446 MINING 626 ZOOLOGY
660 R.A.A.F. ACADEMY
618 MATHEMATICS
100 LEVEL
The Mathematics department offers a range of courses at the 100 level.
Before deciding which courses to take, students should check the pre- requisites for any 200 level courses they may have in mind. If in doubt.
consult the Mathematics department. The courses may be summarized as follows.
131 Analysis and 132 Algebra and Geometry are the basic course units in Mathematics. Although they are listed separately, students are ex- pected to take both. They are intended for students who may wish to study Mathematics as one of their major subjects towards the B.Sc.
degree. These courses are essential for students who intend to take an Honours degree in Mathematics, and are strongly recommended for students who contemplate an Honours degree in Physics, Statistics or Information Science. Students taking 131 and 132 will be expected to have obtained a good result in H.S.0 Pure Mathematics.
171 Calculus and Linear Algebra is a first course for students who regard Mathematics as a minor component of their degree programme. Com- pared with 131 and 132 this course covers less material and does so less thoroughly. Students enrolling for this course should have satis- factorily completed H.S.C. Pure Mathematics or have obtained a good result in H.S.C. General Mathematics.
152 Differential Equations and Dynamical Systems is a first course in the applications of mathematics, and is a prerequisite for similar courses in later years. Students taking this course should have satisfactorily com- pleted H.S.C. Pure Mathematics; either Applied Mathematics or Physics at H.S.C. would be advantageous, but is not essential.
182 General Mathematics (Biological Sciences) is intended for stu- dents who have taken H.S.C. General Mathematics. It aims at giving an understanding of elementary mathematical principles and some basic mathematical techniques for application to the biological and medical sciences. 182 Is a terminal course; it does not qualify as a prerequisite for any 200-level Mathematics course.
183 General Mathematics (Social Sciences) is similar to 182, except that it is concerned with applications of mathematics in the social sciences.
131 ANALYSIS
2 lectures and 1 tutorial per week throughout the year; 5 points.
Conditions: Students taking 131 are expected to take 132 concurrently.
Students taking 131 may not gain credit for 171, 182 or 183.
SYLLABUS Sequences and series: convergent and divergent series, limits (e. N); positive term series. Inequalities. Functions of one real variable:
elementary functions and Inverses; graphs; polar coordinates; continuity (e, 8), differentiation, the Bolzano-Weierstrass Theorem and its applica- tions to continuous functions, first mean-value theorem. Taylor's series, applications, estimate of remainders. Integration: applications of inte- gration; areas, arc-length, solids of revolution. Functions of two real variables: surfaces, partial derivatives, directional derivatives, tangent plane; chain rule.
Additional topics may include one or more of: real numbers, complex numbers, introduction to functions of a complex variable, elementary logic, numerical integration.
BOOKS
Prescribed textbooks:
•Apostol T M Calculus Vol I, Wiley Intern! Student ed 1978
EXAMINATION One 3-hour paper. Prescribed written work may form part of the final assessment.
132 ALGEBRA AND GEOMETRY
Two lectures and one tutorial per week throughout the year. 5 points.
Conditions: Students taking 132 are expected to take 131 concurrently.
Students taking 132 may not gain credit for 171, 182 or 183.
SYLLABUS Group theory: definitions, symmetric groups, permutation groups, order, normal subgroups. Field theory: definitions, examples: real and complex numbers.
Geometry, cardinality of numbers. Vector theory: 3-dimensional vector space, subspaces, linear dependence, basis, dot and cross products, triple products, generalizations. 3-dimensional geometry, equations of lines and planes, quadric surfaces. Linear transformations, inverses, matrices, solution of equations, determinants, geometric illustrations. Eigenvalue concept, central conic, rotations.
Additional topics may include one or more of: geometric topology, linear programming, non-Euclidean geometry, graph theory, number theory.
107
Books
Prescribed textbooks:
• Dinkines F Abstract Mathematical Systems, Appleton Century 1964
•Thomas G B Calculus and Analytic Geometry, 4th ed., Addison- Wesley 1968
EXAMINATION One 3-hour paper. Prescribed written work may form part of the final assessment.
152 DIFFERENTIAL EQUATIONS AND DYNAMICAL SYSTEMS Three lectures and one tutorial per week in first and second terms. 5 points.
Condition: Students taking this course must also take either 131 or 171.
SYLLABUS First-order differential equations: growth and decay models, methods of solution. Second-order equations with constant coefficients.
Simple difference equations. kinematics. Particle dynamics, projectiles, many particle systems. Harmonic oscillator with damping and forcing.
Orbits. Variable mass. Partial differentiation. Continuum models: con- servation laws. Elementary wave motion.
BOO КЅ
Prescribed textbooks:
• Smith R C & Smith P Mechanics, Wiley 1971
EXAMINATION One 3-hour paper. Prescribed written work may form part of the final assessment.
171 CALCULUS AND LINEAR ALGEBRA
3 lectures and 1 tutorial class per week throughout the year, 8 points.
