# Intermediate Mathematics and Statistics Handbook

## Units of Study

Each unit of study has a web page, accessed by following the links on the main Intermediate Mathematics and Statistics web page.

In this chapter, Mathematics units are listed, by semester, in numerical order; then Statistics units are listed, by semester, in numerical order.

Units are designated Normal or Advanced. Entry to an Advanced level unit normally requires a Credit or better in a Normal level prerequisite, or a Pass or better in an Advanced level prerequisite.

Mathematics units are also labelled Applied, or Pure, or both. Although there is no clear distinction between applied mathematics and pure mathematics at the intermediate level, this labelling gives a rough guide as to which senior level units the intermediate level units are most closely allied with.

The unit code for an intermediate unit of study in the School consists of MATH or STAT followed by four digits: for example MATH2063 or STAT2011. The first digit is 1 for junior level units, 2 for intermediate level units, 3 for senior level units. The second digit indicates whether the unit is normal (0 or 1) or advanced (9). In most cases two units which share the same last two digits are mutually exclusive: for example, MATH2061 may not be counted with MATH2961. The one exception is that MATH2068 and MATH2968 are not mutually exclusive. Instead MATH2068 and MATH2988 are mutually exclusive.

Text and reference books are yet to be advised. Except for The Little Blue Book it is suggested that you do not purchase any books until recommendations are made by lecturers.

The Little Blue Book is a compact reference book: it contains definitions, formulas and important results from Junior Mathematics which are used in Intermediate Units. It is recommended that all students have access to this book: it is available from the Coop Bookshop.

## MATH2061 Linear Mathematics and Vector Calculus

(6 credit points, Normal, Pure and Applied)

Prerequisites: One of MATH1011 or MATH1001 or MATH1901 or MATH1906, and one of MATH1014 or MATH1002 or MATH1902, and one of MATH1003 or MATH1903 or MATH1907.

Prohibitions: May not be counted with MATH2067 or MATH2961

Classes: 3 lectures, 1 tutorial and 1 practice class per week.

Assessment: One 2 hr exam, assignments, quizzes.

This unit starts with an investigation of linearity: linear functions, general principles relating to the solution sets of homogeneous and inhomogeneous linear equations (including differential equations), linear independence and the dimension of a linear space. The study of eigenvalues and eigenvectors, begun in junior level linear algebra, is extended and developed. Linear operators on two dimensional real space are investigated, paying particular attention to the geometrical significance of eigenvalues and eigenvectors.

The unit then moves on to topics from vector calculus, including vector-valued functions (parametrised curves and surfaces; vector fields; div, grad and curl; gradient fields and potential functions), line integrals (arc length; work; pathindependent integrals and conservative fields; flux across a curve), iterated integrals (double and triple integrals; polar, cylindrical and spherical coordinates; areas, volumes and mass; Green's Theorem), flux integrals (flow through a surface; flux integrals through a surface defined by a function of two variables, though cylinders, spheres and parametrised surfaces), Gauss' Divergence Theorem and Stokes' Theorem.

## MATH2063/2963 Mathematical Computing and Nonlinear Systems

Prerequisites (MATH2063): One of MATH1011 or MATH1001 or MATH1901 or MATH1906, and one of MATH1014 or MATH1002 or MATH1902, and one of MATH1003 or MATH1903 or MATH1907.

Prerequisites (MATH2963): One of MATH1901 or MATH1906 or Credit in MATH1001, and one of MATH1902 or Credit in MATH1002, and one of MATH1903 or MATH1907 or Credit in MATH1003.

Prohibitions: MATH2063 and MATH2963 may not both be counted.

Classes: 3 lectures, 1 tutorial and 1 computer lab per week.

Assessment: One 2 hr exam, assignments, quizzes.

This unit will introduce students to techniques of mathematical computation as applied to nonlinear systems, using the numerical programming language MATLAB and, where appropriate, computer algebra. This knowledge will be applied to a number of modelling problems, particularly those involving nonlinear mappings and nonlinear ordinary differential equations (ODEs). Throughout the unit of study the essential nonlinear theory will be developed, and the resulting ideas will be explored computationally. This will allow us to explore the modern concepts of chaos using a variety of examples, including the logistic map, the Henon map and the Lorenz equations. No prior knowledge of programming or of the MATLAB language or computer algebra is required.

