Connected spaces
• Continuous functions between metric spaces
• Uniform continuity

• Lipschitz continuity
• Homeomorphisms

📚 Topics Covered

• The real numbers: completeness axiom
• Supremum and infimum
• Sequences and series of real numbers
• Limit superior and limit inferior
• Functions of real variables
• Continuity and uniform continuity on R
• Intermediate value theorem proof
• Extreme value theorem proof
• Monotone functions
• Functions of bounded variation

• Absolutely continuous functions
• Discontinuities classification

📚 Topics Covered

• Derivative as linear approximation
• Differentiability in R^n
• Partial derivatives vs differentiability
• Chain rule proof
• Mean value theorem and generalizations
• Taylor's theorem with remainder
• Inverse function theorem
• Implicit function theorem
• Critical point analysis
• Morse lemma

• Sard's theorem introduction
• Applications to optimization

📚 Topics Covered

• Riemann integral construction
• Riemann integrability conditions
• Properties of Riemann integral
• Fundamental theorem of calculus proof
• Integration techniques review
• Improper Riemann integrals
• Functions of bounded variation
• Riemann-Stieltjes integral
• Convergence theorems limitations
• Counterexamples in integration

• Numerical integration error analysis
• Applications to probability

📚 Topics Covered

• Pointwise convergence
• Uniform convergence
• Weierstrass M-test
• Continuity of uniform limits
• Integration of uniform limits
• Differentiation of uniform limits
• Power series and radius of convergence
• Analytic functions
• Weierstrass approximation theorem
• Stone-Weierstrass theorem

• Fourier series introduction
• Convergence of Fourier series

📚 Topics Covered

• Groups: axioms and examples
• Subgroups and subgroup tests
• Cyclic groups and generators
• Group homomorphisms
• Kernel and image
• Isomorphism theorems
• Normal subgroups and quotient groups
• Lagrange's theorem
• Group actions
• Orbit-stabilizer theorem

• Sylow theorems
• Classification of finite abelian groups

📚 Topics Covered

• Rings: definition and examples
• Ring homomorphisms
• Ideals and quotient rings
• Prime and maximal ideals
• Integral domains
• Principal ideal domains
• Unique factorization domains
• Euclidean domains
• Polynomial rings
• Field of fractions

• Chinese remainder theorem for rings
• Localization of rings

📚 Topics Covered

• Fields: definition and examples
• Field extensions
• Algebraic and transcendental elements
• Degree of field extensions
• Splitting fields
• Algebraic closure
• Finite fields structure
• Cyclotomic fields
• Galois theory introduction
• Fundamental theorem of Galois theory

• Solvability by radicals
• Constructions with ruler and compass

📚 Topics Covered

• Vector spaces over arbitrary fields
• Linear transformations abstract theory
• Dual spaces and dual basis
• Bilinear forms
• Quadratic forms
• Inner product spaces
• Orthogonalization: Gram-Schmidt
• Adjoint operators
• Normal operators
• Spectral theorem

• Singular value decomposition
• Tensor products

📚 Topics Covered

• Modules over rings
• Submodules and quotient modules
• Module homomorphisms
• Free modules
• Finitely generated modules
• Torsion modules
• Structure theorem for finitely generated modules over PID
• Applications to linear algebra
• Jordan canonical form via modules
• Projective and injective modules

• Exact sequences
• Introduction to homological algebra

📚 Topics Covered

• Topological spaces: definition and examples
• Basis and subbasis
• Closed sets and closure operators
• Interior and boundary
• Hausdorff spaces
• Continuous functions
• Homeomorphisms
• Connectedness
• Path connectedness
• Compactness

• Tychonoff's theorem
• Separation axioms

📚 Topics Covered

• Homotopy and homotopy equivalence
• Fundamental group
• Computing fundamental groups
• Van Kampen's theorem
• Covering spaces
• Lifting properties
• Classification of covering spaces
• Introduction to homology
• Simplicial complexes
• Simplicial homology

• Singular homology basics
• Applications to fixed point theorems

📚 Topics Covered

• Complex numbers and complex plane
• Complex functions and analyticity
• Cauchy-Riemann equations
• Harmonic functions
• Complex integration
• Cauchy's theorem
• Cauchy integral formula
• Taylor and Laurent series
• Residues and residue theorem
• Evaluation of real integrals

• Conformal mappings
• Möbius transformations

📚 Topics Covered

• Maximum modulus principle
• Schwarz lemma
• Analytic continuation
• Riemann surfaces introduction
• Entire functions
• Meromorphic functions
• Infinite products
• Weierstrass factorization
• Gamma function
• Riemann zeta function

• Prime number theorem outline
• Applications to number theory

📚 Topics Covered

• Pure mathematics research project
• Original proof development
• Mathematical exposition
• Research paper writing
• Peer review process

📚 Topics Covered

• Classification of PDEs
• First-order linear PDEs
• Method of characteristics
• Quasi-linear PDEs
• General first-order PDEs
• Cauchy problem
• Envelopes and singular solutions
• Conservation laws
• Shock waves
• Weak solutions

• Rankine-Hugoniot conditions
• Applications to traffic flow

📚 Topics Covered

• Classification: elliptic, parabolic, hyperbolic
• Heat equation: derivation and properties
• Separation of variables
• Fourier series solutions
• Maximum principle for heat equation
• Wave equation: d'Alembert's solution
• Energy methods
• Laplace equation: harmonic functions
• Poisson equation
• Green's functions

