Autumn and Spring

Unit(s) of Assessment	Weighting Towards Module Mark( %)
Coursework 1: Diagnostic tests in Autumn semester	0%
Coursework 2: 2 class tests in Autumn semester	50%
Coursework 3: 1-2 class tests in Spring semester	10%
Exam: End of semester	40%
Qualifying Condition(s)  A weighted aggregate mark of 40% is required to pass this module.

This module provides techniques, methods and practise in manipulating mathematical expressions using algebra and calculus, building on and extending the material of A-level syllabus.

• Understand set notation and know the basic properties of real and complex numbers;
• Analyse and manipulate functions and sketch the graph of a function in a systematic way;
• Differentiate functions by applying standard rules;
• Obtain Maclaurin series expansions for a variety of functions;
• Evaluate integrals by means of substitution, integration by parts and partial functions;
• Apply differentiation and integration techniques to a variety of theoretical and practical problems;
• Appreciate that a periodic function can be represented as a sum of sinusoids and compute the coefficients in simple cases;
• Show a good understanding of partial derivatives and the operators div, grad and curl, their propertied and applications;
• Appreciate techniques for optimising a function of more that one variable, and why they are more complex than A-level techniques;
• Appreciate the notation, techniques and applications of multiple integration and the integral theorems of Green, Gauss and Stokes.

• Revision of core material of the A-level syllabus
• Algebraic manipulation. Partial fractions.
• Complex numbers: Cartesian, exponential and polar forms. Argand diagrams. Complex exponentials, Euler’s formula and de Moivre’s theorem.
• Standard functions: exponential, logarithmic, trigonometric and hyperbolic functions.
• Manipulations with functions; odd, even and periodic functions. Limits.
• Techniques and applications of differentiation. Taylor and Maclaurin series, l’Hôpital’s rule.
• Graph and curve sketching
• Parametric and implicit functions: graphs and differentiation.
• Techniques of integration; reduction formulae; arc length, area of surfaces and volumes of  revolution.
• First order ODEs on the line. Separation of variables. Integrating factors for linear first order equations. Homogeneous equations. Initial value problems.
• Introduction to real and complex Fourier series.

• Rules for differentiation of vector functions of scalar.

• Partial differentiation
• Chain rule
• Classification of stationary points for functions of two variable
• Double integrals
• Change of variable formula for double integrals
• Conversion to polar coordinates

• Cylindrical and spherical polars.
• Triple integrals
• The grad operator and its properties
• Method of Lagrange multipliers
• Gradient as normal to a surface
• Tangent plane to a surface
• Line integrals
• Green’s theorem
• The div and curl operators.
• Gauss and Stokes Theorems and applications

40 hours of lectures and tutorials over 12 weeks in the Spring Semester.

K. Stroud : Engineering Mathematics (5th Ed.), Palgrave Macmillan (2001), ISBN 0333919394.

13 October 2008