Mathematics (MAT) is a six-semester subject and Applied Mathematics (MAP) is a four-semester subject. These subjects may be taken as 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 or MAT 1E; 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 or MAT 1E, MAT 2 or MAP 2; MAP 3. See Rule S.23.
See the Departmental Web Page http://www.ru.ac.za/academic/departments/mathematics/ for further details, particularly on the content of courses.
MAT 101 and MAT 102 are each given in both first and second semesters; there are also two semester-credit courses MAT 1E1 and MAT 1E2 (equivalent to MAT 101 and MAT 102 respectively) each offered over a whole year. Students who have not achieved a mark equivalent to at least a Higher Grade D pass at matriculation level will be required to register for MAT 1E1 and MAT 1E2, rather than MAT 101 and MAT 102.
Credit may be obtained in each course separately and, in addition, an aggregate mark of at least 50% will be deemed to be equivalent to a two-credit course MAT 1 or MAT 1E respectively, provided that a candidate obtains the required subminimum in each component, and, in the case of MAT 1, obtains the required aggregate in a single year. Supplementary examinations may be recommended for any of these courses, provided that a candidate achieves a minimum standard specified by the Department.
Aggregated credit in any one of MAT 101 or MAT 1E1 and in either of STA 110 or STA 130 is deemed equivalent to a two-credit course Mathematics 1C, which is an allowed prerequisite for various other courses in the Faculties of Science and Commerce.
Mathematics 1L (MAT 1L) is a semester-credit 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 101 or MAT 1E1 and MAT 102 or MAT 1E2.
As from 2003 students admitted to the Science Faculty into the 4 year or foundation programmes with only a Standard Grade matric Mathematics pass shall be registered for this course irrespective of whether they intend to register for further Mathematics courses or not. Individual consideration can be given to permit such students to register concurrently for MAT IL and one of the slow stream courses MAT IE1 or IE2.
In the Science Faculty, in cases where registration in it is prescribed, credit in MAT IL shall be required over and above the required 18 (or 20) semester credits for the degree. (This is not recommended for Faculties where matric Mathematics is not an entrance requirement.)
MAT 1L : Mathematics Literacy
Arithmetic: 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.
MAT 101 / MAT 1E1 : Fundamental Calculus
Limits, differentiation, curve sketching, maximum - minimum problems, integration, applications, first-order differential equations, partial differentiation.
MAT 102 / MAT 1E2 : Discrete Mathematics
Relations between sets, matrix algebra, linear algebra and linear programming, data analysis, vectors in 2 and 3 dimensions, analytic geometry, complex numbers, induction and algorithms, graph theory, Boolean algebra.
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 subminimum in each component.
Credit in Mathematics (MAT 1 or MAT 1E) 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, Geometry with applications, Differential Geometry.
Year Column 1 Column 2 Column 3 Column 4 2 Transformation Advanced Discrete Mathematical Geometry Calculus Mathematics Programming (M2.1) (M2.2) (AM2.1) (AM2.2) 2 Linear Algebra Applied Numerical Math Foundations (M2.3) Analysis Analysis of Mechanics (M2.4) (AM2.3) (AM2.4) 3 Algebra Complex Numerical Control and (M3.1) Analysis Analysis Optimisation and Transform (AM3.1) (AM3.2) Theory (M3.2) 3 Real Analysis Applied Analysis Logic of Mathematical (M3.3) (M3.4) Computation Modelling (AM3.3) (AM3.4)M2.1 (about 32 lectures) Transformation Geometry
Objectives: This course is a natural sequel to MAT 102, and a good preparation for other courses such as M2.3 (Linear Algebra), M3.1 (Algebra), as well as AM3.2 (Control and Optimization). It conveys the message that Geometry is a powerful means of turning visual images into formal tools for the understanding of other mathematical phenomena.
Syllabus: Transformations on the plane and 3-space (isometries, similarities, affine transformations). Various applications (Frieze groups, wallpaper groups, tessellations, spherical geometry and mapmaking, Poincaré model of hyperbolic geometry, etc.) Other applications to Physics, Computer Graphics and Cartography.
M2.2 (about 32 lectures) Advanced Calculus
Objectives: This is a continuation of MAT 101 without which further studies in Mathematics and Physics are hardly feasible.
Syllabus: Complex numbers. Sequences and series, Taylor series. Advanced integration techniques, improper integrals. Functions of several variables including: tangent planes, extrema, Taylor's Theorem, multiple integrals. Further differential equations and application.
M2.3 (about 32 lectures) Linear Algebra
Objectives: Another fundamental aspect of contemporary mathematical modelling in Physics, Economics, etc.
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) Applied Analysis
Objectives: These concepts are essential to physicists.
Syllabus: The div, grad and curl operations of vector calculus and theorems of Gauss, Green and Stokes; techniques for solving some ordinary differential equations; introduction to some special functions of interest in the natural sciences; Fourier series; applications of the above techniques in solving certain boundary value and initial value problems.
AM2.1 (about 32 lectures) Discrete Mathematics
Objectives: This course is a natural sequel to MAT 102. Further counting techniques are investigated and applied. In particular, a number of applications to probability theory are discovered. Elementary logic and set calculus from MAT 102 is extended and applied to computer science.
