Professor & Acting Head of Department
V Murali, MSc(Madras), MSc(Wales), PhD(Rhodes)
Professor / Associate Professor
To be appointed
MH Burton, BSc(Hons)(Natal), MSc(UCT), PhD(Rhodes)
KJ Koch, MSc(Notal)
AL Pinchuck, MSc(Rhodes)
GJ Shepherd, MSc(Wits)
CC Remsing, MSc(Timisoara), PhD(Rhodes)
Lecturer, Academic Development
M Lubczonok, Masters(Jagiellonian)
Mathematics (MAT) is a six-semester subject and Applied Mathematics (MAP) is a four-semester subject. These subjects may be taken as major subjects for the degrees of BSc, BA, BJourn, BCom, BBusSci, BEcon and BSocSc, and for the diploma HDE(SEC).
To major in Mathematics, a candidate is required to obtain credit in the following courses: MAT1; MAT 2 or MAP 2; MAT 3. See Rule S.23.
To major in Applied Mathematics, a candidate is required to obtain credit in the following courses: MAT 1, MAT 2 or MAP 2; MAP 3. See Rule S.23. The attention of students who hope to pursue careers in the field of Bioinformatics is drawn to the recommended curriculum that leads to postgraduate study in this area, in which Mathematics is a recommended co-major with Biochemistry, and for which two years of Computer Science and either Mathematics or Mathematical Statistics are prerequisites. Details of this curriculum can be found in the entry for the Department of Biochemistry, Microbiology and Biotechnology.
See the Departmental Web Page http://www.ru.ac.za/academic/departments/mathematics/ for further details, particularly on the content of courses.
Mathematics 1 (MAT 1) is given as a year-long unsemesterized two-credit course. Credit in MAT 1 must be obtained by students who wish to major in certain subjects (such as Applied Mathematics, Physics and Mathematical Statistics) and by students registered for the BBusSci degree.
Introductory Calculus (MAT 101) is a required semester-credit course for Pharmacy students, and Discrete Mathematics (MAT 102) is a required semester-credit course for students who wish to major in Computer Science. Either or both of MAT 101 and MAT 102 may be taken for credit purposes in other degree programmes, for example by students who wish to major in Chemistry or Ichthyology. Students who have taken MAT 101 and 102 separately and who obtain a sufficiently high aggregate for the combination may be permitted to continue to second year mathematics or applied mathematics courses at the discretion of the Head of Department.
Note: Students in Pharmacy or in Computer Science will normally be obliged to register for MAT 101 or MAT 102 (respectively). However, students in Pharmacy and Computer Science have the option of taking the two-credit course MAT 1 for their degrees instead.
Supplementary examinations may be recommended for any of these courses, provided that a candidate achieves a minimum standard specified by the Department.
Mathematics 1L (MAT 1L) is a full year course for students who do not qualify for entry into any of the first courses mentioned above. This is particularly suitable for students in the Social Sciences and Biological Sciences who need to become numerate or achieve a level of mathematical literacy. A successful pass in this course will give admission to MAT 1.
MAT 1 (Year course)
This course introduces students to differential and integral calculus as well as linear algebra, complex numbers and discrete mathematics.
Algebra (30-35 lectures in year course)
Vectors; matrices; determinants; linear systems; polar co-ordinates; functions; induction; complex numbers and elementary functions.
Calculus (85-90 lectures in year course)
Limits and continuity; asymptotes. Differentiation: optimization problems from the real world; Intermediate Value Theorem; Rolle's theorem; Mean Value Theorem; rules of curve sketching; l'Hospital's rule; parametric equations. Antiderivative: indefinite integral; Riemann sum and definite integral; Fundamental Theorem. Applications: area, volume, work, first and second moments. Improper integrals. Multivariate calculus: partial differentiation, max/min/saddle point, chain rules, directional derivative, gradient. Elementary ordinary differential equations. Sequences and series; elementary convergence testing; Taylor series.
MAT 101 (Semester course: Introductory Calculus) (about 65 lectures)
Basic concepts; limits and derivatives; uses of differentiation; numerical and graphical methods; integration; differential equations.
MAT 102 (Semester course: Discrete Mathematics) (about 65 lectures)
Logic; sets & functions; mathematical induction; elementary combinatorics; recursion and recurrence relations; linear systems and matrices; determinants; complex numbers.
