|
|
|
Maths Papers
The Department offers 34 Mathematics papers at 100, 200 and 300-level, as well as a selection of 400-level papers. This section includes short descriptions of these papers, together with links to individual illustrated paper pages which feature information about the subject, the paper and assessment procedures.
|
|
 See a flowchart of available papers, their prerequisites and semesters.
Click the paper name below for complete details. Jump to 200 level, 300 level, 400 level
|
|
100 level
Math101 Supplementary Algebra 1 9 points First Semester, Second Semester
This 9-point paper is the algebra half of MATH 160.
Math102 Supplementary Calculus 1 9 points First Semester, Second Semester
This 9-point paper is the calculus half of MATH 160.
Math103 Supplementary Algebra 2 9 points First Semester, Second Semester
This 9-point paper is the algebra half of MATH 170.
Math104 Supplementary Calculus 2 9 points First Semester, Second Semester
This 9-point paper is the calculus half of MATH 170.
Math151 General Mathematics 18 points First Semester, Summer School
This is both a service paper teaching the mathematical methods that are needed in other subjects and also a remedial paper to develop skills to the point where students can move on to MATH 160. It involves calculus and algebra techniques, and covers such topics as linear and quadratic models, linear programming, functional notation, basic differentiation and integration, exponentials and logarithms. MATH 151 is recommended for students with sufficient achievement in NCEA Level 2 Mathematics but insufficient achievement at Level 3 (or equivalent). Students with a reasonable achievement in either of the NCEA Level 3 Mathematics subjects will not normally be accepted for this paper: please check with your course adviser.
Math160 Mathematics 1 18 points First Semester, Second Semester, Summer School
This paper consists of half algebra and half calculus, and is the main entry point to 100-level mathematics. The paper provides the basis for progression to MATH 170 and then to 200-level mathematics, as well as an adequate background to support other subjects.
The algebra half studies vectors and their many uses, matrices and systems of linear equations, complex numbers and polynomials. The calculus half develops differentiation and integration techniques from scratch and provides many applications that use areas, rates of change, simple differential equations and partial derivatives.
Note: This paper is recommended for students with sufficient achievement in NCEA Level 3 Mathematics with Calculus or a strong performance in NCEA Level 3 Statistics with Modelling; those with a weaker background should consider taking MATH 151 first. Advanced placement into MATH 170 may be allowed for suitably qualified students.
Math170 Mathematics 2 18 points First Semester, Second Semester
This paper, half algebra and half calculus, builds on the material introduced in MATH 160. In the algebra half, both vectors and matrices are further investigated extending into linear transformations, rounded off by a section considering numbers, factorization, induction and counting techniques. The calculus part concentrates on integration techniques and applications, with sections on differential equations and special functions. The paper provides the basis for progression to 200-level mathematics as well as a good mathematical background to support other subjects.
Suitably qualified candidates, typically those who have achieved mainly Excellences and Merits in NCEA Level 3 Mathematics with Calculus, may be allowed advanced placement directly into MATH 170. The paper provides the basis for progression to 200-level mathematics as well as a good mathematical background to support other subjects.
|
|
200 level
Math201 Real Analysis 18 points First Semester
MATH 201 is an introduction to the basic techniques of real analysis in the familiar context of single-variable calculus. This paper is compulsory for the Mathematics major.
Math202 Linear Algebra 18 points Second Semester
This is a new paper which replaces MATH 242 (Matrix Algebra). MATH 202 is an introduction to the fundamental ideas and techniques of linear algebra, and the application of these ideas to computer science, the sciences and engineering. This paper is compulsory for the Mathematics major.
Math203 Calculus of Several Variables 18 points First Semester
This paper is an introduction to the mathematics of curves, surfaces and volumes in three-dimensional space, and extends the notions of differentiation and integration to higher dimensions. It replaces MATH 251 Calculus, and is a prerequisite for three level-300 MATH papers.
