Subject: Mathematics

1. Not all prescribed modules will be offered in any given year.

2. Trainee teachers must obtain at least grade ‘D’ in all 100 level modules before taking any 300 level modules unless given exemption by the Head of the Mathematics and Mathematics Education Academic Group.

CAM101 Calculus I
Functions: domain, codomain, range, composition of functions, graphs. Limits and continuity. Differentiation and applications of differentiation. Integration and applications of integration. First and second order ordinary differential equations. Applications of differential equations.

CAM102 Algebra I
Sets, mappings, cardinals. Linear systems, augmented matrix of a linear system. Row echelon form and reduced row echelon form, Gauss-Jordan elimination, homogeneous linear system and general solution. Matrices and determinant. Cofactor expansion, 187

Cramer's rule. Vectors in 2-space and 3-space, norm of a vector, dot product, cross product, lines and planes in 3-space. Euclidean n-space, vector operations; dot product, orthogonality, Cauchy-Schwarz inequality. Linear transformation from 3 " to 3"'  , standard matrices. Eigenvalues and eigenvectors.

CAM103 Finite Mathematics
Counting methods: basic principles, permutations and combinations, generalized permutations and combinations, binomial coefficients and combinatorial identities, the pigeonhole principle. Basic concepts in probability: Sample space. Conditional probability. Stochastic independence. Discrete probability distribution functions.

CAM104 Computational Mathematics
Introduction to computational methods and tools. Numerical solutions of equations in one variable. Approximation of functions. Solving systems of linear equations. Numerical integration and solutions of ordinary differential equations. Numerical simulations.

CAM201 Calculus II
Sequences and series. Power series. Partial derivatives for functions of two or more variables, differentiability and chain rules for functions of two variables, directional derivatives and gradients for functions of two variables, tangent planes and normal lines, maxima and minima of functions of two variables, generalization of the concepts to functions of more than two variables. Double integrals and triple integrals.

CAM202 Algebra II
The basics of logic, proofs in mathematics, mathematical induction. Real vector spaces, subspaces, linear independence. Bases and dimension. Rank and nullity. Linear transformations, kernel and range. Similarity. Eigenvalues and eigenvectors, eigenspaces. Orthonormal bases. Gram-Schmidt process. Least squares solution problem. Diagonalizing quadratic forms; conic section. Groups, subgroups, normal subgroups, cofactors, order, Lagrange's theorem. Group homomorphisms, Cauchy's theorem. Symmetries of plane figures, other applications.

CAM203 Statistics I
Descriptive statistics. Random variables. Probability density functions. Cumulative distribution functions. Mathematical expectations. Sampling and sampling distributions. Estimation and confidence intervals. Hypothesis testing.

CAM204 Number Theory
Factorisation of integers. Linear Diophantine equations. Prime numbers. Congruences. Arithmetic functions. Quadratic residues. Primitive roots. Diophantine equations. Continued fractions.

CAM330 Mathematical Methods I
First order ordinary differential equations: Separable equations and integrating factors, existence and uniqueness theorem, Picard's iterative method and modelling with linear equations. Second order ordinary differential equations: fundamental solutions, Wronskian, linear dependence. Applications of differential equations. Series solutions. Laplace transform methods for initial value problems.

CAM331 Operations Research I
Topics in the theory of linear programming: simplex method, introduction to duality, dual simplex method, sensitivity analysis. Topics in the theory of networks: minimal spanning trees, shortest paths, maximal flows, critical path analysis.

CAM332 Statistics II
Tests concerning variances. Bivariate distributions: marginal and conditional distributions, covariance, independence. Simple linear regression. Non-parametric tests.

CAM333 Probability I
Probability spaces, special discrete and continuous random variables (including bivariate distributions), change of variables, selection of probability generating functions, moment generating functions, Laplace transforms. Limit theorems, Markov chains, random walks.

CAM334 Analysis I
The limit of a sequence, Cauchy condition, limit theorems. The limit of a function (using epsilon-delta), the continuity of a function, intermediate values, boundedness, maximum and minimum principle, existence of roots. Infinite series, convergence tests. Differentiation and Taylor series. Uniform convergence, Weierstrass M-test. Uniform continuity, Riemann integration.

CAM335 Modern Algebra I
Groups, Cauchy’s Theorem, conjugacy and Sylow’s theorem. Quotient groups and fundamental group homomorphism theorems. Rings, commutative rings, integral domains, fields. Ideals, quotient rings. Ring homomorphisms, fundamental homomorphism theorems for rings. Rings of polynomials, irreducible polynomials and the Eisenstein criterion.

CAM430 Mathematical Methods II
Ordinary differential equations involving step functions, impulse functions and discontinuous forcing functions. Numerical methods: Euler, Runge-Kutta and multi-step methods. Predictor-corrector methods. Partial differential equations: Separation of variables, heat conduction and Fourier Series. The wave equation.

CAM431 Operations Research II
Selected topics in the theory of stochastic processes: queuing theory, probabilistic inventory models, project scheduling under uncertainty. Selected topics from advanced linear programming and network theory: least cost flows, integer programming, dynamic programming techniques, non-linear programming.

CAM432 Statistics III
Analysis of variance: completely randomised design, randomised block design, factorial designs. Selected topics from: Estimation theory. Hypothesis testing theory. Selected topics in statistics.

CAM433 Probability II
Selection of topics from Poisson processes, continuous time Markov chains, renewal theory, branching processes, information theory.

CAM434 Analysis II
Topology in 3 . Metric spaces. Open sets and closed sets. Convergence and completeness. Continuity and compactness. Equicontinuity, Arzela-Ascoli Theorem. Topological spaces.

CAM435 Modern Algebra II
Fields, degree of field extension. Constructions with ruler and compass. Finite fields. Algebraically closed fields. The main theorem of Galois theory, cubic equations, symmetric functions, primitive elements, proof of the main theorem. Quartic equations, quintic equations.

CAM436 Graph Theory
Graphs. Euler tours, Hamiltonian cycles, representation of graphs, isomorphisms of graphs, planar graphs. Trees and applications. Selected topics from: Connectivity and matching: Hall's theorem, transversals, Konig's theorem, vertex and edge cuts, Menger’s theorem. Colouring: vertex colouring, Brook's theorem, chromatic polynomials, map colouring and the four colour problem, edge colouring, Vizing's theorem. Planarity: planar graphs, Kuratowski's theorem, Euler's formula, graphs on surfaces, dual graphs. Ramsay theory, matroids, extremal graphs.

CAM437 Geometry
Transformation geometry, symmetries, isometries, and tessellation. Axiomatic geometry, axioms and model, finite geometries, Euclidean geometry. Axioms and models of non-euclidean geometries. Projective geometry, generalised coordinates, duality, cross-ratio, collineation. Hyperbolic geometry, area of triangles, hyperbolic trigonometry, hyperbolic surfaces, path and geodesic.

CAM438 Complex Analysis
Complex numbers , complex functions. Complex differentiation, the Cauchy-Riemann equations. Complex integration over paths, Cauchy integral theorem, Cauchy integral formula. Fundamental theorem of algebra. Laurent series, the residue theorem, evaluation of real definite integrals.

CAM439 Special Topics in Mathematics
This module will tap the expertise of staff available so that interesting topics in the study and understanding of mathematics may be explored and discussed. Examples of topics include: Number fields, Wavelets and applications, Mathematical modelling.