Mathematical Sciences, University of Southampton

To present the basic mathematical methods on matrices and Laplace transforms and further work on ordinary differential equations, complex numbers and calculus.

The aims of this module are to:

• present the basic mathematical methods on matrices and Laplace transforms and further work on ordinary differential equations, complex numbers and calculus.

Objectives (planned learning outcomes)

Having successfully completed the module, you will be able to:

• Demonstrate knowledge and understanding of basic matrices and Laplace transforms, Fourier series, finding roots of complex numbers and the calculation of partial derivatives and be familiar with simple methods of solving ordinary differential equations;
• Critically analyse and solve some mathematical problems;
• Show logical thinking in problem solving;
• Perform calculations in simple situations and work through some longer examples;
• Demonstrate organisational and time-management skills.

Week : (approx)

1 Complex numbers: De Moivre's theorem; roots; logarithm of a complex number.  
2 Matrix algebra: terminology; addition, subtraction and multiplication of matrices; determinants.  
3 Matrix algebra: inverse of matrix using cofactors; sets of linear equations; solution of sets of linear equations using elimination method; inverse of matrix using elimination method.  
4 Matrix algebra: rank; eigenvalues and eigenvectors.  
5 Ordinary differential equations: solution of first order equations (including linear and exact).
6 Ordinary differential equations: linear operators; second order linear inhomogeneous equations with constant coefficients; free and forced oscillations.  
7 Laplace transforms: transforms of standard functions; solution of linear differential equations with constant coefficients.  
8 Laplace Transforms: Transform of Heaviside step function, Second Shift Theorem, application to differential equations.
9 Further calculus: sequences and series; Taylor's series and Maclaurin's series.
10 Further calculus: chain rule for partial differentiation; higher partial derivatives; errors.  
11 Fourier series: periodic signals, whole range Fourier series.
12 Revision.

Teaching methods include

• Three lectures per week, using skeletal notes, and weekly examples sheet together with associated tutorial.

Learning activities include

• Individual work on examples, supported by weekly tutorial.
• Three assignments to be worked through and handed in.

Core Text

G James,  Modern Engineering Mathematics,  Prentice Hall, 2007 ISBN 9780132391443, TA150 JAM  (4 copies in short loan).

K A Stroud, Engineering Mathematics, Palgrave 2007, ISBN 9781403942463, QA100STR (8 copies, 2 in short loan) (Very useful for weaker students) .

Link to module web pages
http://www.soton.ac.uk/~dn/eng1/

Feedback and student support during module study (formative assessment)

The 3 assignments are marked and handed back. 
Solutions to weekly exercise sheets, and weekly tutorial.
Twice weekly timetabled workshops available for additional support.
Past examination papers and solutions available on website

Relationship between the teaching, learning and assessment methods and the planned learning outcomes

The examination is structured into two sections. The first contains 20 short compulsory questions which test basic knowledge of all weekly topics and whether simple calculations can be performed successfully. The second section consists of longer questions (attempt 4 from 6) which test the depth of understanding of topics on the syllabus and the ability to carry out longer pieces of work.
The general skills elements are not explicitly assessed, but their development will reflect on the quality of the overall outcomes.