The first years of all mathematics programmes are designed to give students a thorough grounding in a wide spectrum of mathematical ideas, techniques and tools in order to equip them for the later stages of their course. During first year, as well as consolidating, broadening and extending core material from pre-University study, we initiate a cultural transition to the rigorous development of mathematics which is characteristic at University. Students will develop both their knowledge of mathematics as a subject and their reasoning and communication skills, through lectures, tutorials, seminars, guided self-study, independent learning and project work. This development is addressed in all of our first year modules, although different modules have a different emphasis.
In addition to the above broad aims of the first year, this module focusses on Newton’s application of calculus to describe the motion of objects (Newtonian mechanics). Students will learn the process of building and analysing a mathematical model to answer real-world questions, introducing them to the applied mathematician’s toolkit, used to obtain various types of solutions to a variety of mathematical models, and developing the intuitive aspects of applied mathematics.
Module learning outcomes
Elements of Newtonian mechanics
Fundamental kinematical concepts: position, velocity, acceleration. Examples include Galileo’s inclined plane.
Equations of motion. Newton’s laws of motion. Conservation laws. Energy and momentum.
Motion in one dimension. Conservative systems. Potentials. Nonconservative systems, including models of drag. Mass on a spring (Hooke’s law), harmonic motion and resonance.
Motion in two dimensions. Conservative and nonconservative forces.
Equations of motion in polar coordinates. Examples include, planetary motion and pendula.
Dimensional Analysis and Scaling. Scaling laws and applications, dimensionless numbers, non-dimensionalization, Buckingham's Pi Theorem.
1st order Ordinary Differential Equations (ODEs) in applied contexts. Fixed points, stability, phase lines; exponential and power-law growth and decay; limits to growth including the logistic equation; the Allee effect. Examples from ecology, biology, chemistry, finance. 1st order difference equations in financial contexts and their relation to ODEs.
Coupled pairs of 1st order ODEs in applied contexts. Null clines, fixed points, phase planes, linearization in simple cases. Examples including predator-prey, epidemiology, warfare.
Partial Differential Equations. Wave and heat equations; solution on the finite interval using Fourier sine series. Vibrations of plucked or struck strings; diffusion of heat in a rod. Heat equation on the half-line with a periodic boundary condition [if time permits]. Simple reaction-diffusion systems, including travelling waves [if time permits].
Academic and graduate skills
Academic skills: the application of rigorous mathematical techniques and ideas to the development of mathematics; the power of abstraction as a way of solving many similar problems at the same time; the development and consolidation of essential skills which a mathematician needs in their toolkit and needs to be able to use without pausing for thought.
Many of the techniques and ideas in the mathematical modelling component are ones which graduates employed as mathematicians and in other numerate professions will use from day to day in their work. Mathematics graduates are prized by employers for their ability to think logically and precisely; when combined with a well-developed ability to communicate this makes students much more employable.
% of module mark
University - closed examination Introduction to Applied Mathematics
Special assessment rules
% of module mark
University - closed examination Reassessment: Exam
Current Department policy on feedback is available in the undergraduate student handbook. Coursework and examinations will be marked and returned in accordance with this policy.
The module will draw from a wide variety of sources. Many of them will be available online and on the VLE. There is no one book covering everything. Good sources include:
D. Edwards and M. Hamson, Guide to mathematical modelling (Palgrave, 2001).
K K Tung, Topics in Mathematical Modelling, Princeton 2007.
C.D. Collinson and T. Roper, Particle Mechanics, Arnold (London 1995)/Elsevier (2004);
J. Berry and K. Houston, Mathematical Modelling, Arnold (London 1995).