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Mathematics (offered by the Department of Mathematical Sciences) Telephone number 012 429 6202 1 Introduction All attempts to give a deﬁnition of Mathematics have suﬀered from one shortcoming or another, so we will not try to deﬁne the subject here. There is never any doubt, however, as to whether something is mathematical or not. As aspirant Mathematics students, you have already had some contact with Mathematics and so you know something about the subject. The best that we can do here is to say that a mathematician is someone who asks questions of a certain nature and then attempts to ﬁnd answers to these questions. The aim of the mathematician is to ﬁnd answers to increasingly diﬃcult questions and to systematise and generalise existing Mathematics so that an ever larger class of questions might be answered in a simple fashion. The questions which a mathematician asks might have their origins in a wide variety of disciplines. They might originate in Mathematics itself or in any of the applied sciences such as physics, mechanics, chemistry, economics, etc. In the teaching of Mathematics, we are guided by the following considerations: (i) the work we discuss must link up with the student’s existing knowledge; (ii) the work must be of fundamental importance to both mathematics in its own right and to its ﬁelds of application; (iii) the work must be presented in a systematic manner so that as many problems as possible might be solved with the least possible eﬀort. In view of these considerations, undergraduate Mathematics consists mainly of algebra, analysis and discrete mathematics. These topics are of fundamental importance to both the applications of Mathematics and to further developments of the subject as a discipline in its own right. 1.1 Mathematics on ﬁrst- and second-year level There are ﬁve modules on ﬁrst level, MAT112, 113 and 103 as well as two introductory modules, MAT110 and 111. As from 2001, the modules MAT112 and 113 replaced the modules MAT101 and MAT102. The four modules MAT111, 112, 113 and 103 are compulsory for a ‘full’ ﬁrst-year course in Mathematics and, depending on your results in Mathematics at Matriculation level, it may be required that MAT110 ﬁrst be passed before registering for MAT112, 113 and 103. The ﬁrst-level modules APM111 and 112 in Applied Mathematics contain important applications of the calculus discussed in MAT112 and 113. APM113 contains numerical applications of the linear algebra dealt with in MAT103. Hence you are strongly advised to include at least the abovementioned three APM modules in your curriculum, especially if you intend to take Mathematics on second level or as a major subject. On second level there are seven modules, MAT211, 212, 213, 215, 216, 217 and 219, of which any four form a full second-year course. If you wish to take Mathematics at third level, you must ensure that you satisfy the necessary Prerequisite. If you wish to take Mathematics as a major subject, MAT213 and 211 are compulsory. The other module(s) which you take at second level will depend on your choice at third level. The modules MAT211, 215 and 216 are important for applications in physics and other natural sciences. 1.2 Requirements for admission to postgraduate studies To qualify for admission to studies for the Honours BSc degree in Mathematics a student must hold a Bachelor’s degree and, inter alia, have passed and obtained good results in ONE of the following: (a) four third-level module in Mathematics (b) three third-level module in Mathematics and two third-level module in Applied Mathematics (c) Mathematics III Admission can be refused on grounds of an unsatisfactory undergraduate study record. Further particulars will be found in the departmental brochure on postgraduate studies. 2 General Information Credit for the BSc degree and/or the National Certiﬁcate in Datametrics is given for ﬁve ﬁrst-level MAT modules. To register for one or more of MAT110 and 111 a student must have satisﬁed the requirements of Sc1(1)(b)(i)–(iv) in Part 7 of the Calendar. Students who passed (or were exempted from) one or more of MAT111, 101, 102, 103 prior to 1993, may complete the remaining MAT modules without MAT110. If MAT111 has not yet been passed, it must be completed together with the remaining ﬁrst-level MAT modules. Students who were registered for one or more of MAT111, 101, 102 and 103 prior to 1993 but who did not pass any of them may re-enrol for MAT modules. MAT110 and 111 are NOT available to students who passed all three of MAT101–103 prior to 1990. Such students may now register for second-level MAT modules and, where applicable, second-level APM modules. MAT110 is also NOT available to students who passed all three of MAT101–103 (with or without MAT111) prior to 1993. The use of a pocket calculator is permissible in the examination for MAT217. Credit for a degree is granted for: (i) not more than two of MAT101, 102, 112, 113 (ii) either MAT111 or MAT101, 102 and 103 if all three were passed prior to 1990 (iii) either MAT110 or