Core Mathematics IV module (MA22001)

This module brings together the separate strands of algebra and calculus from the first three Core Mathematics modules to cover two important areas of mathematics.

Vector Calculus is the study of calculus in three dimensions. How can we define differentiation? How do we integrate in 3D? The answers to these questions underpin vast areas of physics and engineering, including Maxwell's equations of electro-magnetism, which we will see for the first time, and the Navier-Stokes equations of fluid mechanics.

Those equations are examples of partial differential equations (PDEs). The second half of the module focuses on mathematical tricks we can use to solve PDEs, by finding different ways to transform them to ordinary differential equations, and then apply methods from previous modules. This includes an introduction to the important area of mathematics known as Fourier Analysis.

Fourier series and Fourier transforms are methods we can use to describe a signal as the sum of many smaller, simpler waves. Once you have studied this, it will open up a vast array of different applications from sound processing to convolutional neural networks.

What you will learn

In this module, you will:

• discuss applications of scalar and vector fields
• learn how to perform line, surface and volume integrals
• learn the method of separation of variables for PDEs
• explore the difference between Fourier series and Fourier transforms

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

• interpret and evaluate grad, div and curl
• calculate the surface area of complicated shapes
• break down a repeating function into an infinite number of sine waves
• solve the heat equation

Assignments / assessment

• Class tests (40%) - short written tests covering different parts of the module
• Final exam, two hours (60%) - written exam covering the whole module

Teaching methods / timetable

You will learn through lectures, covering the content of the online lecture notes and discussing many different example problems, and tutorials, where you will solve problems individually and in groups with support from the lecturers.