MA2MMPNU: Mathematical Methods and Physical Applications

Module code: MA2MMPNU

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

Credits: 20

Level: Level 2 (Intermediate)

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 MA0FMNU AND TAKE MA1LANU (Compulsory)

Co-requisite module(s): IN THE SAME YEAR AS TAKING THIS MODULE YOU MUST TAKE MA2DENU (Compulsory)

Pre-requisite or Co-requisite module(s):

Module(s) excluded:

Placement information: NA

Academic year: 2024/5

Available to visiting students: No

Talis reading list: No

Last updated: 21 May 2024

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 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 Green’s identities we extend the study of initial boundary value problems for 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 solving skills and receive feedback on their formative work.

Study hours

At least 54 hours of scheduled teaching and learning activities will be delivered in person, with the remaining hours for scheduled and self-scheduled teaching and learning activities delivered either in person or online. You will receive further details about how these hours will be delivered before the start of the module.

Please note the independent study hours above are notional numbers of hours; each student will approach studying in different ways. We would advise you to reflect on your learning and the number of hours you are allocating to these tasks.

Semester 1 The hours in this column may include hours during the Christmas holiday period.

Semester 2 The hours in this column may include hours during the Easter holiday period.

Summer The hours in this column will take place during the summer holidays and may be at the start and/or end of the module.

Assessment

Requirements for a pass

Students need to achieve an overall module mark of 40% to pass this module.

Summative assessment

Penalties for late submission of summative assessment

The Support Centres will apply the following penalties for work submitted late:

Assessments with numerical marks

• where the piece of work is submitted after the original deadline (or any formally agreed extension to the deadline): 10% of the total marks available for that piece of work will be deducted from the mark for each working day (or part thereof) following the deadline up to a total of three working days;
• the mark awarded due to the imposition of the penalty shall not fall below the threshold pass mark, namely 40% in the case of modules at Levels 4-6 (i.e. undergraduate modules for Parts 1-3) and 50% in the case of Level 7 modules offered as part of an Integrated Masters or taught postgraduate degree programme;
• where the piece of work is awarded a mark below the threshold pass mark prior to any penalty being imposed, and is submitted up to three working days after the original deadline (or any formally agreed extension to the deadline), no penalty shall be imposed;
• where the piece of work is submitted more than three working days after the original deadline (or any formally agreed extension to the deadline): a mark of zero will be recorded.

Assessments marked Pass/Fail

• where the piece of work is submitted within three working days of the deadline (or any formally agreed extension of the deadline): no penalty will be applied;
• where the piece of work is submitted more than three working days after the original deadline (or any formally agreed extension of the deadline): a grade of Fail will be awarded.

The University policy statement on penalties for late submission can be found at: https://www.reading.ac.uk/cqsd/-/media/project/functions/cqsd/documents/qap/penaltiesforlatesubmission.pdf

You are strongly advised to ensure that coursework is submitted by the relevant deadline. You should note that it is advisable to submit work in an unfinished state rather than to fail to submit any work.

Formative assessment

Formative assessment is any task or activity which creates feedback (or feedforward) for you about your learning, but which does not contribute towards your overall module mark.

Weekly problems sets supported by tutorials.

Reassessment

Additional costs