MAT 1L : Mathematics Literacy
This course helps students develop appropriate mathematical tools necessary to represent and interpret information quantitatively. It also develops skills and meaningful ways of thinking, reasoning and arguing with quantitative ideas in order to solve problems in any given context.
Arithmetic: Units of scientific measurement, scales, dimensions;
Error and uncertainty in measure values.
Fractions and percentages - usages in basic science and
commerce; use of calculators and spreadsheets.
Algebra: Polynomial, exponential, logarithmic and trigonometric functions and their graphs; modelling with functions; fitting curves to data; setting up and solving equations. Sequences and series, presentation of statistical data.
Differential Calculus: Limits and continuity; Rules of Differentiation; Applications of Calculus in curve-sketching and optimisation.
Both Mathematics and Applied Mathematics are offered at the second and third year level. Each consists of four topics as listed below. Two of these successfully completed in one year earn one credit in either Mathematics or Applied Mathematics at the second year level or third year level respectively. The first credit so earned will be called MAT 201 (respectively MAT 301) in the case of Mathematics, and MAP 201 (respectively MAP 301) in the case of Applied Mathematics. Subsequent credits will be MAT 202 (respectively MAT 302) and MAP 202 (respectively MAP 302). The selection of topics is subject to the approval of the Head of Department. Credit may be obtained in each semester-credit separately and, in addition, an aggregate mark of at least 50% will be deemed to be equivalent to a two-credit course MAT 2 (respectively MAT 3) or MAP 2 (respectively MAP 3), provided that a candidate obtains the required sub-minimum in each component.
Credit in Mathematics (MAT 1) is required before a student may register for MAT 201, MAT 202, MAP 201 or MAP 202. Likewise credit for MAT 2 or MAP 2 is required before admission to the third year courses.
Some of the topics below may be prerequisites for others. Not all of the eight topics might be offered in a single year.
Natural combinations are indicated in the table below. If a student chooses the two second year topics in column x, where x = 1 , 2 , 3 or 4, a natural follow-up in the 3rd year would be the topics in column x as well. However, different combinations across the columns are feasible as well. In some cases, and with the permission of the Head of Department, a third-year topic may be done in the second year and vice versa.
Columns 1 and 2 contain standard/classical mathematics which all future mathematicians should know. The topics in column 3 are particularly useful for computer scientists, and the ones in column 4 for physicists and future engineers.
Other topics may be given instead from time-to-time at third year level, depending on the student intake, e.g. Quantum Mechanics, Topology, Differential Geometry and Metric Spaces.
Year Column 1 Column 2 Column 3 Column 4 2 Transformation Advanced Discrete Numerical Methods Geometry Calculus Mathematics with MATLAB (M2.1) (M2.2) (AM2.1) (AM2.2) 2 Differential Linear Algebra Math Foundations Mathematical Equations (M2.3) of Mechanics Modelling (M2.4) (AM2.4) (AM2.3) 3 Algebra Complex Quantum Numerical (M3.1) Analysis Mechanics Analysis (M3.2) (AM3.3) (AM3.1) 3 Real Analysis Applied Analysis Linear Control Optimisation (M3.3) (M3.4) (AM3.2) (AM3.4)M2.1 (about 32 lectures) Transformation Geometry
This course is an elementary (and exciting) introduction to modern geometry.
Syllabus: the Euclidean plane, transformations (colliniations, dilatations), isometries (translations, halfturns, reflections, glide reflections), classification of isometries, similarities (stretches, stretch reflections, stretch rotations), classification of similarities, affine transformations (shears, strains).
M2.2 (about 32 lectures) Advanced Calculus
This course builds on the single-variable calculus covered in Mathematics 1. It covers differential and integral calculus of functions of several variables and vector calculus. Its aim is to acquaint students with the theory, techniques and applications of the calculus of several variables and vector calculus. It is an essential course for students who plan to major in Mathematics, Applied Mathematics or in a related field.
Syllabus: Partial differentiation; directional derivatives and the gradient vector, maximum and minimum values, Lagrange multipliers. Multiple integrals; double and triple integrals in various coordinate systems; Vector calculus: vector fields, line integrals, fundamental theorem for line integrals, Green's theorem, curl and divergence, parametric surfaces and their areas, surface integrals, Stokes' theorem, the divergence theorem.