## MATH2067 Differential Equations and Vector Calculus

(6 credit points, Normal Applied, Engineering Faculty only, available only to students whose degree program requires it)

Prerequisites: One of MATH1011 or MATH1001 or MATH1901 or MATH1906, and one of MATH1002 or MATH1902, and one of MATH1003 or MATH1903 or MATH1907.

Prohibitions: May not be counted with MATH2061 or MATH2961.

Classes: 3 lectures, 1 tutorial and 1 practice class per week.

Assessment: One 2 hr exam, assignments, quizzes.

The unit starts by introducing students to solution techniques of ordinary and partial differential equations (ODEs and PDEs) relevant to the engineering disciplines: it provides a basic grounding in these techniques to enable students to build on the concepts in their subsequent engineering classes. The main topics are Fourier series, second order ODEs, including inhomogeneous equations and Laplace transforms, and second order PDEs in rectangular domains (solution by separation of variables).

The unit moves on to topics from vector calculus, including vectorvalued functions (parametrised curves and surfaces; vector fields; div, grad and curl; gradient fields and potential functions), line integrals (arc length; work; pathindependent integrals and conservative fields; flux across a curve), iterated integrals (double and triple integrals; polar, cylindrical and spherical coordinates; areas, volumes and mass; Green's Theorem), flux integrals (flow through a surface; flux integrals through a surface defined by a function of two variables, though cylinders, spheres and parametrised surfaces), Gauss' Divergence Theorem and Stokes' Theorem.

## MATH2069/2969 Discrete Mathematics and Graph Theory

Prerequisites (MATH2069): 6 credit points of Junior Mathematics.

Prerequisites (MATH2969): 9 credit points of Junior Mathematics at the advanced level or at the normal level with credit.

Prohibitions: MATH2069 and MATH2969 may not both be counted.

Classes: 3 lectures, 1 tutorial and 1 practice class per week.

Assessment: One 2 hr exam, assignments, quizzes.

We introduce students to several related areas of discrete mathematics, which serve their interests for further study in pure and applied mathematics, computer science and engineering. Topics to be covered in the first part of the unit include recursion and induction, generating functions and recurrences, combinatorics, asymptotics and analysis of algorithms. Topics covered in the second part of the unit include Eulerian and Hamiltonian graphs, the theory of trees (used in the study of data structures), planar graphs, the study of chromatic polynomials (important in scheduling problems), maximal flows in networks, matching theory.

## MATH2961 Linear Mathematics and Vector Calculus

(6 credit points, Advanced, Pure and Applied)

Prerequisites: One of MATH1901 or MATH1906 or Credit in MATH1001, and one of MATH1902 or Credit in MATH1002, and one of MATH1903 or MATH1907 or Credit in MATH1003.

Prohibitions: May not both be counted with MATH2061 or MATH2067.

Classes: 4 lectures and 1 tutorial per week.

Assessment: One 2 hr exam, assignments, quizzes.

This unit is an advanced version of MATH2061, with more emphasis on the underlying concepts and on mathematical rigour. Topics from linear algebra focus on the theory of vector spaces and linear transformations. The connection between matrices and linear transformations is studied in detail. Determinants, introduced in first year, are revised and investigated further, as are eigenvalues and eigenvectors. The calculus component of the unit includes local maxima and minima, Lagrange multipliers, the inverse function theorem and Jacobians. There is an informal treatment of multiple integrals: double integrals, change of variables, triple integrals, line and surface integrals, Green's theorem and Stokes' theorem.

## MATH2962 Real and Complex Analysis

Prerequisites: One of MATH1901 or MATH1906 or Credit in MATH1001, and one of MATH1902 or Credit in MATH1002, and one of MATH1903 or MATH1907 or Credit in MATH1003.

Classes: 3 lectures, 1 tutorial and 1 practice class per week.

Assessment: One 2 hr exam, assignments, quizzes.

