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# Course Modules for Fourth Year B.A. Mathematics

## OBJECTIVES:

To develop further the exploration of advanced calculus through a range of topics not covered elsewhere and to include an introduction to differential equations and applications.

## LEARNING OUTCOMES:

Students who successfully complete this module should:

• be able to determine the vector and parameterised equations (as appropriate) of lines and some curves in R2 and of planes and some surfaces in R3;
• be able to analyse curves and surfaces in terms of direction vectors, normal vectors, tangent planes, gradient, directional derivative etc. as appropriate;
• understand the definitions and important theory behind the analysis of functions of several variables, in particular, partial differentiation and multiple integration.

## MODULE CONTENT:

• Real vector spaces of dimension n, norm, inner product and cross product. Lines and planes in 3-dimensional space, curves and surfaces. Cylindrical and spherical coordinates.
• Calculus of several variables, continuity and derivative for multivariable functions. Partial derivatives. Maxima and minima, constrained and unconstrained optimisation. Directional derivative, gradient, divergence and curl. Chain rule. Higher derivatives. Taylor's formula in several variables.
• Inverse functions and implicit functions. Tangent planes and normal lines.
• Definition of multiple integral, Fubini's theorem. Double and line integrals, surface and volume integrals.
• Introduction to differential equations, ordinary and partial differential equations. The Heat Equation, the Wave Equation.

## PRIME TEXT:

McCALLUM, W. G., et al. (2013) Calculus: Multivariable, John Wiley & Sons.

## OTHER RELEVANT TEXTS:

SPIVAK, M. (1968) Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus, Addison-Wesley.

STEWART, J. (1998) Calculus: Concepts and Contexts, London: Pacific Grove.

KLINE, M. (1977) Calculus: An Intuitive and Physical Approach, Wiley.

COURANT, R. & JOHN, F. (1974) Introduction to Calculus and Analysis, Volume 2, Wiley.

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## OBJECTIVES:

Geometry is a core part of mathematical education, because it provides a paradigm of rigorous mathematical reasoning. This module equips students with sound knowledge of euclidean geometry and introduces them to non-euclidean geometry.

## LEARNING OUTCOMES:

Students who successfully complete this module should be able to:

• express, justify and establish geometric results;
• determine geometric quantities using theorems and techniques of euclidean geometry;
• demonstrate understanding of geometry beyond classical euclidean geometry;
• carry out and describe geometric constructions.

## MODULE CONTENT:

• Basic notions and theorems of euclidean geometry.
• Geometric constructions.
• Analytic geometry.
• Transformations and symmetry.
• Vectors and dot product.
• Non-Euclidean geometry.

## PRIME TEXT:

BYER, O., LAZEBNIK, F., SMELTZER, D. (2010) Methods for Euclidean Geometry, MAA.

OSTERMANN, A., WANNER, G. (2012) Geometry by Its History, Springer.

STILLWELL, J. (2005) The Four Pillars of Geometry, Springer.

## OTHER RELEVANT TEXTS:

BARKER, W., HOWE, R. (2007) Continuous Symmetry, From Euclid to Klein. American Mathematical Society.

COXETER, H.S.M. (1961) Introduction to Geometry, John Wiley & Sons.

GARDINER, A.D., BRADLEY, C.J. (2005) Plane Euclidean Geometry: Theory and Problems, UKMT.

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## OBJECTIVES:

To present the algebraic structures of groups, rings and fields in order to foster an understanding of their central importance in modern mathematics and of their relevance to engineering and science.

## LEARNING OUTCOMES:

Students who successfully complete this module should be able to:

• use the vocabulary, symbolism, and basic definitions used in abstract algebra;
• apply the concepts of groups, rings and fields to solve problems in which their use is fundamental to obtaining and understanding the solution.

## MODULE CONTENT:

• Groups; axioms and examples, subgroups, mappings and symmetries, applications of symmetry groups.
• Subgroups; cosets, Lagrange's theorem. Groups of small order; isomorphism.
• Binary codes; application of group codes, error correction.
• Conjugacy; normal subgroups, factor groups, homomorphism, isomorphism. Permutation groups; Cayley's theorem.
• Rings; axioms and examples, polynomial rings.
• Subrings; ideals, quotient rings, ring homomorphisms, isomorphisms.
• Integral domains; integers. Congruences; Fermat's theorem, Euler's theorem, application of Euler's theorem to public key codes.
• Fields; axioms and examples, polynomials over a field.

## PRIME TEXT:

LAURITZEN, N. (2003) Concrete Abstract Algebra, Cambridge University Press.

## OTHER RELEVANT TEXTS:

DURBIN, J.R. (1979) Modern Algebra, Wiley.

FRALEIGH, J.B., A First Course in Abstract Algebra, Addison Wesley.

HUMPHREYS, J.F., PREST, M.Y. (1989) Numbers, Groups and Codes, Cambridge University Press.

IRVING, R. (2004) Integers, Polynomials and Rings: A Course in Algebra, Springer.

KIM, K.H., ROUSH, F.W. (1983) Applied Abstract Algebra, Ellis Horwood.

LEDERMANN, W. (1973) Introduction to Group Theory, Oliver and Boyd.

WHITELAW, T.A. (1988) Introduction to Abstract Algebra, Blackie.

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## OBJECTIVES:

Computers are useful and indispensable tools to carry out calculations in mathematics. Within this capstone module, students will be introduced to a computer algebra system in which numerical and symbolic calculations can be carried out. The main concepts of most mathematics modules taught in this programme will be dealt with from a computational perspective.

## LEARNING OUTCOMES:

Students who successfully complete this module should be able to:

• use a computer algebra system to investigate mathematical concepts and to solve mathematical questions;
• solve equations numerically and symbolically;
• use computers to study problems from various areas of undergraduate mathematics such as number theory, linear algebra, calculus of one or several variables and statistics.

## MODULE CONTENT:

• Introduction to a computer algebra system.
• Calculations in number theory, linear algebra, calculus and in statistics.

## PRIME TEXT:

BARD, G. V. (2015) Sage for Undergraduates American Mathematical Society.

KOSAN, T. (2007) SAGE for Newbies. Available online.

## OTHER RELEVANT TEXTS:

STEIN, W. (2012) Sage for Power Users. Available online.

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