M2.3 (about 32 lectures) Linear Algebra
Linear algebra is the study of vector spaces and linear transformations on them. This is an important branch of mathematics which provides the tools and methods essential for studying many mathematical structures that arise within mathematics and other sciences. This course is essential for students who plan to major in Mathematics, Applied Mathematics or in a related field.
Syllabus: Finite dimensional vector spaces, linear independence, linear maps and their matrix representations, change of base, null space, rank. Eigenvalues and vectors, diagonalization; applications.
M2.4 (about 32 lectures) Differential Equations
Differential equations are used extensively to model physical phenomena. This course develops qualitative theory of ordinary differential equations and analytic techniques for solving them. Partial differential equations are also discussed briefly.
Syllabus: First order ordinary differential equations and their applications; linear differential equations of second order; series solutions; Laplace transforms; systems of first order equations; special functions of mathematical physics.
AM2.1 (about 32 lectures) Discrete Mathematics
This is an introductory course on basic techniques and methods of counting. It builds on the discrete mathematics studied in Mathematics 1. It serves as a bridge between mathematics and computer science.
Syllabus: Review of elementary counting, multisets and multinomial theorem, the principle of inclusion and exclusion, inversion formulae. Counting patterns with group actions. Generating functions.
AM2.2 (about 32 lectures) Numerical Methods with MATLAB.
The student will first be introduced to the MATLAB computational environment and this will be used to develop some of the most useful numerical methods for solving mathematical and scientific problems.
Syllabus: Computations with MATLAB, programming with MATLAB, applications to systems of linear and non-linear equations, optimisation, linear and non - linear regression, interpolation, numerical differentiation, quadrature, numerical solution of systems of non-linear differential equations.
AM2.3 (about 32 lectures) Mathematical Modelling
Mathematical modelling is the simulation of a real-world process using mathematical notions. In this course, a student will learn basic techniques for model formulation, analysis and implementation. The MATLAB environment will be used to implement the necessary numerical methods.
Syllabus: Modelling with: systems of linear and non-linear differential equations, systems of linear and non-linear recurrence relations, dynamical systems, Markov systems. Applications to a variety of real-world situations including: chemical processes, diseases, mechanics, various stochastic processes (such as throwing dice and coins, card games).
AM2.4 (about 32 lectures) Mathematical Foundation of Mechanics
This course provides a sound mathematical foundation of Mechanics. It places great emphasis on a systematic approach to the mathematical formulation of mechanics problems and to the physical interpretation of their mathematical solutions.
Syllabus: Momentum, Angular Momentum, Kinetic Energy, Potential Energy, Centre of Mass Systems, Moment of Inertia. Derivation of Lagrange's equations from NII, generalised coordinates, holonomic constraint, nonholonomic constraint examples. Derivation of Lagrange's equations using Calculus of Variations and Hamilton's Principle. Euler-Lagrange condition in general and examples (extremisation of areas etc.). Using a nonholonomic treatment to derive forces of constraint. Cyclic co- ordinates and constants of motion. Generalised momentum. Hamiltonian formalism. Introduction to Phase Plane methods, isoclinal methods, family portraits.
M3.1 (about 39 lectures) Algebra
Algebra is one of the main areas of mathematics with a rich history. Algebraic structures pervade all modern mathematics. This course introduces students to the algebraic structure of groups, rings and fields. Algebra is a required course for any further study in mathematics.
Syllabus: Sets, equivalence relations, groups, rings, fields, integral domains, homorphisms, isomorphisms, and their elementary properties.
M3.2 (about 39 lectures) Complex Analysis
Building on the first year introduction to complex numbers, this course provides a rigorous introduction to the theory of functions of a complex variable. It introduces and examines complex-valued functions of a complex variable, such as notions of elementary functions, their limits, derivatives and integrals.
Syllabus: Revision of complex numbers, Cauchy-Riemann equations, analytic and harmonic functions, elementary functions and their properties, branches of logarithmic functions, complex differentiation, integration in the complex plane, Cauchy's Theorem and integral formula, Taylor and Laurent series, Residue theory and applications. Fourier Integrals.
M3.3 (about 39 lectures) Real Analysis
Real Analysis is the field of mathematics that studies properties of real numbers and functions on them. The course places great emphasis on careful reasoning and proof. This course is an essential basis for any further study in mathematics.
Syllabus: Topology of the real line, continuity and uniform continuity, Heine-Borel, Bolzano-Weierstrass, uniform convergence, introduction to metric spaces.
