## Department of Mathematics

# Undergraduate Courses

For advice about any of our courses, please email our undergraduate advisors.

## Stage I

Whatever your major, it is important that you choose the Mathematics courses that are right for you. You can find helpful information about how to choose the right Stage I Mathematics course based on your NCEA achievement standards on our Stage 1 Information page.

### MATHS 102

Functioning in Mathematics

**Offered in Summer School, Semesters One and Two**

An introduction to calculus that builds mathematical skills and develops conceptual thinking through active participation in problems that model real life. MATHS 102 makes full use of appropriate technology and prepares students for further study in Mathematics.

**Recommended preparation:** For students who have achieved fewer than 13 credits in Mathematics at NCEA Level 3, or equivalent.

**Restrictions**: MATHS 102 may not be taken concurrently with any other Mathematics course, except MATHS 190 and may not be taken after ENGSCI 111, or any Mathematics course at Stage I or above, except MATHS 190/190G.

**Course Coordinator**

Summer Semester: Garry Nathan

Semester One/Two: Julia Novak

### MATHS 108

General Mathematics 1

**Offered in Summer School, Semesters One and Two**

A general entry to Mathematics for commerce and the social sciences, following Year 13 Mathematics. MATHS 108 covers selected topics in algebra and calculus and their applications, including: linear functions, linear equations and matrices; functions, equations and inequalities; limits and continuity; differential calculus of one and two variables; integral calculus of one variable.

**Recommended preparation**: It is recommended that NCEA students complete the Differentiation Standard 91578 and/or the Simultaneous Equations Standard 91587 at NCEA Level 3.

**Prerequisites**: MATHS 102 or at least 13 credits in Mathematics at NCEA Level 3 or D in CIE A2 Mathematics or C in CIE AS Mathematics or 3 out of 7 in IB Mathematics.

**Restrictions**: MATHS 153, 208, 250, ENGGEN 150, ENGSCI 111. May not be taken with, or after, MATHS 110, 150.

**Course Coordinator**

Summer Semester: Padraic Bartlett

Semester One: Julie de Saedeleer

Semester Tow: Padraic Bartlett / Julia Novak

Coursebook available from the University Book Shop.

### MATHS 110

Mathematics for Science

**Offered in Semester One
Offered from 2017**

A general entry to Mathematics for the physical sciences, following Year 13 Mathematics. MATHS 110 covers selected topics in algebra and calculus and their application to the physical sciences.

**Recommended preparation**: It is recommended that NCEA students complete the Differential Standard 91578 and/or the Simultaneous Equations Standard 91587 at NCEA Level 3*.*

**Prerequisite: **MATH 102 or** **13 credits in Mathematics at NCEA Level 3, or D or better in Cambridge A2 Mathematics, C or better in AS Mathematics, pass in International Baccalaureate Mathematics, or equivalent.

**Restriction: **MATHS 108, 153, 208, 250, ENGGEN 150, ENGSCI 111. May not be taken with, or after, MATHS 150

**Course Coordinator**

James Sneyd

### MATHS 150

Advancing Mathematics 1

**Offered in Semesters One and Two**

The gateway to further mathematics courses for students intending to major in Mathematics, Statistics, Physics, Economics, Finance or Mathematical Biology. It gives an introduction to the use of careful mathematical language and reasoning in the context of calculus of functions of a single variable and of linear algebra in finite dimensional spaces. Recommended preparation for MATHS 250.

**Recommended preparation**: It is strongly recommended that NCEA students have a merit or excellence in the Differentiation Standard 91578 at NCEA Level 3.

**Prerequisites**: B- in MATHS 108 or 110, or A+ in MATHS 102, or any pass in MATHS 208, or at least 18 credits in Mathematics at NCEA Level 3 including at least 9 credits at merit or excellence, or B in CIE A2 Mathematics, or 5 out of 7 in IB Mathematics or equivalent.

**Restrictions**: MATHS 153, ENGGEN 150, ENGSCI 111

**Course Coordinator**

Summer One: Warren Moors

Semester Two: Gabriel Verret

Coursebook available from the Student Resource Centre.

### MATHS 162

Computational Mathematics

**Offered in Semesters One and Two**

An introduction to computational mathematics and programming in MATLAB. The course will introduce some basic concepts in computational mathematics and give applications that include cryptography, difference equations, stochastic modelling, graph theory and Markov chains.

