Module Identifier MA11010 Module Title FURTHER ALGEBRA AND CALCULUS Academic Year 2000/2001 Co-ordinator Dr T P McDonough Semester Semester 2 Other staff Dr R J Douglas, Dr V C Mavron, Dr J M Pearson Pre-Requisite MA10020 Course delivery Lecture 20 x 1 hour lectures Seminars / Tutorials 6 x 1 hour tutorials Workshop 2 x 1 hour workshops (including test) Assessment Exam 2 Hours (written examination) 75% Continuous assessment 25% Resit assessment 2 Hours (written examination) 100%

General description
The aim of this module is to study situations in which functions of several variables arise naturally in Mathematics. Linear functions lead to techniques for the solution of linear equations and elementary matrix theory. Non-linear functions lead to a study of partial derivatives and multiple integrals.

Aims
To establish a clear understanding of the techniques for studying functions of several variables and a facility for recognising when these techniques may be profitably employed.

Learning outcomes
On completion of this module, a student should be able to:

• solve systems of linear equations,
• manipulate matrices according to the laws of matrix algebra,
• evaluate determinants of square matrices,
• determine partial derivatives of functions of several variables and establish identities involving them,
• obtain the stationary points of functions of several variables,
• evaluate multiple integrals in rectangular coordinates,
• evaluate multiple integrals using change of variables.

Syllabus
1. MATRIX ALGEBRA: Matrix operations (addition, scalar multiplication, matrix multiplication, transposition, inversion). Special types of matrices (zero, identity, diagonal, triangular, unimodular, symmetric, skew-symmetric, orthogonal). Row equivalence. LU factorisation.
2. LINEAR EQUATIONS: Systems of linear equations. Coefficient matrix, augmented matrix. Elementary row operations. Gaussian and Gauss-Jordan elimination.
3. DETERMINANTS: Properties of determinants. Matrix cofactors, adjoints and inverses.
4. PARTIAL DERIVATIVES: Functions of several variables. Partial Derivatives. Differentiability and linearisation. The chain rule. Critical points. Change of variables - the Jacobian.
5. MULTIPLE INTEGRALS: Riemann sums and definite integrals. Double and triple integrals in rectangular coordinates. Areas and volumes. Substitution in multiple integrals.