MA2MMP: Mathematical Methods and Physical Applications

Module code: MA2MMP

Module provider: Mathematics and Statistics; School of Mathematical, Physical and Computational Sciences

Credits: 20

Level: 5

When you’ll be taught: Semester 1

Module convenor: Dr Calvin Smith , email: Calvin.Smith@reading.ac.uk

Pre-requisite module(s): BEFORE TAKING THIS MODULE YOU MUST ( TAKE MA1CA AND TAKE MA1LA ) OR ( TAKE MA1LANU AND TAKE MA1CANU ) (Compulsory)

Co-requisite module(s):

Pre-requisite or Co-requisite module(s): BEFORE OR WHILE TAKING THIS MODULE YOU MUST TAKE MA2DE (Compulsory)

Module(s) excluded:

Placement information: NA

Academic year: 2025/6

Available to visiting students: Yes

Talis reading list: Yes

Last updated: 3 April 2025

Overview

Module aims and purpose

The module introduces students to two important mathematical methods: vector calculus and variational principles and demonstrates their application to various areas of physics (including, but not limited to, classical mechanics, elements of electromagnetism, diffusion).

Module learning outcomes

By the end of the module, it is expected that students will be able to:

1. Demonstrate problem solving skills, both ‘by-hand’ and by applying programming skills, and accurately communicate mathematical arguments;
2. Understand and apply the concepts of vector calculus to problems in mathematics and physics;
3. Derive the continuity equation from physical principles and apply this to various contexts, such as advection and diffusion. Solve problems in diffusion physics in a variety of scenarios;
4. Pose and solve problems in the calculus of variations using the Euler equation.

Module content

Five weeks on vector calculus

The concepts of scalar and vector fields in mathematics are introduced, and the concept of differentiation of a real-valued function of a single real variable is extended to introduce the gradient of a scalar field, and divergence and curl of vector fields. Interpretations of these various new operations are discussed and key identities for these differential operators are derived and applied to problem solving. Furthermore, the concept of integration of a real-valued function of a single real variable is extended to line, surface and volume integrals. Key results that illuminate the relationships between the differential and integral operations (e.g. Green’s theorem in the plane, Gauss’ divergence theorem, Stoke’s theorem) are derived and applied to solve problems. This new corpus of knowledge is applied to provide mathematical insights into the field of classical mechanics and electromagnetism.

Two weeks on an application to diffusion problems

Using vector calculus we derive the continuity equation and consider the diffusion equation for boundary conditions of physical interest (e.g. contact with reservoirs, insulated endpoints, etc.)

Three weeks on elementary calculus of variations and analytical mechanics

The concept of a variational principle is introduced to enable the posing of problems involving minimising an integral of an unknown function. Lagrange’s Fundamental Lemma is established and used to derive the Euler equation which in turn is used to solve the so-called ‘simplest problem of the calculus of variations’. The principles of least distance and least time are introduced and used to solve classical problems (e.g. deriving geodesics, the Brachistochrone problem, derivation of Snell’s law). Finally, the topic of classical mechanics is revisited from a variational perspective using Hamilton’s principle of least action to develop the field of analytic mechanics.  

One week on consolidation / revision

Structure

Teaching and learning methods

Module content is delivered via a blend of in-person lectures and the virtual learning environment. In addition, learning is supported by tutorials where students develop problem