**Corequisites**: 15 points from MATHS 108, 110, 150, 153, ENGSCI 111, ENGGEN 150

**Course Coordinator**

Summer One: Graham Donovan

Semester Two: Nicolette Rattenbury

### MATHS 190/MATHS 190G

Great Ideas Shaping Our World

**Offered in Semesters One and Two**

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, knots and fractals, and cryptography.

**Text required**: E. Burger and M. Starbird, The Heart of Mathematics (2nd edition), to be ordered directly from publisher.

**Course Coordinator**

Summer Two: Shayne Waldron

### SCIGEN 101/101G

Communicating for a Knowledge Society

**Offered in Semesters One and Two**

This general education/science course is designed for any student with an interest in learning practical ways to effectively communicate knowledge to a wide range of audiences. It differs from traditional communication courses by concentrating on how to communicate specialist knowledge - the knowledge you learn here at University in your chosen field of expertise.

Eligibility to enrol: Any student from any faculty (including Science students) may take SCIGEN 101G as their general education option. It is advisable to check out the University’s general education website - www.auckland.ac.nz/generaleducation.

For more information visit the course website: web.env.auckland.ac.nz/course_pages/scigen101_101G/

### MATHS 153

Accelerated Mathematics

**Semester One **

MAX (MATHS 153) is designed for students concurrently at high school who enjoy academic challenge and who have shown themselves to be very capable at high school mathematics. This course is part of The University of Auckland Young Scholars Programme.

The Department of Mathematics and the Department of Engineering Science at The University of Auckland jointly offer this course as *MATHS 153 - Accelerated Mathematics*. MATHS 153 is a 15 point course which is academically equivalent to the first year Engineering Mathematics course *ENGSCI 111 - Mathematical Modelling 1* or to the first year Mathematics course *MATHS 150 - Advancing Mathematics 1*. It is also an excellent alternative to *MATHS 108 - General Mathematics 1*.

Students who pass MATHS 153 will be eligible to enrol in any of *ENGSCI 211 - Mathematical Modelling 2*, *MATHS 250 - Advancing Mathematics 2* or *MATHS 208 - General Mathematics 2*, when they become full time students at The University of Auckland.

Enrolment requires permission from Department.

For details please see the MAX webpage

**Restrictions**: MATHS 108, 150, ENGGEN 150, ENGSCI 111.

**Course Coordinator**

Padraic Bartlett

## Stage II

All students intending to major in Mathematics or Applied Mathematics are encouraged to take MATHS 250, MATHS 253 and MATHS 260. Mathematics majors should also take MATHS 255 and Applied Mathematics majors must take MATHS 270.

### MATHS 202

Tutoring in Mathematics

**Offered in Semester Two**

This is a mainly practical course in which selected students learn tutoring skills that are put to use in MATHS 102 and MATHS 108 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 are also introduced.

**Prerequisites**: 30 points from courses in Mathematics and departmental consent required.

It will normally be expected that students will have passed at least 90 points at Stage I including at least 30 points in Mathematics, and that they are enrolling in at least one other Stage II Mathematics course.

**For advice and enrolment**

**Email**: maths202@auckland.ac.nz

**Course Coordinator**

Garry Nathan

### MATHS 208

General Mathematics 2

**Offered in Summer School, Semesters One and Two**

This sequel to MATHS 108 features applications from the theory of multi-variable calculus, linear algebra and differential equations to real-life problems in statistics, economics, finance, computer science, and operations research. Matlab is used to develop analytical and

numerical methods of solving problems.

**Prerequisites**: 15 points from MATHS 108, 110, 150, 153, ENGGEN 150, ENGSCI 111

**Restrictions**: MATHS 208 cannot be taken, concurrently with, or after MATHS 250, 253 or PHYSICS 211.

**Course Coordinator**

Summer Semester:

Semester One/Two: Tanya Evans

Coursebook available from the University Book Shop.

### MATHS 250

Advancing Mathematics 2

**Offered in Semesters One and Two**

This preparation for advanced courses in mathematics is intended for all students who plan to progress further in mathematics. Covers

topics from multivariable calculus and linear algebra that have many applications in science, engineering and commerce, including vector

spaces, eigenvalues, power series, least squares and improper integrals.

**Prerequisites**: 15 points from ENGGEN 150, ENGSCI 111, MATHS 150, 153, or an A in MATHS 108, 110 or a B+ in MATHS 208

Coursebook available from the Student Resource Centre.

