## 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 Semesters One and Two | is not offered in Summer School 2020**

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.

Coursebook available from the Student Resource Centre.

### 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 110 or at least 13 credits in Mathematics at NCEA Level 3 including the Differentiation Standard 91578, or D in CIE A2 Mathematics or C in CIE AS Mathematics or 3 out of 7 in IB Mathematics, or equivalent.

**Restrictions**: ENGGEN 150, ENGSCI 111, MATHS 120, 130, 150, 153, 208, 250.

Coursebook available from the University Book Shop.

### MATHS 110

Mathematics for Science

**Offered in Semester One
**

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: **MATHS 102 or 108 or at least 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: **ENGEN 150, ENGSCI 111, MATHS 150, 153, 208, 250 ENGGEN 150, ENGSCI 111. More than 15 points from MATHS 120 and 130

### MATHS 120

Algebra

**Offered in Semesters One and Two
**

MATHS 120, alongside MATHS 130 and 162, forms a foundation for further study in mathematics at the University of Auckland. It is essential for students intending to major in Mathematics, Applied Mathematics, Statistics, Physics, or for anyone who wants a strong mathematical component to their degree.

MATHS 120 is an introduction to

**linear algebra;**more generally, along with Maths 130 it is an introduction to

**mathematical thinking and problem-solving**. Students who successfully complete this course will be able to understand and write logical mathematical arguments, and will be comfortable reading and using mathematical language and notation. In particular, they will be confident with algebra involving complex numbers, able to solve systems of linear equations of several variables, capable of a wide variety of vector and matrix operations, and will understand these operations from both an algebraic and geometric perspective.

**Recommended preparation**: It is strongly recommended that NCEA students have a merit or excellence in the Differentiation Standard 91578 and the Integration Standard 91579 at NCEA Level 3.

**Prerequisites**: MATHS 208, or B- or higher in MATHS 108, or A- or higher in MATHS 110, or A+ or higher in MATHS 102, 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**: Nil

### MATHS 130

Calculus

**Offered in Semesters One and Two**

Maths 130, alongside Maths 120 and Maths 162, forms a foundation for further study in mathematics at the University of Auckland. It is essential for students intending to major in Mathematics, Applied Mathematics, Statistics, Physics, or for anyone who wants a strong mathematical component to their degree.

Maths 130 is a rigorous course in **single-variable calculus;** more generally, along with Maths 120 it is an introduction to **mathematical thinking and problem-solving**. Students who successfully complete this course will be able to understand and write logical mathematical arguments, and will be comfortable reading and using mathematical language and notation. In particular, they will be capable of using the language of sets and functions to describe mathematical objects, know how to calculate limits rigorously, understand how to differentiate and integrate functions and the theorems behind techniques for doing so, and will be able to use these operations to optimize and study the behavior of a wide variety of objects.

**Recommended preparation**: It is strongly recommended that NCEA students have a merit or excellence in the Differentiation Standard 91578 and the Integration Standard 91579 at NCEA Level 3.

**Prerequisites**: MATHS 208, or B- or higher in MATHS 108, or A- or higher in MATHS 110, or A+ or higher in MATHS 102, 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**: Nil

### 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**: ENGGEN 150 or ENGSCI 111 or MATHS 108 or 120 or 150, or 153

**Restriction: **MATHS 199

### 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.

### MATHS 199

Advancing in Mathematics

**Offered in Semester One**

Offered in Semester One

This course is part of the Young Scholars Programme and is designed for high-achieving students currently at high school. The numerical computing environment MATLAB is used to study beautiful mathematics from algebra, analysis, applied mathematics and combinatorics. Students will learn to write mathematical proofs and create mathematical models to find solutions to real-world problems.

Enrolment requires permission from Department. For details, please see the MAX webpage.

**Prerequisite:** Departmental approval

### 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. We advise you to check the University’s general education website.

## Stage II

All students intending to major in Mathematics need to take MATHS 250, alongside at least two other courses from MATHS 253, 254, 260, 270.

