Advanced Mathematics Course

Lesson 1 • Logic and Propositional Calculus

Covers truth tables, logical connectives, and quantifiers as tools for precise mathematical argument. Establishes the formal language used throughout all subsequent chapters.

Lesson 2 • Set Theory and Operations

Introduces sets, subsets, and operations including union, intersection, and complement. Provides the structural vocabulary underlying functions, relations, and number systems.

Lesson 3 • Proof Techniques and Strategies

Teaches direct proof, proof by contradiction, contrapositive, and induction. Learners apply each method to number-theoretic and algebraic claims.

Lesson 1 • Polynomial Rings and Factorization

Studies polynomial rings over fields, irreducibility criteria, and unique factorization. Connects abstract factorization theory to practical root-finding methods.

Lesson 2 • Divisibility and Modular Arithmetic

Covers divisibility rules, the Euclidean algorithm, and congruences. Forms the computational backbone for cryptographic and algebraic applications introduced later.

Lesson 3 • Rings, Ideals, and Quotient Structures

Defines rings, integral domains, and ideals, then constructs quotient rings. Bridges group theory to polynomial algebra and field extensions.

Lesson 4 • Field Extensions and Galois Theory

Explores algebraic and transcendental extensions, splitting fields, and Galois groups. Culminates in the unsolvability of the general quintic by radicals.

Lesson 1 • Riemann Integration Theory

Constructs the Riemann integral via upper and lower sums and proves the fundamental theorem of calculus. Establishes integrability conditions for bounded functions.

Lesson 2 • Limits, Continuity, and Uniform Continuity

Defines epsilon-delta limits and continuity, then distinguishes pointwise from uniform continuity. Proves the extreme value and intermediate value theorems.

Lesson 3 • Metric Spaces and Topological Concepts

Generalises analysis to metric spaces, defining open sets, compactness, and connectedness. Prepares students for functional analysis and advanced geometry.

Lesson 4 • Differentiation and Mean Value Theorems

Develops the derivative rigorously and proves Rolle's, mean value, and Taylor's theorems. Connects analytic properties to geometric interpretations of functions.

Lesson 1 • Multiple Integration Techniques

Develops double and triple integrals using Fubini's theorem and coordinate transformations. Covers polar, cylindrical, and spherical coordinate systems.

Lesson 2 • Surface Integrals and Integral Theorems

Computes surface integrals and flux, then proves Stokes' and Divergence theorems. Unifies the integral theorems as instances of a generalised Stokes' theorem.

Lesson 3 • Vector Fields and Line Integrals

Introduces gradient, divergence, and curl operators and evaluates line integrals of vector fields. Establishes path independence and conservative field conditions.

Lesson 4 • Optimisation in Multiple Variables

Applies second-derivative tests and Lagrange multipliers to locate extrema of multivariable functions. Connects optimisation theory to applied modelling problems.

Lesson 1 • Vector Spaces and Subspaces

Defines abstract vector spaces over fields, bases, and dimension. Connects the abstract definition to concrete coordinate spaces and function spaces.

Lesson 2 • Matrix Decompositions and Applications

Covers LU, QR, and singular value decompositions and their computational roles. Applies SVD to low-rank approximation and principal component analysis.

Lesson 3 • Eigenvalues and Diagonalisation

Computes eigenvalues and eigenvectors via the characteristic polynomial and diagonalises matrices when possible. Introduces the minimal polynomial and Cayley-Hamilton theorem.

Lesson 4 • Inner Product Spaces and Orthogonality

Defines inner products, norms, and orthogonal complements, then applies Gram-Schmidt orthogonalisation. Proves the spectral theorem for symmetric operators.

Lesson 1 • First-Order Differential Equations

Solves separable, linear, exact, and Bernoulli equations and analyses existence and uniqueness. Introduces direction fields and qualitative behaviour of solutions.

Lesson 2 • Laplace Transform Methods

Uses the Laplace transform to convert ODEs to algebraic equations and handles discontinuous forcing. Covers convolution and the transfer function concept.

Lesson 3 • Series Solutions and Special Functions

Applies power series and Frobenius methods near ordinary and regular singular points. Introduces Bessel and Legendre equations as canonical examples.

Lesson 4 • Linear Systems and Higher-Order ODEs

Converts higher-order linear ODEs to first-order systems and solves via eigenvalue methods. Applies variation of parameters and undetermined coefficients.

Lesson 1 • Residue Theorem and Applications

Computes residues at poles and applies the residue theorem to evaluate definite integrals. Covers the argument principle and Rouche's theorem.

Lesson 2 • Conformal Mappings and Applications

Studies angle-preserving maps, Mobius transformations, and the Riemann mapping theorem. Applies conformal maps to solve Laplace's equation on irregular domains.

Lesson 3 • Complex Numbers and Analytic Functions

Reviews complex arithmetic, introduces the Cauchy-Riemann equations, and defines analyticity. Establishes the connection between complex differentiability and harmonic functions.

Lesson 4 • Taylor and Laurent Series

Expands analytic functions in Taylor series and meromorphic functions in Laurent series. Classifies isolated singularities as removable, poles, or essential.

Lesson 1 • Fourier and Laplace Transforms for PDEs

Uses Fourier and Laplace transforms to solve PDEs on unbounded domains. Derives the heat kernel and analyses dispersion relations.

Lesson 2 • Classification and Well-Posedness of PDEs

Classifies second-order linear PDEs as elliptic, parabolic, or hyperbolic and states well-posedness conditions. Introduces canonical forms and characteristic curves.

Lesson 3 • Fourier Series and Convergence

Derives Fourier series coefficients and analyses pointwise and uniform convergence. Proves Parseval's identity and discusses Gibbs phenomenon.

Lesson 4 • Separation of Variables Method

Applies separation of variables to heat, wave, and Laplace equations on standard domains. Connects eigenfunction expansions to Sturm-Liouville theory.