MAT101, 102, 103 if all three were passed prior to 1993 (iv) not more than one of MAT216, APM201 and 211 (v) not more than one of MAT215, APM202, and 212 (vi) not more than two of MAT215 (or APM212) or MAT216 (or APM211) and MAT214 (or 203) 12 (vii) (viii) (ix) (x) (xi) (xii) not more than two of MAT201, 202 and 211 either MAT212 or COS201 passed prior to 1991 either MAT214 or MAT203 either MAT311 or MAT303 and/or 304 either MAT217 or APM214 either MAT218 or APM215 3 Mathematics as a Major Subject If you intend to obtain a BSc degree with Mathematics as a major subject, please note that: Because some of the most important non-trivial applications of mathematics are found in the ﬁeld of mechanics and physics, we strongly recommend that you include at least all the ﬁrst-level modules in Applied Mathematics and/or Physics in your curriculum. If you choose Mathematics as a major subject, you are also strongly advised to include modules in Statistics and Computer Science in your curriculum. A basic knowledge of these Subjects is important for your training as a mathematician. If you plan to take a SECOND major together with Mathematics as a major, you will ﬁnd that almost any subject oﬀered as a major for the BSc degree can be sensibly combined with Mathematics, eg Applied Mathematics, Physics, Chemistry, Statistics, Astronomy, Computer Science, Geography, etc. Further recommendations are given in this chapter. If you plan to take Mathematics as your ONLY major subject, you should seriously consider including more than the required minimum of four third-level MAT modules in your curriculum. We further recommend that you include all the ﬁrst- and second-level modules in at least one other subject. Note that your curriculum must contain a total of EIGHT third-level module (see Sc5(1) in Part 7 of the Calendar). NB Should you require any further information or advice regarding the composition of your curriculum, please write to the Registrar or, if possible, discuss the matter in person with the staﬀ of the Department of Student Admissions and Registrations (Tel. 0861 670 411), or one of the Provincial Centres or Regional Oﬃces. Compulsory modules for a major subject combination: NB At least two further second-level MAT modules will be required, depending on your choice at third level. It is strongly recommended that you ﬁrst pass the second-level modules before attempting the corresponding third-level module. First level: MAT111, 103 and TWO of (MAT101, 102 prior to 2002), 112, 113 Second level: MAT211, 213 Third level: FOUR of the following: (a) MAT301 (b) MAT302 (c) MAT305, 215 (d) MAT306, 216 (e) MAT307, 212 (f) MAT311, 215 (g) APM301, MAT217 REQUIREMENTS FOR THE BSc DEGREE and A pass in Mathematics (not Mathematical Literacy) with a rating of 5 or higher (NSC) or at least 50% (D-symbol) in Mathematics HIGHER GRADE or 80% (A-symbol) on STANDARD GRADE at Matriculation level prior to 2008, or equivalent. Students who do not meet the requirements may register for MAT111. 4 Syllabus NB All modules in this subject are oﬀered as YEAR MODULES except MAT103N, MAT111N, MAT112P and MAT113Q, which are oﬀered as SEMESTER modules. ACCESS MODULE MAT011K Access to Mathematics (year module) Advice: This module may NOT be taken towards degree studies – see Sc1(1)(c) in Part 7 of the Calendar. Purpose: to enable students to demonstrate the understanding of the real number system, ratio, proportion, percentage, integral exponents, scientiﬁc notation and estimation, roots, units, algebraic expressions, sequences, linear and quadratic equations and inequalities, systems of equations in two unknowns, exponents, logarithms, functions, straight lines, parabolas, hyperbolas, circles, introduction to elementary statistics, basic geometry (angles, triangles, quadrilaterals) and calculation of areas and volumes. FIRST-LEVEL MODULES (NQF LEVEL 5) MAT110M Precalculus mathematics A (year module)* Prerequisite: Mathematics as in Sc1(1)(b)(i)–(iv) in Part 7 of the Calendar. Students who obtained at least 75% in MAT011 or at least 50% in Mathematics HIGHER GRADE, 80% in Mathematics STANDARD GRADE at matriculation level will not be required to take MAT110 on ﬁrst level. 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. 13 MAT111N Precalculus mathematics B (S1 and S2)* Prerequisite: Mathematics as in Sc1(1)(b)(i)–(iv) in Part 7 of the Calendar. Purpose: to acquire practical experience of vectors, polar co-ordinates, the complex number system, theory of polynomials, systems of linear equations, sequences, mathematical induction, binomial theorem, conic sections. MAT103N Linear algebra (S1 and S2)* Prerequisite: ONE of the following: (a) Mathematics HIGHER GRADE passed with at least 50% (D symbol) or 80% (A symbol) on STANDARD GRADE at Matriculation level (b) Mathematics at Matriculation level passed prior to diﬀerentiation (c) An equivalent examination in Mathematics – see Sc1(1)(b)(iv) in Part 7 of the Calendar. (d) MAT110 (e) MAT011 passed with at least 75% Purpose: to obtain knowledge of systems of linear equations, Gaussian elimination and