MATHEMATICAL METHODS - 2026/7

Module Overview

This module introduces students to a range of mathematical methods and illustrates their application to real-world problems. Students will be introduced to residue calculus, integral transforms, variational calculus, and asymptotic approximations. Students will revise and extend their knowledge of methods of solving ordinary and partial differential equations.

Module provider

Mathematics & Physics

Module Leader

PRINSLOO Andrea (Maths & Phys)

Number of Credits: 15

ECTS Credits: 7.5

Framework: FHEQ Level 6

Module cap (Maximum number of students): N/A

Module Availability

Semester 1

Prerequisites / Co-requisites

N/A

Module content

Indicative content includes: 

• Complex Methods:  

• Revision topics:  Functions of a complex variable.  Differentiability and the Cauchy-Riemann equations. Holomorphicity. Contour integration. Cauchy's theorem. Taylor series. Analytic continuation. Laurent series. Classification of isolated singularities. 

• Core topics:  Residues at simple and higher order poles. Cauchy's residue theorem. Application to trigonometric integrals. Application to integrals over the real line by completing the contour. Jordan's lemma. 

• Extension topic:  Summing series using residue calculus. 

• Fourier and Laplace transforms:  

• Revision topic:  Fourier series. 

• Core topics:  Fourier transforms and inverse Fourier transforms using residue calculus. Properties. Convolutions. The Dirac delta. Laplace transforms. Inverse Laplace transforms using residue calculus. Properties. 

• Extension topic:  Fourier transform of Gaussian functions using residue calculus. 

• Ordinary Differential Equations: 

• Revision topics:  First order separable ODEs. First order linear ODEs using an integrating factor. Linear ODEs with constant coefficients (homogeneous and inhomogeneous). Cauchy-Euler ODEs. 

• Core topics:  Sturm-Liouville differential equations and eigenvalue problems. Chebyshev, Legendre and Laguerre differential equations. The Bessel differential equation. Solving initial value problems using Laplace transforms. Solving ODEs using Fourier transforms. Green's functions. 

• Extension topics:  Orthogonal functions (relative to a weight function), generalised series, self-adjoint differential equations, and orthogonal polynomials. Generating functions of orthogonal polynomials. 

• Partial Differential Equations: 

• Revision topic:  Method of separation of variables. 

• Core topics:  Solving PDEs using Fourier transforms. The heat equation and wave equation in 1+1D. Laplace equation in 2D. Laplace equation in 3D and spherical harmonics. 

• Extension topics: Conformal maps.  The Schrödinger equation for the Hydrogen atom. 

• Variational Methods:  

• Revision topics:  Optimisation of multivariable real functions. Method of Lagrange multipliers. 

• Core topics:  Functional variation and the Euler-Lagrange equations. Principle of least action and Lagrangian mechanics. Constrained functional variation. 

• Extension topic:  Sturm-Liouville problems from constrained functional variation. 

• Asymptotic Approximations:  

• Core topics:  Introduction to asymptotic and perturbative expansions. Solving algebraic equations using regular and singular perturbative expansions. Solving differential equations using perturbative expansions. Asymptotic evaluation of integrals. Method of stationary phase. Method of steepest descent. 

• Extension topics: Matched asymptotic expansions.  Method of multiple time scales. Van der Pol equation. 

Assessment pattern

Alternative Assessment

N/A

Assessment Strategy

The assessment strategy is designed to provide students with the opportunity to demonstrate: 

• Knowledge and understanding of mathematical concepts and methods. 
• The ability to identify and use the appropriate method to solve mathematical problems. 

Thus, the summative assessment for this module consists of: 

• One in-semester test (50 minutes), worth 20% of the module mark, corresponding to Learning Outcomes 1, 2, 3 and 6. 
• A synoptic examination (2 hours), worth 80% of the module mark, corresponding to Learning Outcomes 1 to 5. 

Formative assessment: 

• There will be a number of formative unassessed courseworks over an eleven-week period, designed to consolidate student learning. 

Feedback:  

• Students will receive feedback on unassessed coursework and the in-semester test. This feedback is timed such that feedback from the earlier coursework will assist students with preparation for the in-semester test. The feedback from all coursework and the in-semester test will assist students with preparation for the final synoptic examination. This feedback is complemented by verbal feedback at tutorials, which are designed to promote student engagement with mathematical problems.

Module aims

• Provide students with an introduction to residue calculus, integral transforms, variational calculus, and asymptotic approximations.
• Develop students' understanding of methods of solving ordinary and partial differential equations.
• Introduce students to variational methods, and methods of solving problems approximately as perturbative expansions.
• Enable students to identify and apply appropriate methods to solve mathematical problems, including problems with real-world applications.

Learning outcomes

Methods of Teaching / Learning

The learning and teaching strategy is designed to: 

• Introduce students to residue calculus, integral transforms, variational calculus, and asymptotic approximations. 
• Provide students with knowledge of advanced methods used to solve ordinary and partial differential equations, variational methods, and approximation methods. 
• Provide students with experience identifying and using the appropriate method to solve a variety of mathematical problems. 

The learning and teaching methods include: 

• Three one-hour lectures for eleven weeks, with module notes provided to complement the lectures. These lectures provide a structured learning environment and opportunities for students to ask questions and to practice methods taught. 
• A weekly one-hour tutorial for eleven weeks for guided discussion of solutions to problem sheets (provided to students in advance). These tutorials will reinforce students' understanding of mathematical concepts and methods, and enable them to engage in solving mathematical problems. 
• Formative unassessed coursework designed to enable students to consolidate learning. Feedback on unassessed coursework will be provided to guide students on their progress and understanding. 
• Video recordings of certain core topics covered in lectures. These recordings are intended to give students an opportunity to review parts of lectures which they may not fully have understood and should not be seen as an alternative to attending lectures. 
• Additional video recordings of revision topics and extension topics. Extension topics may also be covered via problems discussed in tutorials using a flipped learning approach. 

Indicated Lecture Hours (which may also include seminars, tutorials, workshops and other contact time) are approximate and may include in-class tests where one or more of these are an assessment on the module. In-class tests are scheduled/organised separately to taught content and will be published on to student personal timetables, where they apply to taken modules, as soon as they are finalised by central administration. This will usually be after the initial publication of the teaching timetable for the relevant semester.

Reading list

https://readinglists.surrey.ac.uk
Upon accessing the reading list, please search for the module using the module code: MAT3056

Other information

Please note that the information detailed within this record is accurate at the time of publishing and may be subject to change. This record contains information for the most up to date version of the programme / module for the 2026/7 academic year.