Math262 Ordinary Differential Equations 18 points Second Semester
Mathematical techniques useful for solving problems arising in the physical, health and life sciences, and commerce. Topics include the analytical solution of ordinary differential equations, Laplace transforms, systems of linear ordinary differential equations, and nonlinear dynamical systems.
Math272 Discrete Mathematics 18 points First Semester
Graph theory and algorithms; combinatorial counting techniques; sets, relations, modular arithmetic and applications to cryptography. There is an emphasis on both proof techniques and practical algorithms.
|
|
300 level
Math341 Linear Algebra 18 points First Semester
This is a course in linear algebra and its applications. Topics include the theory of vector spaces and inner product spaces, linear transformations (the structure-preserving maps between vector spaces), diagonalisation of transformations (some linear transformations, just like some matrices, are diagonalisable), the spectral theorem for normal operators, and the singular value decomposition (a very important application used in engineering, computer science and the sciences).
Math342 Modern Algebra 18 points Second Semester
This paper introduces the modern algebraic concepts of a group (a set with a standard operation, usually called multiplication), a ring (a set with two operations, usually called addition and multiplication), and polynomial rings and field extensions. These concepts occur throughout modern mathematics and this paper looks at their properties and some applications.
Math351 Vector Calculus 18 points Second Semester
Vector calculus provides a fundamental tool for describing spatial variations, and lies at the heart of modelling in engineering and physics, and multivariate systems in all disciplines. Topics considered include space curves and their properties, the theory of vector fields, triple integrals and surface integrals.
Math353 Analysis 18 points Second Semester
The elements of real analysis, including sequences and series, limits, continuity, differentiation and integration, followed by an introduction to metric and normed spaces.
Math361 Numerical Analysis 18 points First Semester
Numerical solution of problems including systems of non-linear equations, interpolation, integration, differentiation and differential equations. Analysis of the convergence and stability of the methods with a study of their errors.
Math362 Mathematical Methods 2 18 points First Semester
An assortment of mathematical techniques of value in solving problems arising in the physical, health and life sciences, and commerce. Topics include, e.g., boundary value problems, initial value problems, ordinary and partial differential equations.
Math372 Applications of Mathematics 18 points Not available from 2013
This paper consists of three modules on different topics, all internally assessed.
|
|
400 level
Not all papers shown here will be available in any one year. Additional papers may be offered. Check with the Director of Studies for a confirmed list of papers that are running, their prerequisites and semester.
Math401 Graph Theory 18 points First Semester
This modern, developing subject has many applications, especially in computer science. The topics offered will depend to some extent on the interests of the students, but will cover such areas as: Colouring problems (vertex and edge problems; the Four-Colour Theorem; snarks and where to find them); Matchings (covering a graph with independent edges; Hall’s Marriage Theorem; Tutte’s matching theorem); Symmetry (vertices and edges of a graph fall into equivalence classes according to the automorphisms of the graph, studying these offers simplification of many proofs).
Math402 Groups, Rings & Fields (Galois Theory) 18 points Second Semester
Galois Theory is a showpiece of algebra bringing together ideas from groups, rings, fields and vector spaces. It is concerned with the solutions of polynomial equations and how these relate to the coefficients. For example, there is a well known formula for solving a quadratic in terms of its coefficients (and a less well known one for a cubic or quartic). However by considering a group associated with permutations of the roots, Galois showed that it was not possible to find such a formula for a general quintic. In order to tackle some Galois theory we need to learn more about groups, rings (in particular quotient groups and a quotient ring) and fields (in particular those that lie between the rationals and the complex numbers), which are of interest in their own right. As well as the historical aspects there are many contemporary uses of such algebra.
Math403 Hilbert Spaces 18 points First Semester
This course extends the techniques of linear algebra and real analysis to study problems of an intrinsically infinite-dimensional nature. A Hilbert space is a vector space with an inner product that allows length and angles to be measured; the space is required to be complete (in the sense that Cauchy sequences have limits) so that the techniques of analysis can be applied. Hilbert spaces arise frequently in mathematics, physics, and engineering, often as infinite-dimensional function spaces. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis (with applications to signal processing and heat transfer) and many areas of pure mathematics including topology, operator algebra and even number theory.