Condition: Students taking 171 may not gain credit for 131, 132, 182 or 183.
SYLLABUS Mathematical induction. Complex numbers, complex exponen- tial function. Introduction to group theory. Cyclic and permutation groups. Subgroups, cosets, Lagrange's Theorem. Isomorphism, groups of small order. Vectors: vector algbera, applications to lines and planes.
Determinants. Matrix algebra, eigenvalues and eigenvectors and their applications to conics and quadric surfaces. Representatation of linear transformations by matrices. Calculus of functions of one real variable:
differentiation, integration and applications. Trigonometric functions and their inverses. Exponential and logarithmic functions. Hyperbolic func- tions and their inverses. Differential equations: simple first order; second order with constant coefficients. Functions of two real variables, partial derivatives, chain rule, double integrals.
BOO КЅ
Prescribed textbook:
• Thomas G B Calculus and Analytic Geometry, Addison-Wesley 1968 EXAMINATION Two 3-hour papers. Prescribed written work may form part of the final assessment.
182 GENERAL MATHEMATICS (BIOLOGICAL SCIENCES) 3 lectures and 1 tutorial class per week throughout the year; 8 points.
Conditions: Students taking 182 may not gain credit for 131, 132, 152, 171 or 183.
Note that 182 is a terminal course; it should not be taken by students who contemplate continuing with Mathematics in their second year.
SYLLABUS Vectors, matrices, determinants. Finite series. Introduction to probability. Graph sketching and curve fitting.
Differentiation and Integration of standard elementary functions. Dif- ferential and difference equations. Functions of two variables, partial dif- ferentiation. Applications will Include population regulation, logarithmic growth, competing species, genetics of population ecologies, regulation and control biology.
BOOKS
Prescribed textbook:
•Batschelet E Introduction to Mathematics for Life Scientists, Springer 1973
EXAMINATION Two 3-hour papers. Prescribed written work may form part of the assessment.
183 GENERAL MATHEMATICS (SOCIAL SCIENCES)
3 lectures and 1 tutorial class per week throughout the year; 8 points.
Conditions: Students taking 183 may not gain credit for 131, 132, 152, 171 or 182.
Note that 183 is a terminal course; it should not be taken by students who contemplate continuing with Mathematics in their second year.
SYLLABUS Vectors, matrices, determinants. Elements of linear pro- gramming and game theory. Introduction to probability. Graph sketching and curve fitting. Differentiation and integration of standard elementary functions. Differential and difference equations. Functions of two vari- ables, partial differentiation. Applications will include economic and sociological models.
BOOKS
Prescribed textbook:
Williams D R Modern Mathematics for Business Decision-making, Wadsworth
EXAMINATION Two 3-hour papers. Prescribed written work may form part of the assessment.
200 LEVEL
Mathematics at 200 level comprises a variety of courses, which are organized in units each consisting of 24 lectures together with practice classes. Some of these course-units are offered at two different levels, called ordinary and advanced; while some units are offered at one level only. Advanced level and undesignated units are worth 4 points each, while ordinary level units are worth 3 points each. Students are free to 109
take any combination of units subject only to the prerequisites and conditions prescribed. A student intending to major in Mathematics should normally take a minimum of 6 units of 200-level Mathematics.
Students who are not certain which units to take should consult the Mathematics department.
Note: (i) Several of the advanced courses have 131 and/or 132 as pre- requisite; (ii) Where courses are offered at both advanced and ordinary levels, students contemplating an Honours degree in Mathematics, Mathematical Statistics or Physics should take most, if not all, at advanced level.
(iii) The department may waive the formal prerequisite for a particular unit in the case of a student of sufficient merit who undertakes a pres- cribed course of reading, possibly followed by an informal examination, to bring him to the required standard.
In addition to the above, there are three projects (227, 228 and 229), on which no lectures are given, and which should be done mainly in the summer vacation prior to the student's second year. It is recommended that students proposing to major in Mathematics should take one of these projects.
227 NUMERICAL PROJECT
Vacation work; 2 points. Assessment by written work.
Intending students should obtain the instructions and exercises from the Mathematics department in November or December, and should hand in their work complete not later than 31 March.
Condition: Students taking this unit may not gain credit for 622-202.
SYLLABUS Numerical exercises on summation of series, difference tables.
interpolation, integration, differential equations, curve fitting, simultaneous linear equations and determinants. Calculating machines will be available, for use in the Mathematics department, on request.
Book
Prescribed textbook:
Wilkes M V A Short Introduction to Numerical Analysis, CUP 1971
228 NUMBER THEORY PROJECT
Vacation work; 2 points. Assessment by written work.
Intending students should obtain the instructions and exercises from the Mathematics department in November or December, and should hand in their work complete not later than 31 March.
SYLLABUS Exercises on prime numbers, factorization, congruences, quadratic residues, continued fractions, Diophantine approximation, quadratic forms.
Book
Prescribed textbook:
Davenport H The Higher Arithmetic, Hutchinson 1970