Occurrence | Teaching cycle |
---|---|
A | Spring Term 2019-20 to Summer Term 2019-20 |
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.
Subject content
Elements of Newtonian mechanics
Fundamental kinematical concepts – position; velocity; acceleration
The inclined plane
Conservation laws – Energy and Momentum
Newton's laws of motion – a physicist's perspective and a mathematician's perspective
Forces in one and two dimensions -- Free motion; Hooke's law; Models of friction; Uniform gravity
Mathematical Modelling
A selection of the following topics will be covered:
Introduction to mathematical modelling – Stages of the modelling process; examples; working in groups on real-world problems; presenting models in talks and written reports
Modelling with functions – Sketching the graphs of simple functions; Finding expressions for functions with specified behaviour; Transforming functions; Finding roots and extrema; Using Maple for plotting and analysing functions; Modelling competition with Cournot equillibria
Modelling with difference equations – Deriving difference equations from the description of a model; Solving first-order and homogeneous higher-order linear difference equations as well as sets of coupled first-order linear difference equations; Using spreadsheets to solve difference equations; Finding fixed points of first-order non-linear difference equations and determine their stability and drawing cobweb diagrams; Investigating Chaos in the logistic map
Modelling with differential equations – Deriving first-order differential equations for mathematical models; Reducing the number of parameters by non-dimensionalisation; Finding analytic solutions to simple first-order differential equations; Euler's method for numerical solution of first-order differential equations; Qualitative understanding of the solutions of first-order differential equations in terms of flow diagrams; Finding fixed points of first-order differential equations and their stability; Changing parameters and observing bifurcations and hysteresis effects; Investigating coupled first-order differential equations; Formulating and solving second-order differential equations; Parameter-dependence of the solutions
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.
Task | Length | % of module mark |
---|---|---|
Essay/coursework Coursework |
N/A | 5 |
University - closed examination Introduction to Applied Mathematics |
3 hours | 95 |
None
Task | Length | % of module mark |
---|---|---|
University - closed examination Reassessment: Exam |
N/A | 100 |
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 Moodle (the VLE). There is no one book covering everything.