Module Code: MAT1015 
Module Title: CALCULUS 

Module Provider: Mathematics

Short Name: MS114

Level: HE1

Module Coordinator: GOURLEY SA Dr (Maths)

Number of credits: 30

Number of ECTS credits: 15




Module Availability 
Autumn / Spring 


Assessment Pattern 
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.




Module Overview 



Prerequisites/Corequisites 
None 


Module Aims 
This module provides techniques, methods and practise in manipulating mathematical expressions using algebra and calculus, building on and extending the material of Alevel syllabus. 


Learning Outcomes 
At the end of the course a student should have revised strongly reinforced the skills found in the Alevel 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 Alevel techniques;
 Appreciate the notation, techniques and applications of multiple integration and the integral theorems of Green, Gauss and Stokes.




Module Content 
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




Methods of Teaching/Learning 
Teaching is by lectures, tutorials and tests. Learning takes place
through lectures, tutorials, tests and exercise sheets:
40 hours of lectures and tutorials over 11 weeks in the Autumn semester.
40 hours of lectures and tutorials over 11 weeks in the Spring Semester. 


Selected Texts/Journals 
Reading:
Susan J. Colley : Vector Calculus, Pearson / Prentice Hall (2005), ISBN 0131858742.
J. Gilbert and C. Jordan : Guide to Mathematical Methods, Palgrave Macmillan (2002), ISBN 0333794443.
Further Reading:
Steven G. Krantz : Calculus Demystified, McGrawHill (2002), ISBN 0071393080.
Robert A. Adams : Calculus – A Complete Course (5th Ed.), AddisonWesley (2002), ISBN 0201791315.
K. Stroud : Engineering Mathematics (5th Ed.), Palgrave Macmillan (2001), ISBN 0333919394. 


Last Updated 
21 March 2011 


