 # MATHEMATICS

(1) Students must have studied Mathematics (not Mathematics Literacy) at Matriculation or Grade 12 level
(2) Re-enrolment cannot exceed 2 years
Major combinations:
NQF Level: 5: MAT1512, MAT1503
NQF Level: 6: MAT2611, MAT1613, MAT2613 and at least two further 2nd year NQF Level 6 MAT or APM modules.
NQF Level: 7: FIVE of the following: MAT3701 MAT3702 MAT3705 MAT3706 MAT3707 MAT3711 APM3701

Calculus B - MAT1613
Under Graduate Degree Year module NQF level: 6 Credits: 12
Module presented in English
Pre-requisite: MAT1512 (or XAT1512) Co-requisite: MAT1512 (or XAT1512)
Purpose: To enable students to obtain basic skills in differentiation and integration, and build on the knowledge provided by module MAT1512. More advanced techniques and further basic applications are covered. Together, the modules MAT1512 and MAT1613 constitute a first course in Calculus which is essential for students taking Mathematics as a major subject
Numerical Methods for Civil Engineers II (Offered from 2021 Semester 2) - APM4814
Diploma Semester module NQF level: 8 Credits: 12
Module presented in English
Purpose: The purpose of this module is to enable students develop competencies and skills in solving engineering problems using numerical methods.
Partial Differential Equations I - MAT4847
Honours Year module NQF level: 8 Credits: 12
Module presented in English Module presented online
Purpose: To introduce the learner to analytical techniques of solving partial differential equations of mathematical physics, amongst others the Laplace equation, the wave equation and the heat or diffusion equation. On completing the module the learner will be able to formulate mathematical models and give physical interpretation of mathematical results; using standard techniques such as the method of separation of variables, Fourier series, orthogonal functions and Integral transforms. More advanced techniques are treated in the sister module Partial Differential Equations II.
Linear Algebra 2 - MAT2611
Under Graduate Degree Semester module NQF level: 6 Credits: 12
Module presented in English
Pre-requisite: MAT1503 (or XAT1503)
Purpose: To understand and apply the following linear algebra concepts:vector spaces, rank of a matrix, eigenvalues and eigenvectors, diagonalisation of matrices, orthogonality in Rn, Gram-Schmidt algorithm, orthogonal diagonalisation of symmetric matrices, linear transformations, change of basis and their matrix representations.
Engineering Mathematics IV - EMT4801
Baccalareus Technologiae Degree Year module NQF level: 8 Credits: 12
Module presented in English Module presented online
Purpose: Convergence of series; power series; complex analysis; Laplace transforms revisited; Z-transforms and difference equations; transfer functions; state space representation.
Partial Differential Equations II - MAT4848
Honours Year module NQF level: 8 Credits: 12
Module presented in English Module presented online
Co-requisite: MAT4847
Purpose: To provide the students with knowledge of advanced analytical techniques of solving partial differential equations of mathematical physics. On completing this module the students will be able to construct and solve partial differential equations using advanced methods which amongst others include zeros of Sturm-Liouville eigenfunctions, Rayleigh quotient, method of eigenfunction expansion using Green's functions and method of characteristics. Prior knowledge of the sister module Partial Differential Equations I is assumed.
Introduction to Discrete Mathematics - MAT2612
Under Graduate Degree Semester module NQF level: 6 Credits: 12
Module presented in English
Pre-requisite: COS1501 (or XOS1501) or MAT1512 (or XAT1512) or MAT1503 (or XAT1503)
Purpose: To acquaint students with the theory and applications of the following aspects of discrete mathematics: counting principles, relations and digraphs, (including equivalence relations), functions, the pigeonhole principle, order relations and structures (e.g. partially ordered sets, lattices, Boolean algebras), the principle of induction.
Honours Research in Mathematics - HRMAT82
Honours NQF level: 8 Credits: 36
Module presented in English Module presented online
Purpose: The purpose of this module is to learn how to do research in mathematics, and how to present the research. This is demonstrated during a research project under supervision of an academic supervisor, and usually involves the preparation of a document describing the research according to the norms and expectations in mathematics.
