2007 Course descriptions
From MathsDept
Aims to promote an understanding of arithmetic. Topics include number work and estimation; ratio, proportion and percentage; the metric system. Algebra is introduced. Emphasis is on numeracy and skills. Students become confident users of simple technology and develop their problem solving abilities.
Aims to extend algebra skills and problem solving capability. Topics include elementary geometry, and trigonometry in relation to right-angled triangles; a study of linear graphs and co-ordinate geometry; a study of functions, in particular quadratic and exponential functions. Applications to science and commerce are considered.
Aims to promote an understanding of number work and estimation with emphasis on the structure of number systems; ratio, proportion and percentage in relation to growth functions; and the metric system. Algebra is introduced with an emphasis on algebraic manipulation skills.
Aims to extend algebra skills and problem solving capability. Topics include trigonometry, both in relation to right-angled triangles and in terms of trigonometric functions; linear graphs and co- ordinate geometry; quadratic, cubic and exponential functions. There is an introduction to differential calculus with applications to optimisation problems.
Aims to build confidence in using mathematics while demonstrating the role mathematics plays in understanding and guiding human activity. The course is taught thematically and students experience how fundamental mathematical ideas occur in modelling diverse features of our society such as the environment (eg, air pollution) and medicine (eg, burns, drug dosages). For students who have no formal mathematical background For students who have not studied Mathematics at NCEA Level 3 (or equivalent). This course may not be taken with or after any other Mathematics course at Stage I or above
MATHS 108 and 150. Syllabus includes: mathematical modelling; graphs and calculus of polynomial, trigonometric, exponential and logarithmic functions; graphs of rational and piecewise-defined functions; trigonometry. For students who have achieved fewer than 12 credits in Calculus or Statistics at NCEA Level 3, or who have achieved at least 18 credits in Mathematics at NCEA Level 2 (or equivalent). This course may not be taken with or after any other Mathematics course at Stage I or above except MATHS 101.
Selected topics in algebra and calculus and their applications including: sets, real numbers, integers; linear algebra including matrices, linear functions, linear equations; functions, equations and inequalities; limits and continuity; differential calculus of one and several variables; integral calculus of one variable.
The standard first year Mathematics course for students with a good mathematics background who are considering a major or a minor in Mathematics, or a major in any science, economics or finance. Functions and their inverses. Limits and derivatives of functions of 1 and 2 variables. Integration of functions of a single variable, differential equations. Vectors, lines and planes. Systems of linear equations. Dot and cross product, matrix algebra and determinants.
A version of MATHS 150 for Year 13 students
An introduction to mathematical modelling and scientific computing, and to a selection of mathematical techniques in the context of applications. Students will learn how to formulate mathematical models and how to solve them using numerical and other methods. This is a core course for students who wish to advance in Applied Mathematics. Concurrent or prior enrolment in MATHS 108 or MATHS 150 or equivalent is recommended.
Mathematics contains many powerful and beautiful ideas that have shaped the way we understand our world. This course explores some of the grand successes of mathematical thinking. No formal mathematics background is required, just curiosity about topics such as infinity, paradoxes, cryptography, knots and fractals.
This is a mainly practical course in which selected students learn tutoring skills that are put to use in MATHS 102 tutorials. In a small interactive class, students learn to mark, to question strategically and to facilitate learning. The theory and issues of mathematics education as a research field is also introduced.
A sequel to the course MATHS 108 covering: further matrix and vector algebra (solution of linear systems, least squares, eigenvalues, vector spaces), calculus of series and Taylor approximation, multivariable calculus and optimization, differential equations and difference equations, and the use of symbolic computing with applications.
Note: Students who have passed MATHS 108 with a B+ or better and who intend to advance in Mathematics are strongly advised to take MATHS 250 as an advancing course or MATHS 260 as a complementary Mathematics course if no Mathematics major is intended, rather than MATHS 208.
Vector spaces and subspaces, linear transformations, linear independence, bases, coordinates. Eigenvalues and eigenvectors, Markov processes. L’Hopital’s rule, improper integrals, integration using partial fractions and trig substitutions. Sequences, series, convergence tests. Taylor and Maclaurin expansions. Functions of two variables. Partial derivatives and tangent planes, optimisation.
Inner product spaces and applications. Orthogonal diagonalization and quadratic forms. Differential calculus for functions of several variables. Multiple integrals. Vector valued functions and space curves. Vector calculus. Green’s theorem. Series.