Syllabus: Review of elementary counting, multisets and multinomial theorem, the principle of inclusion and exclusion, inversion formulae. Counting patterns with group actions. Matching problems, optimal assignments.
AM2.2 (about 32 lectures) Mathematical Programming.
Objectives: This is a course in mathematical programming with MATLAB. The course will introduce the student to programming for the purpose of solving scientific or mathematical problems. The basics of MATLAB will be introduced and MATLAB knowledge and skills will be developed throughout the course. This will be achieved by means of weekly practical sessions in the laboratory where the students will gain hands-on experience under supervision. Knowledge of programming is not a prerequisite for this course. The main objective is to acquire the ability to construct exploratory environments on a computer.
Syllabus: MATLAB basics, matrix manipulation, control flow, graphics, solution of equations, sequences and recurrence relations, Leslie matrices and their applications, Monte Carlo simulation, introduction to numerical methods. Various applications including: population dynamics and chaos, kinematics, the investigation of stochastic processes such as throwing dice and coins.
AM2.3 (about 32 lectures) Numerical Analysis
Objectives: 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: Computing and use of Matlab, solving nonlinear equations, solving sets of equations, interpolation and curve fitting, numerical differentiation and integration, numerical solution of ODE's, approximation of functions.
AM2.4 (about 32 lectures) Mathematical Foundation of Mechanics
Objectives: To provide a sound mathematical foundation for an important part of Physics and much of Engineering.
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
Objectives: Algebraic structures pervade all of modern mathematics. No mathematician can afford not to know group theory. Coding theory is an application of modern algebra. Group structures occur in the theoretical study of most natural sciences.
Syllabus: Induction. Properties of integers and polynomials, division algorithms, primes, unique factorization, congruences. Group theory, Lagrange's theorem, quotient groups.
M3.2 (about 39 lectures) Complex Analysis and Transform Theory
Objectives: These are classical aspects of mathematical analysis and fundamental to Physics.
Syllabus: Complex Analysis: analytic functions, Cauchy-Riemann conditions,
branch cuts, Taylor and Laurent series, contour integration. Cauchy residue
theorem, evaluation of some real value integrals using the Cauchy residue
Fourier Analysis: Derivation of Fourier's identity, Fourier transform pairs, elementary properties of Fourier transforms, convolution theorems, use of contour integrals in evaluating Fourier transforms and inverses.
Laplace Transform: Definition, elementary properties of the Laplace transform. Inverse Laplace transform and Bromwich's contour integral formula.
M3.3 (about 39 lectures) Real Analysis
Objectives: Continuity and convergence of processes are important considerations in our daily experiences. These can be described in exact mathematical terms. This course leads to basic topological concepts.
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
Objectives: The solution of differential equations is important in many areas of application, most particularly engineering fields.
Syllabus: A selection of techniques used in the solution of certain ordinary differential equations, systems of ODE, partial differential equations and systems of PDE. These techniques include the method of characteristics, Fourier transforms and Laplace transforms. Introduction to dynamical systems.
AM3.1 (about 39 lectures) Numerical Analysis
Objectives: To study and apply advanced techniques in 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) Control and Optimization
Objectives: This course is intended to be an elementary introduction to two related, but distinct, areas of mathematical investigation. The approach is modern and geometric, and application-oriented. These techniques are much used in modern engineering design as well as problems in economics.
Syllabus: Linear Control Systems: Solution of an uncontrolled system: spectral form, exponential form. Solution of a controlled system. Time varying systems. Relationship between state space and classical form. Controllability. Observability. Linear feedback. Optimization: Optimization of functions of a single and several variable(s). Constrained optimization. Linear programming, geometric interpretation. The simplex method. Optimal Control: Calculus of variations. Pontryagin's principle.
AM3.3 (about 39 lectures) Logic of Computation
Objectives: Mathematical logic is the basis of computer science and machine languages. A good foundation as well as an introduction to fuzzy concepts are provided in this course.
Syllabus: Introduction to logic: conjunction, disjunction, complementation,
implication and entailment of propositional calculus.
Two-valued logic : Boolean lattices, algebras, unique complementation, De Morgan laws, relations.
Many-valued logic : Power set operation of fuzzy sets : Zadeh's extension principle, various t-norms on the unit-interval and their associated complements; lattice-valued fuzzy sets, fuzzy relations.
Introduction to Quantum logic : quantum bits, gates, state spaces and computational basis.
AM3.4 (about 39 lectures) Mathematical Modelling
Objectives: This is a natural sequel to AM2.2 which is a prerequisite for this course. A wide range of real-world problems will be simulated by mathematical models.
Syllabus: Linear and non-linear differential equations, numerical solution to non-linear differential equations with MATLAB, solution of systems of linear differential equations using exponential matrices and MATLAB, linear difference equations, non-linear difference equations with MATLAB. Curve fitting: linear and non-linear regression. Various applications, including: mathematical theories of war, spread of diseases, chemical processes and reactions, dynamical systems and chaos, population dynamics, problems in dynamics and kinetics, stochastic processes.
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.