Autumn / Spring

Unit(s) of Assessment	Weighting Towards Module %
Autumn class tests (2)	20
Autumn exam	30
Spring class test	13
Spring exam	37
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.

At the end of the course a student should have revised strongly reinforced the skills found in the A-level syllabus and should be able to: 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 Taylor & 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.

Single Variable Calculus: Algebraic manipulation. Partial fractions. Complex numbers: Cartesian, exponential and polar forms. Argand diagrams. Complex exponentials, Euler’s formula and de Moivre’s theorem. Functions of a complex variable. Standard functions: exponential, logarithmic, trigonometric and hyperbolic functions. Manipulations with functions; odd, even and periodic functions. Limits. Techniques and applications of differentiation. l’Hôpital’s rule. Power series, convergence tests, Taylor and Maclaurin series Parametric and implicit functions: graphs and curve sketching 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. Second order linear ODEs with constant coefficients. Introduction to real and complex Fourier series.   Vector functions of a scalar: Rules for differentiation of vector functions of scalar.   Calculus in 2 dimensions: 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   Calculus in 3 dimensions: 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

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

21 March 2011