**Course Coordinator**

Summer Two: Shayne Waldron

### MATHS 253

Advancing Mathematics 3

**Offered in Semesters One and Two**

The standard sequel to MATHS 250. It covers topics in linear algebra and multi-variable calculus including linear transformations, quadratic forms, double and triple integrals and constrained optimisation. It is a preparation for a large number of Stage III courses in mathematics and statistics, and for many advanced courses in physics and other applied sciences.

**All students intending to advance in mathematics should take this course.**

**Prerequisites**: MATHS 250 or an A+ in MATHS 208.

**Restrictions**: PHYSICS 211.

**Course Coordinator**

Summer One: Arkadii Slinko

Semester Two: Jeroen Schillewaert

### MATHS 255

Principles of Mathematics

**Offered in Semesters One and Two**

An introduction to mathematical thinking and communication: how to organise arguments logically and prove results. Rigorous notions are developed using topics that are central to the foundations of algebra and analysis including set theory, logic, abstract vector spaces and elementary number theory. An essential course for all students advancing in pure mathematics.

**Prerequisites**: MATHS 250, or an A in MATHS 208, or an A in MATHS 150, MATHS 153, ENGGEN 150 or ENGSCI 111 and a concurrent enrolment in MATHS 250 or ENGSCI 211.

**Course Coordinator
**Semester Two: Padraic Bartlett

### MATHS 260

Differential Equations

**Offered in Semesters One and Two**

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.

A core course for Applied Mathematics.

**Prerequisites**: MATHS 208 or 250 or ENGSCI 211 or a concurrent enrolment in MATHS 250.

**Text required**: Blanchard, Devaney and Hall, Differential Equations.

**Course Coordinator
**Semester Two: Gemma Mason

### MATHS 270

Numerical Computation

**Offered in Semesters One and Two**

Many mathematical models occurring in science and engineering cannot be solved exactly using algebra and calculus. In this course students are introduced to computer based methods that can be used to find approximate solutions to these problems. The methods covered in the course are powerful yet simple to use. This is a core course for students who wish to advance in Applied Mathematics.

**Prerequisites**: 15 points from MATHS 108, 110 150, 153, ENGGEN 150, ENGSCI 111, and 15 points from MATHS 162, COMPSCI 101, 105, INFOSYS 110, 120 (recommended MATHS 162)

**Text required**: Turner, Guide to Scientific Computing.

**Course Coordinator
**Semester Two: Steve Taylor

### COMPSCI 225

Discrete Structures in Mathematics and Computer Science

**Offered in Semesters One and Two**

Introduction to logic, principles of counting, mathematical induction, recursion, relations and functions, graphs and trees, and algorithms. This course is especially suited for students of computer science and others who are interested in logic and the foundations of mathematics.

**Prerequisites**: 15 points from MATHS 108, 150, 153, COMPSCI 101, PHIL 101.

**Restriction**: Cannot be taken after MATHS 255.

Coursebook available from the Student Resource Centre.

**Course Coordinator
**Semester Two: Padraic Bartlett

## Stage III

Students wishing to pursue graduate studies in Mathematics are encouraged to take MATHS 320, MATHS 332 and MATHS 361. Those majoring in Applied Mathematics must take MATHS 340, MATHS 361 and MATHS 363.

### MATHS 302

Teaching and Learning Mathematics

**Offered in Semester One**

For people interested in thinking about the social, cultural, political, economic, historical, technological and theoretical ideas that influence Mathematics Education, who want to understand the forces that shaped their own Mathematics Education, or who are interested in teaching. Students will develop their ability to communicate ideas in essay form.

**Recommended preparation**: At least 45 points from courses in Mathematics or Statistics.

### MATHS 315

Mathematical Logic

**Offered in Semester Two**

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. Recommended for anyone studying high-level computer science or mathematical logic.

**Prerequisites**: COMPSCI 225 or MATHS 255 or PHIL 222.

**Course Coordinator
**Semester Two: Sina Greenwood

### MATHS 320

Algebraic Structures

**Offered in Semester Two**

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.

**Prerequisites**: MATHS 255 or 328, or an A– pass in MATHS 253.

**Course Coordinator
**Semester Two: Jianbei An

### MATHS 326

Combinatorics

**Offered in Semester One**

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.

**Prerequisites**: MATHS 255, or COMPSCI 225 and a B+ in MATHS 208, or COMPSCI 225 and any pass in MATHS 250.

### MATHS 328

Algebra and Applications

**Offered in Semester One**

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.

**Prerequisites**: MATHS 255, or B+ pass in COMPSCI 225 and one of MATHS 208, 250, 253.

Coursebook available from the Student Resource Centre.

