# Mathematical skills for Masters-level short courses in the School of Engineering

### Introduction

The short course you are currently reviewing uses mathematical skills learned during undergraduate engineering mathematics modules.

If you're considering applying for this short course, please review this page as a guide to the level of mathematics required.

The standard textbook used by many institutions for undergraduate engineering mathematics courses in the UK is K. A. Stroud’s *Engineering Mathematics* [1]. This textbook is split into two parts:

- The first part deals with foundation issues of mathematics and contains 12 sections. It reviews some of the main topics learned up to Higher and A-Level in the UK.
- The second part contains 28 separate learning programmes on a range of topics required by engineers. It constitutes the new material that is learned during a university undergraduate degree.

One or two more relevant advanced topics are contained in a second volume, *Advanced Engineering Mathematics* [2], also by K. A. Stroud.

The next section of this page sets out a range of topics that may be used during this short course. A brief description of each topic is given, together with the relevant section in references [1] and [2].

### Mathematical content

The following topics are of particular importance:

**Foundation Mathematics:** Arithmetic; algebra; equations; graphs; linear equations and simultaneous linear equations; polynomial equations; partial fractions; trigonometry; binomial series; differentiation; integration; functions.

- All of Part 1, pages 1–434 of reference [1].

**Complex Numbers:** Uses of the number *j = √-1*, particularly in the description of random functions and hyperbolic functions. Applied in dynamics and control theory problems.

- Programmes 1 and 2, pages 437–494 of reference [1].

**Determinants:** Solution methods for simultaneous equations. Used for aspects of solid mechanics, structural engineering and control.

- Programme 4, pages 521–554 of reference [1].

**Matrices:** More advanced solution methods for groups of simultaneous equations. Used in reliability theory; solid mechanics; finite element methods; control theory.

- Programme 5, pages 555–589 of reference [1].

**Scalars, Vectors and Tensors:** Description of physical quantities in engineering systems. Required for numerical subjects; Tensors required for aspects of solid mechanics.

- Programme 6, pages 591–618 of reference [1] for Scalars and Vectors.

**Differentiation and Partial Differentiation:** Rates of change form the basis of much of mechanics and the dynamics of engineering systems.

- Programme 7–11, pages 619–728 of reference [1].

**Integration - Direct and Approximate Solutions:** Application of integration to engineering problems including area under a function; mean values of functions; RMS values of functions; surfaces of revolution; volumes of revolution; locating centroids; moments of inertia; second moments of area.

- Programmes 15 and 16 for integration theory, pages 823–884 of reference [1].
- Applications in programmes 18–20, pages 901–979 of reference [1].
- Approximate integration in programme 21, pages 981–1000 of reference [1].

**Multiple Integrals:** Multiple integrals used to describe multi-dimensional systems including structural analysis; reliability theory.

- Programme 23, pages 1025–1049 of reference [1].

**Polar Co-ordinates:** Use of polar co-ordinate system to describe behaviour of engineering systems.

- Programme 22, pages 1001–1024 of reference [1].

**Differential equations and Laplace transforms:** Used for the description of dynamic behaviours of engineering systems. Laplace Transforms used as a solution method for certain types of differential equations.

- Programmes 24–26, pages 1051–1138 of reference [1].
- Programmes 2–3, pages 46–122 of reference [2].

**Fourier transforms:** Mapping of time domain functions into frequency domain using Fourier pairs. Use of both real and imaginary components for signals analysis; random functions and dynamics.

- Programmes 6 and 7, pages 172–276 of reference [2].

**Probability theory and statistics:** Probability calculus used to describe physical and knowledge-based uncertainties in engineering systems. Used in reliability theory. Statistics used in failure data analysis and collection of data from engineering systems.

- Programmes 27 and 28 of reference [1].

The above list should not be taken as definitive, but as a guide to prospective applicants in understanding the level of maths skills required in some of the Masters-level courses delivered by the School of Engineering.

Applicants should be prepared to have this knowledge assessed if they have not done so already through an undergraduate level course. In this case, the list should be used as a guide for self-study; a task for which *Engineering Mathematics* [1] is well designed.

### References

[1] K. A. Stroud. *Engineering Mathematics*. Palgrave Macmillan, Houndmills, Basingstoke, 6th edition, 2007. ISBN 978-1403942463.

[2] K. A. Stroud. *Advanced Engineering Mathematics*. Palgrave Macmillan, Houndmills, Basingstoke, 5th edition, 2011. ISBN 978-0230275485.