homogeneous systems; matrix algebra, partitioning of matrices, matrix inverses and elementary matrices; determinants, Laplace expansion and cofactor matrices; vector geometry, orthogonality, distance and the dot product, planes and the cross product, and least squares polynomial ﬁtting. MAT112P Calculus A (S1 and S2)* Prerequisite: ONE of the following: (a) Mathematics HIGHER GRADE passed with at least 50% (D symbol) or 80% (A symbol) on STANDARD GRADE at Matriculation level (b) Mathematics at Matriculation level passed prior to diﬀerentiation (c) An equivalent examination in Mathematics see Sc1(1)(b)(iv) in Part 7 of the Calendar (d) MAT110 (e) MAT011 passed with at least 75% 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 MAT113. MAT113Q Calculus B (S1 and S2)* Prerequisite: MAT112P Purpose: to enable students to continue to obtain basic skills in diﬀerentiation and integration, and build on the knowledge provided by module MAT112. More advanced techniques and further basic applications are covered. Together, the modules MAT112 and MAT113 constitute a ﬁrst course in Calculus which is essential for students taking Mathematics as a major subject. SECOND-LEVEL MODULES (NQF LEVEL 6) MAT211R Linear algebra* Prerequisite: MAT103 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, least squares polynomial ﬁtting, linear transformations, change of basis, invariant subspaces and direct sums, block triangular form. MAT212S Introduction to discrete mathematics* Prerequisite: Any ONE of COS101, MAT101, 102, 103, 112, 113 Advice: For parts of this module the following modules contain useful background: MAT103, COS101. 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 (eg partially ordered sets, lattices, Boolean algebras), the principle of induction. MAT213T Real analysis* NB This module is purely theoretical and is intended mainly for students who wish to take Mathematics as a major subject. Prerequisite: Any TWO of MAT101, 102, 112, 113 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. MAT215V Calculus in higher dimensions Prerequisite: MAT111 (or 103) and any TWO of MAT101, 102, 112, 113 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. THIRD-LEVEL MODULES (NQF LEVEL 7) Prerequisite: Any TWO MAT or APM modules on second level MAT301S Linear algebra* Advice: Linear algebra, as dealt with in MAT211 is assumed as known in this module. Purpose: to acquire a basic knowledge concerning inner product spaces, invariant subspaces, cyclic subspaces, operators and their canonical forms. MAT302T Algebra* Advice: Aspects of linear algebra, as dealt with in MAT211, are used in this module. 14 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. MAT305W Complex analysis* Prerequisite: MAT213 or MAT215 Advice: Real analysis and Calculus in Higher dimensions, as dealt with in MAT213 and MAT215 respectively are assumed as known in this module. 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. MAT306X Ordinary differential equations* Advice: Aspects of linear algebra, as dealt with in MAT211, are used in this module. The content of module MAT216 (or APM211) is assumed as known. 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. MAT307Y Discrete mathematics: Combinatorics* Advice: For parts of this module the following modules contain useful background: MAT212, COS201. 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. MAT311U Real analysis Advice: Thorough knowledge of the content of MAT213 and MAT215 is essential for this module. 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. 5 Practical Work and Admission Requirements Practical work in MAT219 mainly comprises the writing of computer programs. The programs have to be developed on suitable computers using prescribed computer packages. Access to a suitable computer is an admission requirement for all modules with a practical component. Students can gain access as follows: (i) by purchasing a computer for their own use; or (ii) by using a computer belonging to a study group, friend, computer bureau, or employer; or (iii) by reserving time on a computer at one of Unisa’s microcomputer laboratories in Pretoria, Polokwane, Cape Town, and Durban. The minimum conﬁguration of a ‘suitable’ computer is described as follows: An IBM or IBM-compatible personal computer which is year 2000-compliant and has the following minimum conﬁguration: Pentium 75 MHz processor or faster VGA or higher graphics Windows 95 or later version Hard disk 1 Gigabyte or bigger 16 MB RAM (32 MB or higher recommended by some software) A CD-ROM drive A 3.5 inch high-density (1.44 MB) diskette drive A printer that can print both text and graphics (minimum A4 paper size) A compiler for any suitable programming language is an additional requirement for APM213 and APM311. NB Unisa CANNOT supply any of the commercial software packages mentioned. Students are required to either obtain their own copy of the software, or make use of the microcomputer laboratories. Full particulars of the microcomputer laboratories are supplied in a tutorial letter sent to students upon registration. 15