Matrix Theory and Linear Algebra I - MAT4857
Honours Year module NQF level: 8 Credits: 12
Module presented in English Module presented online
Purpose: This module comprises an introduction to the theory of matrices designed to cover those topics most frequently encountered in physical applications. Topics include matrices and linear systems, equivalence (row and column) and its canonical forms, vector spaces, determinants, linear transformations, the Cayley-Hamilton theorem, the Gram-Schmidt orthogonalisation process, Schur's triangularisation theorem, nilpotent operators, similarity and the Jordan canonical form.
Real Analysis I - MAT2613
Under Graduate Degree Semester module NQF level: 6 Credits: 12
Module presented in English
Pre-requisite: MAT1613
Purpose: To enable students to master and apply the fundamental concepts and techniques of real analysis as they occur in an elementary discussion of the real number system, sequences and series; limits, continuity and differentiability of functions; the Bolzano-Weierstrass property, continuous and uniformly continuous functions, the mean value theorem, Taylor's theorem; the Riemann integral, the fundamental theorem of calculus, improper integrals, and the power series.
Measure Theory and Integration - MAT4831
Honours Year module NQF level: 8 Credits: 12
Module presented in English Module presented online
Purpose: To enable the learner to master the fundamental concepts of the following: measures on abstract sigma-algebras, outer measures, measurable functions, Lebesgue integral, convergence theorems, product measures and Fubini's theorem.
Matrix Theory and Linear Algebra II - MAT4858
Honours Year module NQF level: 8 Credits: 12
Module presented in English Module presented online
Co-requisite: MAT4857
Purpose: This module is a continuation of the introductory matrix theory module MAT4857. It presents an alternative matrix development of the Jordan canonical form as well as a treatment of the classical canonical form for similarity over an arbitrary field. Introductions to matrix analysis and numerical linear algebra, two interesting and important areas of further study, are also presented. Topics include further development of the Jordan form, solution of the matrix equation AX = XB, polynomial matrices, Lagrange interpolation, the classical canonical form, spectral and polar decompositions, matrix analysis, systems of linear differential equations, numerical methods, iterative methods for solving AX = K.
Calculus in Higher Dimensions - MAT2615
Under Graduate Degree Semester module NQF level: 6 Credits: 12
Module presented in English
Pre-requisite: MAT1512 (or XAT1512) or MAT1503 (or XAT1503)
Purpose: To gain clear knowledge and an understanding of vectors in n-space, functions from n-space to m-space, various types of derivatives (grad, div, curl, directional derivatives), higher-order partial derivatives, inverse and implicit functions, double integrals, triple integrals, line integrals and surface integrals, theorems of Green, Gauss and Stokes.
Group Theory - MAT4833
Honours Year module NQF level: 8 Credits: 12
Module presented in English Module presented online
Purpose: To give students a sound understanding of group theory and some basic understanding of representation theory of finite groups. The module starts off with basic concepts in set theory which lead to the Well-ordering principle, Hausdorff maximality principle, Axiom of choice and Zorn's lemma. It then goes on to focus on permutation groups, Cayley's theorem and applications, group actions, Sylow's theorem and applications. The latter part of the module is dedicated to the representations and characters of finite groups.
Mathematics II (Engineering) - MAT2691
Diploma Semester module NQF level: 6 Credits: 12
Module presented in English
Pre-requisite: MAT1581
Purpose: Differentiation: partial differentiation, series; integration solutions of first-order differential equations; numerical methods; statistics.
Rings and Fields - MAT4834
Honours Year module NQF level: 8 Credits: 12
Module presented in English Module presented online
Co-requisite: MAT4833
Purpose: To concentrate on ring theory, especially integral domains, their fields of quotients, rings of polynomials and ideal structure in general. Considerable emphasis is placed on problem solving.
Numerical Methods for Civil Engineers A - APM3715
Advanced Diploma Semester module NQF level: 7 Credits: 12
Module presented in English
Purpose: This module is compulsory and core for the Advanced Diploma in Civil Engineering. The purpose of this module is to enable students develop competencies and skills in solving engineering problems using numerical methods.