M3.4 (about 39 lectures) Applied Analysis
This course deals with the basic theory of partial differential equations (elliptic, parabolic and hyperbolic) and dynamical systems. It presents both the qualitative properties of solutions of partial differential equations and methods of solution.
Syllabus: First-order partial equations, classification of second-order equations, derivation of the classical equations of mathematical physics (wave equation, Laplace equation, and heat equation), method of characteristics, construction and behaviour of solutions, maximum principles, energy integrals. Fourier and Laplace transforms, introduction to dynamical systems.
AM3.1 (about 39 lectures) Numerical Analysis
Many mathematical problems cannot be solved exactly and require numerical techniques. These techniques usually consist of an algorithm which performs a numerical calculation iteratively until certain tolerances are met. These algorithms can be expressed as a program which is executed by a computer. The collection of such techniques is called "numerical analysis".
Syllabus: Systems of non-linear equations, two-dimensional cubic splines, numerical algebra, numerical computation of eigenvalues, numerical solution of ordinary and partial differential equations, finite differences, the finite element method, discrete Fourier transform algorithms.
AM3.2 (about 39 lectures) Linear Control
This course is an elementary introduction to modern linear control, from an application-oriented mathematical perspective.
Syllabus: Mathematical formulation of the control problem, linear dynamical systems, linear control systems (controllability, observability, linear feedback, realization), stability and control, optimal control (Pontryagin's principle, linear regulators with quadratic costs).
AM3.3 (about 39 lectures) Quantum Mechanics
The course develops a quantum theory as a set of axioms or postulates within a suitable mathematical framework by reducing the known facts from real experiment to "thought experiment" type arguments which can then be used to write the laws of quantum mechanics in an easily acceptable yet mathematically correct way. The working theory thus obtained is then used to analyse a quantum system of interest, such as angular momentum.
Syllabus: Phenomenology and experimental background; Quantum statics; Quantum dynamics; Application to a typical real world Quantum system.
AM3.4 (about 39 lectures) Optimisation
This is an introductory course on the mathematical theory of optimisation. In this course, students will learn the theory, computational techniques and algorithms used in many areas of applied mathematics, economics and engineering.
Syllabus: Basic definitions and examples; Unconstrained optimisation: First and second order conditions; algorithms for unconstrained optimisation: one dimensional search methods, gradient methods, Newton methods, conjugate direction methods, quasi-Newton methods. Constrained optimisation: Lagrange's Theorem and Method, Kuhn-Tucker Theorem; algorithms for constrained optimisation. Linear Programming: Simplex Method, two-phase method, duality, sensitivity analysis; Convex optimisation: Convexity and optimality.
Each of the two courses consists of ten topics of about 26 lectures each, plus tutorial assignments or practical work. A Mathematics Honours course usually requires the candidate to have majored in Mathematics, whilst Applied Mathematics Honours usually requires the candidate to have majored in Applied Mathematics. The topics are selected from the following general areas covering a wide spectrum of contemporary Mathematics and Applied Mathematics:
Topology; Algebra and applications (e.g. Coding Theory and Cryptography); Mathematical Logic and Set Theory; Fuzzy Sets with applications to Topology, Algebra and Control Theory; Measure Theory and Functional Analysis; Lattice Theory; Combinatorics; Category Theory; Differential Equations; Differential Geometry; Aspects of mathematical education (such as olympiads, elementary mathematics from an advanced viewpoint); Group representations; Quantum Field Theory; Nonlinear Dynamics and Chaos Theory; Mathematics of networking; Wavelet Analysis; Numerical Analysis; Neural Networks; Information and Control; Fuzzy linear programming; Mathematical modelling.
Two or three topics from those offered at the third-year level in either Mathematics or Applied Mathematics may also be taken in the case of a student who has not done such topics before. Two or three topics may also be replaced by an appropriate project.
With the approval of the Heads of Department concerned, the course may also contain topics from Education, and from those offered by other departments in the Science Faculty such as Physics, Computer Science, and Statistics.
On the other hand, the topics above may also be considered by such Departments as possible components of their postgraduate courses.
Suitably qualified students are encouraged to proceed to these degrees under the direction of the staff of the Department. Requirements for these degrees are given in the General Rules.
A Master's degree in either Mathematics or Applied Mathematics is taken by a combination of course work and a thesis. Normally four examination papers and/or essays are required apart from the thesis. The whole course of study must be approved by the Head of Department.