Module Identifier MA11010
Module Title FURTHER ALGEBRA AND CALCULUS
Co-ordinator Professor John Gough
Semester Semester 2
Other staff Ms Brenda M Hughes, Professor John Gough
Pre-Requisite MA10020
Course delivery Lecture   22 Hours. (22 x 1 hour lectures)
Seminars / Tutorials   5 Hours. (5 x 1 hour tutorials)
Assessment
Assessment TypeAssessment Length/DetailsProportion
Semester Exam2 Hours (written examination)  100%
Supplementary Assessment2 Hours (written examination)  100%

#### Learning outcomes

On completion of this module, a student should be able to:
1. solve systems of linear equations,
2. manipulate matrices according to the laws of matrix algebra,
3. evaluate determinants of square matrices,
4. determine partial derivatives of functions of several variables and establish identities involving them,
5. obtain the critical points of functions of several variables,
6. evaluate multiple integrals in rectangular coordinates,
7. evaluate multiple integrals using change of variables.

#### Brief 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.

#### Content

1. MATRIX ALGEBRA: Matrix operations (addition, scalar multiplication, matrix multiplication, transposition, inversion). Special types of matrices (zero, identity, diagonal, triangular, symmetric, skew-symmetric, orthogonal). Row equivalence.
2. LINEAR EQUATIONS: Systems of linear equations. Coefficient matrix, augmented matrix. Elementary row operations. Gaussian and Gauss-Jordan elimination.
3. DETERMINANTS: Properties of determinants. Computation of determinants.
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 integrals in rectangular coordinates; areas. Substitution in multiple integrals.

Books
** Recommended Text
H Anton & C Rorres (2000) Elementary Linear Algebra: Applications Version 8th. J Wiley 0471170526
Weir, M D, Haas, J and Giordano, F R (2005) Thomas' Calculus 11/e. Addison Wesley 0321243358
** Supplementary Text
D.W.Jordan & P.Smith (1994) Mathematical Techniques: an introduction for the engineering, physical and mathematical sciences Oxford University Press 0198562683
J Stewart (2001) Calculus : concepts and contexts 2/e. Brooks/Cole 0534377181
Lay, David C. (Nov. 2002) Linear Algebra and Its Applications 3/e. Addison-Wesley 0321149920
R L Finney & G B Thomas (1994) Calculus 2nd. Addison-Wesley 0201549778
T S Blyth & E F Robertson Basic Linear Algebra Springer 3540761225

#### Notes

This module is at CQFW Level 4