Set Theory - MAT4835
Honours Year module NQF level: 8 Credits: 12
Module presented in English Module presented online
Purpose: To introduce the student to set-theoretic principles and fundamental constructions involving sets, from an intuitive but axiomatic point of view, and to provide a foundation which is essential for the understanding of Modern Mathematics.
Mathematics III (Engineering) - MAT3700
Baccalareus Technologiae Degree,Diploma Semester module NQF level: 7 Credits: 12
Module presented in English
Pre-requisite: MAT2691(Not applicable to 90128, 90133, 90134 and 90142)
Purpose: Students completing this module will be able to solve first-order ordinary differential equations and second order ordinary differential equations using the method of undermined coefficients, solve any order differential equations using d-operators and laplace transforms, to find the eigenvalues and eigenvectors of matrix and write the Fourier series of a function. This module will assist students to develop their mathematical knowledge and analytical skills to support and advance their studies in the field of engineering.
Topology - MAT4836
Honours Year module NQF level: 8 Credits: 12
Module presented in English Module presented online
Purpose: To introduce the student to the general framework of Topology. The module provides a detailed discussion of convergence (in terms of nets and filters), continuity, compactness and compatifications, local compactness, regularity, and complete regularity. All these topics are fundamental to an understanding of Modern Abstract Analysis.
Linear Algebra III - MAT3701
Under Graduate Degree Semester module NQF level: 7 Credits: 12
Module presented in English
Pre-requisite: MAT2611
Purpose: To acquire a basic knowledge concerning inner product spaces, invariant subspaces, cyclic subspaces, operators and their canonical forms.
Introduction to Category Theory - MAT4837
Honours Year module NQF level: 8 Credits: 12
Module presented in English Module presented online
Purpose: To provide an introduction to an area of mathematics which attempts to study those parts that are concerned with objects and special functions between them. The module presents the notion of a category, examines special constructions in categories, and considers the concept of a functor between categories as well as natural transformations between them. Concrete instances of the general categorical concepts, as they appear in established areas of mathematics, will be given and discussed.
Fundamental Mathematics - MAT1501
Diploma Year module NQF level: 5 Credits: 12
Module presented in English Module presented online
Purpose: Students credited with this module have the basic skills which can be applied in the natural sciences and engineering sciences. They will have the understanding of basic ideas of algebra and very basic calculus, which are crucial in problem solving. This module will be useful to students who have studied Mathematics at matriculation level but who do not satisfy the minimum requirements for direct admission to undergraduate study, as stated in the calendar.
Abstract Algebra - MAT3702
Under Graduate Degree Semester module NQF level: 7 Credits: 12
Module presented in English
Pre-requisite: MAT2611 Co-requisite: MAT3701
Purpose: To enable students to master and practise the applications of the concepts, results and methods necessary to construct mathematical arguments and solve problems independently as they occur in an elementary treatment of algebraic structures, groups, homomorphism theorems,factor groups, permutation groups, the main theorem for Abelian groups, Euclidean rings, divisibility in Euclidean rings, fields, finite fields, and the characteristics of a field.
Category Theory - MAT4838
Honours Year module NQF level: 8 Credits: 12
Module presented in English Module presented online
Purpose: To introduce and develop the concept of adjoint functors, limits, Monads and Algebras. Special attention shall be given to concrete instances of these occurrences in established areas of mathematics.
Linear Algebra I - MAT1503
Year module NQF level: 5 Credits: 12
Module presented in
Purpose: To enable students to understand and apply the following basic concepts in linear algebra: non-homogeneous and homogeneous systems of linear equations, Gaussian and Jordan-Gauss elimination, matrices and matrix operations, elementary determinants by cofactor expansion, inverse of matrix using the adjoint, Cramer's rule, evaluating determinants using row/column reduction, properties of the determinant function, vectors in 2- , 3- and n- space, dot product, projections, cross product, areas of parallelograms and volumes of parallelepipeds determined by vectors, lines and planes in 3-space and complex numbers.