Analysis is one of the fundamental topics underlying much of mathematics including differential equations, dynamical systems, differential geometry, topology and Fourier analysis. Starting off with an axiomatic description of the real number system, this first course in analysis concentrates on the limiting behaviour of infinite sequences and series on the real line and the complex plane. These concepts are then applied to sequences and series of functions, looking at pointwise and uniform convergence. Particular attention is given to power series leading into the theory of analytic functions and complex analysis. Topics in complex analysis include elementary functions on the complex plane, the Cauchy integral theorem, Cauchy integral formula, residues and related topics with applications to real integrals.

## MATH2916 Working Seminar A (SSP)

(3 credit points, Advanced, Pure and Applied)

Prerequisites: By invitation, High Distinction average over 12 credit points of Advanced Junior Mathematics.

Classes: 1 one-hour seminar per week.

Assessment: A one-hour presentation and a 15 to 20 page essay.

The main aim of this unit is to develop the students' written and oral presentation skills. The material will consist of a series of connected topics relevant to modern mathematics and statistics. The topics are chosen to suit the students' background and interests, and are not covered by other mathematics or statistics units. The first session will be an introduction on the principles of written and oral presentation of mathematics. Under the supervision and advice of the lecturer(s) in charge, the students present the topics to the other students and the lecturer in a seminar series and a written essay in a manner that reflects the practice of research in mathematics and statistics.

## MATH2065 Introduction to Partial Differential Equations

(6 credit points, Normal, Applied)

Prerequisites: One of MATH1011 or MATH1001 or MATH1901 or MATH1906, and one of MATH1014 or MATH1002 or MATH1902, and one of MATH1003 or MATH1903 or MATH1907.

Classes: 3 lectures, 1 tutorial and 1 practice class per week.

Assessment: One 2 hr exam (70%), mid-semester test (20%), assignments (10%).

This is an introductory course in the analytical solutions of partial differential equations and boundary value problems. The techniques covered include separation of variables, Fourier series, Fourier transforms and Laplace transforms.

## MATH2068/2988 Number Theory and Cryptography

(6 credit points, Normal/Avanced, Pure)

Prerequisites (MATH2068): 6 credit points of Junior Mathematics.

Prerequisites (MATH2988): 9 credit points of Junior Mathematics at the advanced level or at the normal level with credit.

Classes: 3 lectures, 1 tutorial and 1 computer lab per week.

Assessment: One 2 hr exam (75%), tutorial and lab participation (5%), assignments (20%).

Cryptography is the branch of mathematics that provides the techniques for confidential exchange of information sent via possibly insecure channels. This unit introduces the tools from elementary number theory that are needed to understand the mathematics underlying the most commonly used modern public key cryptosystems. Topics include the Euclidean Algorithm, Fermat's Little Theorem, the Chinese Remainder Theorem, Möbius Inversion, the RSA Cryptosystem, the Elgamal Cryptosystem and the Diffie-Hellman Protocol. Issues of computational complexity are also discussed.

## MATH2070/2970 Optimisation and Financial Mathematics

(6 credit points, Normal/Avanced, Applied)

Prerequisites (MATH2070): MATH1011 or MATH1001 or MATH1901 or MATH1906, and MATH1014 or MATH1002 or MATH1902.

Prerequisites (MATH2970): MATH1901 or MATH1906 or credit in MATH1001, and MATH1902 or credit MATH1002.

Assumed knowledge: MATH1003 for MATH2070, MATH1903 (or MATH1907 or Credit in MATH1003) for MATH2970.

Classes: 3 lectures, 1 tutorial and 1 computer lab per week.

Assessment: One 2 hour exam (70%), assignments (10%), quizzes (10%), project (10%).

Problems in industry and commerce often involve maximising profits or minimising costs subject to constraints arising from resource limitations. The first part of this unit looks at the important class of linear programming problems and their solution using the simplex algorithm, and the minimisation of functions of several variables with constraints, including Lagrange multipliers, Kuhn-Tucker theory and quadratic programming.

## MATH2965 Introduction to Partial Differential Equations

Prerequisite: MATH2961 or Credit in MATH2061.

Classes: 3 lectures, 1 tutorial and 1 practice class per week.