### 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, 150, 153, ENGSCI 111, ENGGEN 150, or MATHS 120 and MATHS 130, or B- or higher in MATHS 110.

**Restrictions**: Cannot be taken, concurrently with, or after MATHS 250, 253.

Coursebook available from the University Book Shop.

### MATHS 250

Algebra and Calculus 2

**Offered in Semesters One and Two**

Designed for all students who plan to progress further in mathematics, this course follows directly from MATHS 120 and MATHS 130. Covering topics from multivariable calculus and linear algebra, which have many applications in science, engineering and commerce. Students will learn mathematical results and procedures as well as the underpinning ideas and mathematical proofs.

**Prerequisites**: MATHS 120 and 130, or 15 points from ENGGEN 150, ENGSCI 111, MATHS 150, 153, or a B+ in MATHS 208.

Coursebook available from the Student Resource Centre.

### MATHS 253

Algebra and Calculus 3

**Offered in Semesters One and Two**

A sequel to MATHS 250, further developing and bringing together linear algebra and calculus. Students will learn about quadratic forms, projections, spectral decomposition, methods of multicriteria optimisation, double, triple and line integrals, Green’s theorem and applications.

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

**Prerequisites**: MATHS 250

**Restrictions**: Nil

### MATHS 254

Fundamental Concepts of Mathematics

**Offered in Semesters One and Two**

Fundamentals of mathematics important to many branches of the subject and its applications. Topics include equivalence relations, elementary number theory, counting techniques, elementary probability, geometry, symmetry and metric spaces. This is an essential course for all students advancing beyond Stage II in pure mathematics, and highly suitable for other students in the mathematical sciences.

**Corequisite:** MATHS 250

**Restriction:** MATHS 255

### 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.

### 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**: MATHS 120 and 130, or 15 points from ENGGEN 150, ENGSCI 111, MATHS 108, 110, 150, 153 and 15 points from COMPSCI 101, 105, 130, INFOSYS 110, 120, MATHS 162, 199.

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

### 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 COMPSCI 101, 107, 120, MATHS 120, 150, 153, PHIL 101.

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

Coursebook available from the Student Resource Centre.

## 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**: B+ or higher in COMPSCI 255 or MATHS 254 or 255 or PHIL 222.

### 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 250 and MATHS 254.

### 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 254 or 255, or COMPSCI 225 and a B+ in MATHS 208, or COMPSCI 225 and 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 250, and MATHS 254 or 255, or a B+ or higher in COMPSCI 225 and 15 points from 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 250 and MATHS 254 or 255 or an A or higher in MATHS 253 and MATHS 260

### MATHS 333

Analysis in Higher Dimensions

**Offered in Semester One**

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. 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. Recommended preparation: MATHS 253.

**Prerequisite**: MATHS 250.

**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 350

Topology

**Offered in Semester Two**

Aspects of point-set, set-theoretic and algebraic topology including: properties and construction of topological spaces, continuous functions, axioms of separation, countability, connectivity and compactness, metrisation, covering spaces, the fundamental group and homology theory. Recommended preparation: MATHS 333.

**Recommended preparation: **MATHS 333

**Prerequisite:** MATHS 332 and approval from the undergraduate advisors (email ugadvice@math.auckland.ac.nz).

**Restriction:** MATHS 750

### 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. Covers: linear PDEs and analytical methods for their solution, weak solutions. Recommended preparation: MATHS 253.

**Prerequisites**: MATHS 250 and 260

**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**

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. Recommended preparation: MATHS 253, 361.

**Prerequisites**: MATHS 250 and 260

**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.)

### 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 399

Capstone: Mathematics

**Offered in Semester Two**

An exploration of the role of mathematics in society and culture, and the activities performed by mathematicians as teachers, critics, and innovators. Students will develop their skills in communication, critical thinking, teaching, and creative problem solving.

**Prerequisite:** MATHS 250 and 30 points at Stage III in Mathematics

### 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.