In second year, we offer you the opportunity to expand your knowledge and skill in the 'core' areas of mathematics and statistics and, at the same time, to begin to specialise towards Applied Mathematics, Pure Mathematics, or Statistics.

Some of the courses you take are determined by your choice of Majors/Minors (in the Science degree program 3970 or combined degree programs) or a study plan (in Advanced Science). Other courses are 'free electives' which you can choose for yourself.

The information given here is to help you choose your electives, starting with a list of courses together with links to a fuller description of each course. Many of the courses are available at Higher level as well as Ordinary, so we offer advice on whether to take Higher courses. Finally, there is a guide to the differences between Applied, Pure and Statistics and how this may affect your choice of courses.

### What do we offer?

#### Course List for Mathematics and Statistics Majors etc.

The following is a list of the courses we offer Mathematics and Statistics majors at second year. Each item in the list is linked to a course description which tells you what the course is about and why it is worth studying. The last digit of the course number tells you how many units of credit it is worth: last digit 0 means 3 units of credit and last digit 1 means 6 units of credit.

**NOTE:** MATH2260 is not taught every year.

- MATH2011 Several Variable Calculus
- MATH2111 Higher Several Variable Calculus
- MATH2121 Mathematical Methods for Differential Equations
- MATH2221 Higher Mathematical Methods for Differential Equations
- MATH2241 Introduction to Atmosphere and Ocean Dynamics
- MATH2281 Biomathematics
- MATH2301 Mathematical Computing
- MATH2400 Finite Mathematics
- MATH2501 Linear Algebra
- MATH2521 Complex Analysis
- MATH2601 Higher Linear Algebra
- MATH2621 Higher Complex Analysis
- MATH2801 Theory of Statistics
- MATH2901 Higher Theory of Statistics
- MATH2831 Linear Models
- MATH2931 Higher Linear Models
- MATH2871 Data Management for Statistical Analysis
- MATH2881 Quantitative Risk

### Higher level or not?

If you did well in your first year of mathematics (70 or better) and enjoyed it, you should consider taking at least some of your second (and later) year courses at the Higher level.

These courses are more demanding, but more rewarding. You will begin to understand *why* Mathematics and Statistics works, rather than merely how. This gives you a more solid foundation for using mathematics and statistics in other areas, as well as for further studies in Mathematics or Statistics. It also makes it easier for you to continue to an honours degree in Mathematics or Statistics, if you decide that you want to.

### Applied, Pure or Statistics?

Students at UNSW can study subjects from applied mathematics, pure mathematics, and statistics. You may already know which of these areas you want to specialise in; if not, don't worry, and keep your options open. Some more information is given below.

#### Applied Mathematics

Applied Mathematics is concerned with the use of mathematics to help solve problems in the real world. This means that applied mathematics is involved in innumerable human activities.

Three important areas of applied mathematics are *modelling, design and analysis***.**

In *mathematical modelling* we try to incorporate the essential features of a complex system (such as ocean currents, the national economy, or the human nervous system) into a series of manageable equations. A good model will show how the system works, will predict the future, and will suggest further aspects for study. In the course MATH2241 Introduction to Atmosphere and Ocean Dynamics you can learn about modelling in many contexts, including ecology and economics.

In *design* you might consider questions such as "What is the best way to design a robot, or the automatic control system of an aircraft?". This leads into control theory, which you can study as part of the third year course MATH3161 Optimization.

Analysis in applied mathematics includes analysis of how best to achieve accuracy in numerical calculations, or to use a computer to solve differential equations and other types of equations, or to use a supercomputer to evaluate the gigantic integrals that arise in atomic physics. The first such course is MATH2301 Mathematical Computing.

Applied Mathematics is concerned with all of these things and with many more.

#### Pure Mathematics

Pure Mathematics is mainly concerned with understanding and developing the concepts and principles underlying mathematics, rather than with the mechanics of calculating specific numerical answers or finding exact formulae for the solution of various problems. While an Applied Mathematician might be looking for the exact value of the sum of a series or for a rapidly converging approximation to the solution of a differential equation, the Pure Mathematician is more likely to be concerned with whether the series does really converge or whether the differential equation does have a (unique) solution and whether that solution behaves in a particular way.

It is amazing how often the same mathematical principles are found to underlie and unify topics which seem at first to have very little in common. For example, the same mathematical techniques (Fourier analysis, integral geometry and group theory) underlie both medical imaging and mineral exploration, while the idea of chaos links heart attacks with turbulence in fluids and the stability or otherwise of complex ecosystems. Oversimplifying, you could say that pure mathematicians develop the underlying ideas, applied mathematicians apply them to specific problems (taking into account features which a special to each problem) and statisticians try to quantify what happens when random variations are introduced into the picture.

Three "core" second year courses in Pure Mathematics show you the basic principles which support enormous areas of mathematics and its applications. These are MATH2011 Several Variable Calculus, MATH2501 Linear Algebra and MATH2520 Complex Analysis (or their Higher counterparts). These courses are vital for anyone who wants to go further in mathematics and statistics.

There are two other Pure courses which are valuable to all mathematicians but particularly relevant if your interests are at all related to computer science. The first is MATH1081 Discrete Mathematics, which you may have already done in first year. This provides the background in logic and abstract modes of thought which is very necessary in (theoretical) computer science. The other one is MATH2400 Finite Mathematics, which shows how to develop codes based on the arithmetic of integers and finite fields.

#### Statistics

The fundamental aim of Statistics is to draw conclusions from sets of data. This leads to questions about how experiments should be designed, the number of observations to be taken and the manner in which they are to be taken. These, and other questions relating to collecting and analysing of data, have led to a vast array of statistical theories and models whose validity is confirmed through probability theory.

The core second year course in Statistics is MATH2801 Theory of Statistics (or its Higher counterpart MATH2901) which introduces you to the mathematical foundations of probability and some fundamental statistical ideas. Most majors and study plans in mathematics require you to do this course. If you are interested in Statistics, you should also take the course MATH2831 Linear Models (or MATH2931) which deals with how inferences can be made under the assumption that variables are linearly related.

There is also MATH2871 Data Management for Statistical Analysis (developed in collaboration with SAS), which deals with the querying and manipulation of large datasets needed before their statistical analysis.