### MATHS 332

Real Analysis

**Offered in Semester Two
**

An essential course for every student intending to advance in 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.

**Prerequisites**: MATHS 255, or an A in both MATHS 253 and 260, or MATHS 250 and a concurrent enrolment in MATHS 255

**Course Coordinator
**Semester Two: Warren Moors

### MATHS 333

Analysis in Higher Dimensions

**This course will not be offered in 2017
This course will be offered in Semester One from 2018**

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 mathematics.

**Prerequisite**: MATHS 332

**Strongly Recommended**: MATHS 253, 255.

### MATHS 340

Real and Complex Calculus

**Offered in Semester Two**

Calculus plays a fundamental role in mathematics, answering deep theoretical problems and allowing us to solve very practical problems. This course 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.

**Prerequisite**: MATHS 253.

**Texts required**: K-T Tang, Mathematical Methods for Engineers and Scientists 1 and 2. (Both texts are available as e-books in the library.)

**Course Coordinator
**Semester Two: Vivien Kirk

### MATHS 361

Partial Differential Equations

**Offered in Semester One**

Partial differential equations (PDEs) are used to model many important applications of phenomena in the real world such as electric

fields, diffusion and wave propagation. An introduction to linear PDEs and analytical methods for their solution. The course will also cover weak solutions.

**Prerequisites**: MATHS 260 and 253, or PHYSICS 211

**Texts required:** K-T Tang, Mathematical Methods for Engineers and Scientists 1 and 2. (Both texts are available as e-books in the library.)

### MATHS 362

Methods in Applied Mathematics

**Offered in Semester Two**

The course covers a selection of techniques including the calculus of variations, asymptotic methods and models based on conservation laws. These methods are fundamental in the analysis of traffic flow, shocks, fluid flow, as well as in control theory, and the course is recommended for students intending to advance in Applied Mathematics.

**Prerequisites**: MATHS 260 and 253, or PHYSICS 211.

**Recommended preparation**: MATHS 340, 361.

**Texts required**:

Holmes, Introduction to the Foundations of Applied Mathematics.

K-T Tang, Mathematical Methods for Engineers and Scientists 3

(Both texts are available as e-books in the library.)

**Course Coordinator
**Semester Two: Steve Taylor

### MATHS 363

Advanced Modelling and Computation

**Offered in Semester One**

In real-world situations, the interesting and important variables are often not directly observable. To address this problem, mathematical models and quantities that are observable are usually employed to carry out inference on the variables of interest. This course is an introduction to fitting of models to (noisy) observational data and how to compute estimates for the interesting variables. Numerical methods for partial differential equations, which are commonly used as models for the observations, will also be covered. Matlab is used extensively.

**Prerequisite**: MATHS 260 and 270.

**Texts required:**

Holmes, Introduction to Numerical Methods in Differential Equations

Holmes, Introduction to the Foundations of Applied Mathematics.

(Both texts are avilable as e-books in the library.)

### MATHS 383

Foundations of Algebraic Geometry

**This course will be offered in Semester One in 2017, it will not be offered in 2018.**

Algebraic geometry is a branch of mathematics studying zeros of polynomials. The fundamental objects in algebraic geometry are algebraic varieties i.e., solution sets of systems of polynomial equations.

Examples of the most studied classes of algebraic varieties are (plane) algebraic curves such as lines, circles, parabolas, elliptic curves and so on. Algebraic geometry uses abstract techniques to solve geometrical problems about varieties and occupies a central place in modern mathematics with multiple connections with diverse fields inside and outside of mathematics.

**Prerequisites**: At least one of MATHS 320, 326, 328, 332, 333, or A+ in MATHS 255

### MATHS 384

Complex Analysis

**This course will be offered in Semester Two in 2017, it will not be offered in 2018.**

An introductory course to functions of one complex variable, including Cauchy’s integral formula, the index formula, Laurent series and the residue theorem. Many applications are given including a three line proof of the fundamental theorem of algebra. Complex analysis is used extensively in engineering, physics and mathematics.

**Prerequisites**: MATHS 332

**Recommended preparation**: MATHS 333

### STATS 370

Financial Mathematics

**Offered in Semester Two**

This course is suitable for Finance majors who want to learn more about the more mathematical aspects of the subject and for Mathematics or Statistics majors wanting to learn about Finance.

**Prerequisite**: 15 points in Stage 2 Mathematics and 15 points in Stage 2 Statistics or BIOSCI 209.