Applied mathematics course

Lesson 1 • Mathematical Logic and Proof Techniques

Introduces propositional logic, quantifiers, and standard proof strategies. Develops rigorous argumentation skills used throughout the course.

Lesson 2 • Algebraic Manipulation and Equations

Reinforces polynomial, rational, and exponential algebra with systematic equation-solving. Provides computational fluency needed for calculus and linear algebra.

Lesson 3 • Sets, Relations, and Functions

Defines sets, mappings, and relational structures as the language of modern mathematics. Connects directly to algebraic and analytic frameworks ahead.

Lesson 4 • Coordinate Geometry Essentials

Covers the Cartesian plane, distance, slope, and conic sections. Bridges algebraic and geometric thinking for later analytic and vector work.

Lesson 1 • Limits and Continuity

Defines limits rigorously and characterises continuous functions. Establishes the analytic foundation for differentiation and integration.

Lesson 2 • Differentiation Rules and Applications

Derives and applies power, product, quotient, and chain rules. Connects derivatives to optimisation, curve sketching, and rate problems.

Lesson 3 • Integration Techniques

Introduces antiderivatives, the Fundamental Theorem, and key integration methods. Builds computational skill for area, accumulation, and differential equations.

Lesson 4 • Applications of Integration

Applies definite integrals to area, volume, arc length, and physical quantities. Reinforces the connection between integration and real-world measurement.

Lesson 1 • Network Flows and Optimisation

Introduces max-flow min-cut, shortest path, and matching algorithms on graphs. Applies discrete optimisation to logistics, scheduling, and resource allocation.

Lesson 2 • Recurrence Relations and Generating Functions

Solves linear recurrences and introduces generating functions for sequence analysis. Bridges combinatorics with algorithmic complexity and dynamic programming.

Lesson 3 • Graph Theory Fundamentals

Defines graphs, trees, paths, and connectivity with key algorithms. Connects graph structures to network modelling and optimisation problems.

Lesson 4 • Counting Principles and Combinatorics

Covers permutations, combinations, and the inclusion-exclusion principle. Provides the counting foundation for probability and algorithm analysis.

Lesson 1 • Determinants and Their Applications

Defines determinants via cofactor expansion and row operations. Uses determinants to assess invertibility, compute volume, and apply Cramer's rule.

Lesson 2 • Vectors and Vector Spaces

Defines vectors, linear combinations, span, and basis in abstract and concrete settings. Provides the geometric and algebraic vocabulary for all matrix work.

Lesson 3 • Matrix Operations and Systems

Covers matrix arithmetic, row reduction, and solution of linear systems. Connects matrix algebra to the vector space concepts introduced earlier.

Lesson 4 • Eigenvalues and Eigenvectors

Derives eigenvalues from the characteristic polynomial and interprets eigenvectors geometrically. Prepares students for diagonalisation and spectral applications.

Lesson 1 • Partial Derivatives and Gradients

Defines partial derivatives, directional derivatives, and the gradient vector. Connects these tools to the geometry of surfaces and rates of change in multiple directions.

Lesson 2 • Vector Calculus Fundamentals

Introduces line integrals, surface integrals, and the major integral theorems. Connects vector fields to physical concepts such as flux and circulation.

Lesson 3 • Multivariable Optimisation

Applies second-order conditions and Lagrange multipliers to locate extrema under constraints. Extends single-variable optimisation to higher-dimensional settings.

Lesson 1 • Discrete and Continuous Distributions

Covers key probability distributions including binomial, Poisson, normal, and exponential. Connects distribution choice to real-world data-generating processes.

Lesson 2 • Regression and Correlation Analysis

Develops simple and multiple linear regression models with diagnostic checks. Applies regression to prediction and relationship quantification in applied contexts.

Lesson 3 • Probability Foundations

Defines sample spaces, events, and axioms of probability. Establishes the formal framework underlying all statistical reasoning in the chapter.

Lesson 4 • Descriptive Statistics and Data Summaries

Covers measures of centre, spread, and shape for univariate and bivariate data. Provides tools for summarising datasets before formal inference.

Lesson 1 • Laplace Transform Methods

Uses the Laplace transform to convert ODEs into algebraic equations for efficient solution. Particularly useful for piecewise forcing functions and engineering applications.

Lesson 2 • Second-Order Linear Equations

Solves homogeneous and non-homogeneous second-order ODEs with constant coefficients. Models oscillatory and damped systems common in applied settings.

Lesson 3 • First-Order Differential Equations

Covers separable, linear, and exact first-order ODEs with initial value problems. Establishes solution techniques and qualitative analysis for simple dynamic models.

Lesson 4 • Systems of Differential Equations

Analyses coupled ODE systems using matrix methods and phase plane techniques. Connects eigenvalue theory from linear algebra to dynamic system behaviour.

Lesson 1 • Principles of Mathematical Modelling

Defines the modelling cycle: problem identification, assumption setting, formulation, and validation. Establishes a systematic framework for applying mathematics to real problems.

Lesson 2 • Model Communication and Reporting

Covers technical writing, visualisation, and presentation of mathematical results to diverse audiences. Prepares students to deliver actionable insights from complex models.

Lesson 3 • Optimisation Models in Practice

Applies linear programming, nonlinear optimisation, and integer programming to applied scenarios. Connects calculus and discrete methods to decision-making under constraints.

Lesson 4 • Dynamical Systems and Simulation

Models time-evolving systems using ODEs, difference equations, and agent-based approaches. Connects differential equations and discrete maths to simulation-based analysis.