Complex Analysis - MAT3705
Under Graduate Degree Semester module NQF level: 7 Credits: 12
Module presented in English
Pre-requisite: MAT2615
Purpose: To introduce students to the following topics in complex analysis: functions of a complex variable, continuity, uniform convergence, complex differentiation, power series and the exponential function, integration, Cauchy's theorem, singularities and residues.
Functional Analysis I - MAT4841
Honours Year module NQF level: 8 Credits: 12
Module presented in English Module presented online
Purpose: To enable learners to master and apply the fundamental concepts of linear and metric spaces. The metric (or topological) structure of a space involves the concepts of continuity, convergence, compactness and completeness. The structure of Banach spaces, linear operators defined on Banach spaces and linear functions defined on Banach spaces with range contained in the set of complex numbers are studied. The latter functions are called functionals. We also concentrate on a specific Banach space, the Hilbert space, where orthogonality, ortonormality, separability and classes of bounded linear operators (defined by using the Hilbert-adjoint operator) are studied.
Precalculus Mathematics A - MAT1510
Under Graduate Degree Year module NQF level: 5 Credits: 12
Module presented in English
Purpose: To acquire the knowledge and skills that will enable students to draw and interpret graphs of linear, absolute value, quadratic, exponential, logarithmic and trigonometric functions, and to solve related equations and inequalities, as well as simple real-life problems.
Ordinary Differential Equations - MAT3706
Under Graduate Degree Semester module NQF level: 7 Credits: 12
Module presented in English
Pre-requisite: APM2611
Purpose: To enable students to master the fundamental concepts and apply the methods for the solution of homogeneous and non-homogeneous systems of differential equations, as well as Gronwall's inequality, qualitative theory, and the linearisation of nonlinear systems.
Functional Analysis II - MAT4842
Honours Year module NQF level: 8 Credits: 12
Module presented in English Module presented online
Co-requisite: MAT4841
Purpose: To enable learners to master and apply the more advanced theory of normed and Banach spaces. The four so-called 'corner stones' of functional analysis, namely the Hahn-Banach theorem, the Banach-Steinhaus theorem, the open mapping theorem and the closed graph theorem are studied. Spectral theory of bounded linear operators in normed spaces as well as some spectral theory in Banach algebras are studied since this is one of the main branches of modern functional analysi The spectral properties of compact linear operators are also studied.
Precalculus Mathematics B - MAT1511
Under Graduate Degree Year module NQF level: 5 Credits: 12
Module presented in English
Purpose: Students credited with this module will have understanding of basic ideas of algebra and to apply the basic techniques in handling problems related to: the theory of polynomials, systems of linear equations, matrices, the complex number system, sequences, mathematical induction, and binomial theorem
Ordinary Differential Equations I - MAT4843
Honours Year module NQF level: 8 Credits: 12
Module presented in English Module presented online
Purpose: To enable the learner to apply the investigative techniques of the qualitative theory of nonlinear ordinary differential equations. Dynamical systems, modelled in terms of nonlinear differential equations, have many applications in the physical, biological and social sciences. However, it is not in general possible to obtain an analytical solution to an arbitrary differential equation - and even when an analytical solution can be found, it is sometimes very difficult to "see" the main feature of the solution from it. In the qualitative study of differential equations, quite detailed information on the nature of the solution to a differential equation is obtained without constructing an exact solution. Contents: Phase diagrams, periodic solutions, limit cycles, energy balance and harmonic balance, stability, Lyapunov methods.
Discrete Mathematics: Combinatorics - MAT3707
Under Graduate Degree Semester module NQF level: 7 Credits: 12
Module presented in English
Pre-requisite: MAT2612
Purpose: To enable students to understand and apply the following concepts: (a) In graph theory: isomorphism, planar graphs, Euler tours, Hamilton cycles, colouring problems, trees, networks; (b) In enumeration: basic counting principles, distributions, binomial identities, generating functions, recurrence relations, inclusion-exclusion.
Calculus A - MAT1512
Under Graduate Degree Year module NQF level: 5 Credits: 12
Module presented in English
Purpose: To equip students with those basic skills in differential and integral calculus which are essential for the physical, life and economic sciences. Some simple applications are covered. More advanced techniques and further applications are dealt with in module MAT1613.