• Boundary value problems
• Sturm-Liouville theory

📚 Topics Covered

• Fourier transform and PDEs
• Heat equation via Fourier transform
• Laplace transform for PDEs
• Hankel transforms
• Integral transform methods
• Duhamel's principle
• Green's function method
• Eigenfunction expansions
• Bessel functions
• Legendre polynomials

• Spherical harmonics
• Special functions in PDEs

📚 Topics Covered

• Finite difference methods
• Stability and convergence
• CFL condition
• Implicit vs explicit schemes
• Crank-Nicolson method
• Finite element method basics
• Weak formulation
• Galerkin method
• Basis functions
• Assembly process

• Spectral methods introduction
• Multigrid methods

📚 Topics Covered

• Nonlinear heat equation
• Burger's equation
• KdV equation and solitons
• Nonlinear Schrödinger equation
• Reaction-diffusion equations
• Pattern formation
• Traveling waves
• Bifurcation in PDEs
• Variational methods
• Fixed point theorems for PDEs

• Existence and uniqueness
• Regularity theory basics

📚 Topics Covered

• Floating point arithmetic
• Condition numbers and stability
• LU decomposition
• Cholesky decomposition
• QR decomposition
• Singular value decomposition
• Iterative methods: Jacobi, Gauss-Seidel
• Conjugate gradient method
• GMRES and Krylov methods
• Preconditioning

• Sparse matrix techniques
• Parallel algorithms

📚 Topics Covered

• Polynomial interpolation
• Lagrange interpolation
• Newton's divided differences
• Hermite interpolation
• Spline interpolation
• B-splines and NURBS
• Least squares approximation
• Orthogonal polynomials
• Chebyshev approximation
• Rational approximation

• Padé approximants
• Wavelets introduction

📚 Topics Covered

• Newton-Cotes formulas
• Gaussian quadrature
• Adaptive quadrature
• Romberg integration
• Monte Carlo integration
• Quasi-Monte Carlo methods
• Multidimensional integration
• Numerical differentiation
• Richardson extrapolation
• Automatic differentiation

• Complex step differentiation
• Applications to optimization

📚 Topics Covered

• Unconstrained optimization
• Gradient descent variants
• Newton's method for optimization
• Quasi-Newton methods: BFGS, L-BFGS
• Conjugate gradient for optimization
• Trust region methods
• Line search strategies
• Constrained optimization
• Lagrange multipliers numerical
• Penalty and barrier methods

• Interior point methods
• Sequential quadratic programming

📚 Topics Covered

• Fast Fourier Transform (FFT)
• Fast multiplication algorithms
• Fast matrix multiplication
• Divide and conquer strategies
• Multigrid methods
• Fast multipole method
• Hierarchical matrices
• Randomized algorithms
• Compressed sensing basics
• Sparse recovery

• Low-rank approximations
• Tensor decompositions

📚 Topics Covered

• Population dynamics models
• Predator-prey systems
• Competition models
• Epidemic models: SIR, SEIR
• Age-structured models
• Spatial models and diffusion
• Pattern formation in biology
• Biochemical networks
• Neural models
• Evolutionary game theory

• Phylogenetic trees
• Systems biology approaches

📚 Topics Covered

• Time value of money
• Portfolio theory
• CAPM model
• Options and derivatives
• Black-Scholes equation
• Greeks and hedging
• Monte Carlo methods in finance
• Interest rate models
• Credit risk models
• Value at Risk

• Stochastic volatility
• Jump diffusion models

📚 Topics Covered

• Principal component analysis
• Singular value decomposition applications
• Matrix factorizations for data
• Kernel methods
• Support vector machines math
• Neural network mathematics
• Backpropagation derivation
• Optimization in machine learning
• Regularization theory
• Statistical learning theory

• Information theory basics
• Compressed sensing applications

📚 Topics Covered

• Conservation laws in fluids
• Navier-Stokes equations
• Potential flow
• Boundary layers
• Turbulence introduction
• Computational fluid dynamics basics
• Elasticity theory
• Stress and strain tensors
• Constitutive equations
• Wave propagation in solids

• Fracture mechanics basics
• Multiphysics modeling

📚 Topics Covered

• Choosing research topic
• Literature review methods
• Research question formulation
• Mathematical writing
• LaTeX for mathematics
• Proof techniques review
• Computational experiments
• Data visualization for mathematics
• Collaboration tools
• Version control for research

• Presenting mathematics
• Peer review process

📚 Topics Covered

• Algebraic geometry introduction
• Differential geometry basics
• Lie groups and Lie algebras
• Representation theory
• Category theory basics
• Homological algebra
• K-theory introduction
• Operator theory
• Harmonic analysis
• Ergodic theory

• Mathematical physics topics
• Current research areas

📚 Topics Covered

• Graduate school preparation
• GRE Mathematics subject test
• Research opportunities
• Industry applications of mathematics
• Quantitative careers overview
• Academic career paths
• Mathematical consulting
• Teaching mathematics
• Open problems in mathematics
• Mathematical communities

• Conferences and workshops
• Lifelong learning in mathematics