An introduction to logic and proof; sets, relations and operations on sets; natural numbers, congruencies; algebra of polynomials; complex numbers; examples of groups. Fundamental concepts of calculus in a rigorous setting: real numbers, sequences and convergence, continuity, uniform continuity, theorems on derivatives, Taylor’s theorem.
The study of differential equations is central to mathematical modelling of systems that change. This course develops methods for understanding the behaviour of solutions to ordinary differential equations. Qualitative and elementary numerical methods for obtaining information about solutions are discussed, as well as some analytical techniques for finding exact solutions in certain cases. Some applications of differential equations to scientific modelling are discussed. This is a core course for applied mathematics.
An introduction to algorithms that are used to solve frequently-occurring problems in computation. The problems covered include linear and nonlinear systems of equations, interpolation, quadrature and ordinary differential equations. The use of a high level programming language in scientific computing is also taught. Prerequisite: MATHS 108 or 150 or equivalent, and a computing course such as COMPSCI 101 or MATHS 162 or equivalent Restriction: MATHS 267
A broad-based study of mathematics education in New Zealand which includes: social-political, gender, curriculum, assessment, technology and psychological issues in mathematics teaching and learning.
A study of some of the topics occurring in the history of mathematics which facilitate the understanding of modern mathematics. These include: concepts of number, geometry, algebra, and the differential and integral calculus.
Logic addresses the foundations of mathematical reasoning. It models the process of mathematical proof by providing a setting and the rules of deduction. Builds a basic understanding of first order predicate logic, introduces model theory and demonstrates how models of a first order system relate to mathematical structures. The course is recommended for anyone studying high level computer science or mathematical logic.
This is a framework for a unified treatment of many different mathematical structures. It concentrates on the fundamental notions of groups, rings and fields. The abstract descriptions are accompanied by numerous concrete examples. Applications abound: symmetries, geometry, coding theory, cryptography and many more. This course is recommended for those planning graduate study in pure mathematics.
Combinatorics is a branch of mathematics that studies collections of objects that satisfy specified criteria. An important part of combinatorics is graph theory, which is now connected to other disciplines including bioinformatics, electrical engineering, molecular chemistry and social science. The use of combinatorics in solving counting and construction problems is covered using topics that include algorithmic graph theory, codes and incidence structures, and combinatorial complexity.
The goal of this course is to show the power of algebra and number theory in the real world. It concentrates on concrete objects like polynomial rings, finite fields, groups of points on elliptic curves, studies their elementary properties and shows their exceptional applicability to various problems in information technology including cryptography, secret sharing, and reliable transmission of information through an unreliable channel.
A standard course for every student intending to advance in pure mathematics. It develops the foundational mathematics underlying calculus, it introduces a rigorous approach to continuous mathematics and fosters an understanding of the special thinking and arguments involved in this area. The main focus is analysis in one real variable with the topics including real fields, limits and continuity, Riemann integration and power series.
By selecting the important properties of distance many different mathematical contexts are studied simultaneously in the framework of metric and normed spaces. Examines carefully the ways in which the derivative generalises to higher dimensional situations. These concepts lead to precise studies of continuity, fixed points and the solution of differential equations. A recommended course for all students planning to advance in pure mathematics.
Calculus plays a fundamental role in mathematics, answering deep theoretical problems and allowing us to solve very practical problems. Extends the ideas of calculus to two and higher dimensions, showing how to calculate integrals and derivatives in higher dimensions and exploring special relationships between integrals of different dimensions. It also extends calculus to complex variables.
A selection of topics providing an introduction to a range of concepts in geometry and general topology, with emphasis on visualizable aspects of these subjects. Topics include some or all of the following: axiom systems, affine geometry, Euclidean and non- Euclidean geometry, projective geometry, symmetry, convexity, the geometric topology of manifolds, and algebraic structures associated with topological spaces.
This is an introductory course in partial differential equations (PDEs), covering Fourier series, Fourier integrals, boundary value problems and separation of variables, with application to the solution of second order PDEs in one, two and three dimensions.
Further techniques used in modern applied mathematics, including vector calculus, complex variables, the calculus of variations, and Green’s functions for ODEs. The emphasis throughout the course is on application of these techniques.
Numerical methods and mathematical modelling. Topics will include numerical linear algebra and differential equations, bifurcations in ordinary differential equations, traffic flow and nonlinear waves. Matlab will be used extensively.