Real Analysis II - MAT3711
Under Graduate Degree Semester module NQF level: 7 Credits: 12
Module presented in English
Pre-requisite: MAT2613
Purpose: To enable students to understand metric spaces, continuity, convergence, completeness, compactness, connectedness, Banach's fixed point theorem and its applications, the Riemann-Stieljes integral, normed linear spaces, and the Stone-Weierstrass theorem.
Ordinary Differential Equations II - MAT4844
Honours Year module NQF level: 8 Credits: 12
Module presented in English Module presented online
Co-requisite: MAT4843
Purpose: To introduce the learner to the behaviour and analysis of nonlinear systems, in particular nonlinear and forced oscillations. Solutions to linear differential equations can only behave in a fairly limited number of ways, but the presence of nonlinear elements may introduce totally new phenomena. Seemingly simple nonlinear differential equations can lead to unexpectedly complex solution structures. This module introduces analytical approximation methods as well as qualitative methods for analysing the behaviour of solutions to the nonlinear systems. Contents: Perturbation methods, forced oscillations, harmonic and subharmonic response, stability of periodic solutions, bifurcation, structural stability, chaos.
Precalculus (Engineering) - MAT1514
Diploma Semester module NQF level: 5 Credits: 12
Module presented in English
Purpose: Students credited with this module will understand the basic ideas of algebra. The focus is on building strong algebraic and trigonometric skills that will support the development of analytical skills that is crucial in problem solving in more advanced mathematics and related subjects. Students learn the definitions and laws of linear, absolute value, quadratic, exponential, logarithmic and trigonometric functions in order to draw and interpret graphs, solve equations and apply these to simple real-life problems.
Mathematics III B (Engineering) - MAT3714
Advanced Diploma Semester module NQF level: 7 Credits: 12
Module presented in English
Purpose: Students completing this module will be able to solve first-order ordinary differential equations and second order ordinary differential equations using the method of undetermined coefficients, solve any order differential equations using d-operators and Laplace transforms, to find the eigenvalues and eigenvectors of a matrix and write the Fourier series of a function. This module will assist students to develop their mathematical knowledge and analytical skills to support and advance their studies in the field of engineering.
Graph Theory I - MAT4845
Honours Year module NQF level: 8 Credits: 12
Module presented in English Module presented online
Purpose: To introduce learners to Graph Theory, starting with the basic concepts and elementary theory. Learners will be guided to construct proofs and will gain experience in independent problem solving. Topics covered in this module include: Subgraphs, degree sequences, structure of graphs, trees and connectivity. In this module and in Graph Theory II, graphs are studied from the viewpoint of pure mathematics, but the concepts studied have applications in Computer Science, Chemistry, Biology and other areas.
Mathematics I (Engineering) - MAT1581
Diploma Semester module NQF level: 5 Credits: 12
Module presented in English
Recommendation: Students are recommended to register for MAT1581 in the 2nd semester, once they have done the first part of MAT1510
Purpose: Algebra; trigonometry; calculus; complex numbers; co-ordinate geometry; analytic geometry; matrices; determinants.
Advanced Engineering Mathematics - APM4813
Honours Year module NQF level: 8 Credits: 12
Module presented in English
Purpose: This module will be useful to students in engineering who need to be equipped with a scope of knowledge regarding more advanced mathematical concepts and techniques, necessary in various fields of engineering. In particular convergence tests for series of a single variable and their applications to power series, the basic theory of complex functions and their differentiation and integration, a more advanced knowledge of the theory of Laplace transforms (including transfer functions and stability criteria), Z-transforms as a discrete-time analog of the Laplace transform, and the state space approach to dealing with MIMO systems.
Graph Theory II - MAT4846
Honours Year module NQF level: 8 Credits: 12
Module presented in English Module presented online
Co-requisite: MAT4845
Purpose: To enable learners to further their knowledge and understanding of Graph Theory, to gain deeper insight into higher mathematics and to improve their problem solving skills and their ability to reason logically. Topics covered in this module include planar graphs, graph colourings trees and Hamiltonian graphs.