Core Mathematics IV module (MA22001)

Core Mathematics IV brings together the algebra and calculus developed across the Core Mathematics sequence, introducing some of the most powerful tools used in advanced mathematics and mathematical modelling. You will study vector calculus and Fourier analysis, two areas that are central to pure mathematics, physics, engineering, and applied sciences. 

You will explore curves, surfaces, scalar fields, and vector fields, learning how mathematics describes quantities that vary through space. How do we differentiate and integrate in three dimensions? Concepts such as gradient, divergence, and curl help answer this question, allowing you to analyse fields that appear in fluid flow, heat transfer, electromagnetism, climate modelling, and many other scientific models. 

You will also study line, surface, and volume integrals, and see how the Divergence Theorem and Stokes’ Theorem connect local behaviour to global effects. You will explore Fourier series, learning how complex repeating functions can be broken down into simpler waves and used to solve partial differential equations, including Laplace’s equation. 

This module completes the Core Mathematics sequence, developing your ability to model complex systems, analyse mathematical and physical structures, and solve advanced problems. By the end of the sequence, you will be well prepared for honours-level study in mathematics, physics, and related quantitative fields. 

What you will learn

In this module, you will:

• study curves, surfaces, and fields in three-dimensional space 
• explore gradient, divergence, curl, and vector potentials 
• calculate line, surface, and volume integrals 
• apply the Divergence Theorem and Stokes’ Theorem 
• use orthogonal coordinates in mathematical problems 
• study Fourier series and periodic functions 
• use separation of variables to solve key partial differential equations

By the end of the module, you will be able to:

• use vector calculus to analyse mathematical and physical systems 
• apply major theorems to solve problems involving fields and flows 
• represent functions using Fourier series 
• solve introductory partial differential equations 
• connect mathematical methods to applications in physics, biology, and engineering

Assignments / assessment

• Coursework (40%) 
• Final, written exam (60%) 

Teaching methods / timetable

You will learn through a combination of lectures and tutorials designed to build confidence and problem-solving ability. 

Lectures introduce the key ideas, methods, and worked examples, supported by clear online lecture notes. 

In tutorials, you will apply these ideas by solving problems individually and in groups, with guidance and feedback from lecturers.