AJEE Online
ISSN 1324-5821
EDITORIAL
Engineering Mathematics
Teaching engineering mathematics is a difficult balancing act. The lecturer is required to teach a large amount of mathematics necessary for the students to understand later engineering subjects, while including enough engineering applications to keep the students motivated. If too much time is spent on explaining the application, then insufficient time is available for the student to learn the mathematics. The familiar student question “Why are we doing this?” conveys the all-too-common frustration that students have with mathematics: a perceived lack of connection between mathematics and their engineering courses, and indeed their daily lives. Yet often these same students in later years invariably comment that they wish they had worked harder on their mathematics subjects because they really need it for their fourth-year subjects.
Engineering mathematics is often taught with only a token nod to the engineering applications in which the mathematics will be used. Indeed, most classic textbooks titled Advanced Engineering Mathematics, make only minor efforts to include real-world applications within their texts. Finding applications relevant to undergraduate mathematics subjects is difficult as most real-world problems are too long and involved to discuss in lectures.
An aim of this special issue is to provide some accessible examples of modern engineering and applied mathematics that can be used by lecturers within an engineering mathematics subject. The papers included here can be used as motivating examples, or as student projects and exercises. The topics cover a wide range of applications and mathematical techniques. Some of these applications add a new twist to conventionally taught material, while other applications illustrate a type of mathematics often overlooked in standard degrees. The topics considered here have arisen from a variety of sources, often as real-world problems posed by companies to Mathematics in Industry Study Groups (examples are given in the websites listed below). These papers also illustrate that modern mathematics research can be related to current undergraduate topics.
Some of the problems considered here are:
- how to design a timetable for a train network (Albrecht et al, 2009), which highlights an important application of matrices and optimisation methods
- where to put drainage points on roads to minimise contamination of water systems (Barry & Thompson, 2009), which shows the process of mathematical modelling, and gives examples of quadratics, Taylor series, inequalities and result interpretation
- how far apart to put cooling rods in setting concrete (Fowkes & Bassom, 2009), which illustrates diffusion equations, and that clever modelling and scaling can give useful results without complex mathematical calculations
- how long does it take a single or layered material to heat up (Hickson et al, 2009), which extends the traditionally taught separation methods for diffusion equations, giving simple and elegant results understandable by students
- how to design a sluice gate (Binder, 2009), which uses fluid mechanics to give an example of how phase planes for a couple of ordinary differential equations can provide useful insight into a practical problem
- how a differential equation model of tissue growth can be related to basic probability methods, hence nicely illustrating the inter-linked nature of mathematics (Binder & Landman, 2009)
- how to model the stability of a ship (Jovanoski & Robinson, 2009), which demonstrates key elements of second-order, forced differential equations
- how to model the random diffusion of discrete objects, such as moving people or growing tissue (Simpson et al, 2009), which links probability theory and partial differential equations
- how to model complex phenomena such as self-organisation in granular materials using vector calculus and mechanics (Tordesillas et al, 2009).
This special issue arose through discussions with the Australian and New Zealand Industrial and Applied Mathematics society (ANZIAM), the Australian Mathematical Science Institute (AMSI) and the Australasian Association of Engineering Education (AAEE). We would like to express our thanks to these bodies and to Phil Broadbridge, Les Dawes and Roger Hadgraft. One of the goals of this issue is to promote further communication between the engineering and applied mathematics communities, and raise awareness of the role of applied mathematics within engineering education. We hope this issue assists in this process.
Steven Barry
National Centre Epidemiology and Population Health, ANU, Canberra, ACT
Antoinette Tordesillas
Department of Mathematics and Statistics, The University of Melbourne, Victoria
REFERENCES
Albrecht, A. R., Howlett, P. G. & Coleman, D. 2009, “Application of origin-destination matrices to the design of train services”, Australasian Journal of Engineering Education, Vol. 15, No. 2, pp. 95-104.
Barry, S. I. & Thompson, C. J. 2009, “Modelling sediment drainage from roads to rivers”, Australasian Journal of Engineering Education, Vol. 15, No. 2, pp. 69-76.
Binder, B. J. 2009, “A non-linear dynamical system: Flow past a sluice gate”, Australasian Journal of Engineering Education, Vol. 15, No. 2, pp. 27-34.
Binder, B. J. & Landman, K. A. 2009, “Tissue growth and the Pólya distribution”, Australasian Journal of Engineering Education, Vol. 15, No. 2, pp. 35-42.
Fowkes, N. & Bassom, A. P. 2009, “Piped water cooling of concrete: An exercise in scaling”, Australasian Journal of Engineering Education, Vol. 15, No. 2, pp. 51-58.
Hickson, R. I., Barry, S. I. & Sidhu, H. S. 2009, “Critical times in one- and two-layered diffusion”, Australasian Journal of Engineering Education, Vol. 15, No. 2, pp. 77-84.
Jovanoski, Z. & Robinson, G. 2009, “Ship stability and parametric rolling”, Australasian Journal of Engineering Education, Vol. 15, No. 2, pp. 43-50.
Mathematics in Industry Study Groups, n. d., www.anziam.org.au/MISG and www.math-in-industry.org.
Simpson, M. J., Hughes, B. D. & Landman, K. A. 2009, “Diffusing populations: Ghosts or folks”, Australasian Journal of Engineering Education, Vol. 15, No. 2, pp. 59-68.
Tordesillas, A., Kirszenblat, D. & Mangelsdorf, C. 2009, “Taming the complexity of granular materials with vector calculus”, Australasian Journal of Engineering Education, Vol. 15, No. 2, pp. 85-94.