Math404 Rings and Modules 18 points First Semester
Vector spaces have an important generalization: instead of using numbers as scalars, we allow the scalars to be the elements of any ring R. This gives us an R-module. Linear (in)dependence and spanning subsets can be defined for R-modules, but whereas every vector space has a basis, every R-module has one if and only if R is a division ring. This illustrates how modules are used to distinguish classes of rings. The course looks at various ways of constructing modules over different rings, and at how different types of modules, in particular free, projective and injective modules, can be used to characterize several classes of rings.
Math405 Foundations of Probability Theory 18 points First Semester
The course gives a rigorous introduction to the mathematical foundations of probability theory. Topics include probability measures, integration theory, random variables, independence, Gaussian random variables, characteristic functions, various convergence notions of random variables and convergence theorems such as the Law of Large Numbers and the Central Limit Theorem.
Math406 Complex Analysis 18 points Second Semester
Differentiation and integration are extended to functions of a complex variable. Topics include analyticity, conformality, complex integration, harmonic and subharmonic functions.
Math407 General Relativity 18 points First Semester
The course begins with a brief overview of things to come. Then we will discuss the principle of equivalence in some detail because it is the foundation on which general relativity is built. Next we go into mathematical necessities like tensor calculus and apply it to Lorentzian geometry and Minkowski space-time. This provides a geometric foundation for special relativity. The main part of the course is usually taken up by an introduction to differential geometric concepts like parallel transport and curvature. After this the course is all downhill to derive the field equations of GR and study particular classes of solutions; in particular, the Schwarzschild solution with its black hole will be described.
Math408 Numerical Analysis 18 points Not available in 2012
This paper builds on the numerical techniques and basic theory introduced in MATH 361. Numerical analysis is both a user of and a stimulator for many branches of mathematics and this is reflected in the fact that the paper draws on many areas of algebra and calculus to support the methods discussed. Topics are chosen from Gaussian quadrature, the calculus of difference operators, difference schemes for solving parabolic and elliptic partial differential equations, iterative methods for solving linear systems, optimization of functions of several variables, and approximation theory.
Math409 Lie Groups, Analytical Mechanics and Field Theory 18 points First Semester
This paper is concerned firstly with continuous groups, especially those important in physics, of which the rotation group is the most familiar. It then relates tensors to the spinors that are central to quantum mechanics, e.g. in Dirac’s equation. It gives an introduction to Lagrangian and Hamiltonian mechanics, pointing out the links with quantum mechanics, and ends with some Lagrangian field theory.
Math410 Fluids and Oceanography 18 points Second Semester
Fluid mechanics underpins many facets of our lives, from aircraft flight to atmospheric winds. This course develops the equations of fluid motion and applies them to real-world examples. The focus is on incompressible flows of air and water at subsonic speeds. Topics include potential flow, aerodynamic lift and drag, vortices, viscous flows, boundary layers and ocean surface waves. Developing problem-solving skills and a grasp of the physical concepts along with the mathematical techniques are important parts of this course.
Math411 Operator Algebras 18 points Second Semester
The object of this course is to introduce students to the basic theory of C*-algebras which is a research strength in the Department.
Math412 Advanced Topics in General Relativity 18 points Second Semester
This paper builds upon the previous course on General Relativity (MATH 407), which is a prerequisite. It consists of a selection of advanced topics in General Relativity. The main themes are the structure of spherical bodies (stars), cosmological models, gravitational waves and black holes.
Math480 Honours Project 36 points Full year
A 36-point project in an agreed topic, supervised by one or more staff.
Math485 Honours Project 36 points Full year
A 36-point project in an agreed topic, supervised by one or more staff.
Math495 MSc Preparation 18 points Full year
MSc Preparation.
|
|  |