Assessment: One 2 hr exam (80%), assignments (20%).

The course begins with a brief introduction to ordinary differential equations including variation of parameters and series solution techniques.

The major part of the course deals with partial differential equations and boundary value problems using Fourier Series, separation of variables, Laplace and Fourier transforms and other orthogonal expansion procedures. Applications include the heat equation, Laplace’s equation and the wave equation.

The course concludes with a brief introduction to the solution of first order PDEs using the method of characteristics and deals with applications such as traffic flow.

## MATH2968 Algebra

Prerequisites: 9 credit points of advanced level Junior Mathematics (or normal level with credit), including MATH1902 or MATH1002.

Classes: 3 lectures, 1 tutorial and 1 practice class per week.

Assessment: One 2 hour exam (60%), quizzes (30%), assignments (10%).

This unit provides an introduction to modern abstract algebra, via linear algebra and group theory. It extends the linear algebra covered in Junior Mathematics and MATH2961, and proceeds to a classification of linear operators on finite dimensional spaces. Permutation groups are used to introduce and motivate the study of abstract group theory. Topics covered include actions of groups on sets, subgroups, homomorphisms, quotient groups and the classification of finite abelian groups.

## MATH2917 Working Seminar B (SSP)

(3 credit points, Advanced, Pure and Applied)

Prerequisites: By invitation, High Distinction average over 12 credit points of Advanced Junior Mathematics.

Classes: 1 one-hour seminar per week.

Assessment: A one-hour presentation and a 15 to 20 page essay.

The main aim of this unit is to develop the students' written and oral presentation skills. The material will consist of a series of connected topics relevant to modern mathematics and statistics. The topics are chosen to suit the students' background and interests, and are not covered by other mathematics or statistics units. The first session will be an introduction on the principles of written and oral presentation of mathematics. Under the supervision and advice of the lecturer(s) in charge, the students present the topics to the other students and the lecturer in a seminar series and a written essay in a manner that reflects the practice of research in mathematics and statistics.

## STAT2011 Statistical Models

(6 credit points, Normal)

Prerequisites: MATH1001 or MATH1011 or MATH1901 or MATH1906, and MATH1005 or MATH1015 or STAT1021 or MATH1905 or ECMT1010.

Prohibition: May not be counted with STAT2911.

Classes: 3 lectures, 1 tutorial and 1 computer lab per week.

Assessment: One 2 hour exam, assignments, quizzes, computer practical reports, and a one-hour computer practical class assessment task.

This unit provides an introduction to univariate techniques in data analysis and the most common statistical distributions that are used to model patterns of variability. Common discrete random variable models, like the binomial, Poisson and geometric, and continuous models, including the normal and exponential, will be studied. The method of moments and maximum likelihood techniques for fitting statistical distributions to data will be explored. The unit will have weekly computer classes where candidates will learn to use a statistical computing package to perform simulations and carry out computer intensive estimation techniques like the bootstrap method.

## STAT2911 Probability and Statistical Models

Prerequisites: MATH1903 or MATH1907 or credit in MATH1003, and and MATH1905 or credit in MATH1005 or MATH1015 or ECMT1010.

Prohibition: May not be counted with STAT2011.

Classes: 3 lectures, 1 tutorial and 1 computer lab per week.

Assessment: One 2 hour exam, assignments, quizzes, computer practical reports, and a one-hour computer practical class assessment task.

This unit is essentially an advanced version of STAT2011 with an emphasis on the mathematical techniques used to manipulate random variables and probability models. Common random variables including the Poisson, normal, beta and gamma families are introduced. Probability generating functions and convolution methods are used to understand the behaviour of sums of random variables. The method of moments and maximum likelihood techniques for fitting statistical distributions to data will be explored. The unit will have weekly computer classes where students will learn to use a statistical computing package to perform simulations and carry out computer intensive estimation techniques like the bootstrap method.

## STAT2012 Statistical Tests

(6 credit points, Normal)

Prerequisites: MATH1005 or MATH1015 or MATH1905 or ECMT1010.

Prohibition: May not be counted with STAT2912.

Classes: 3 lectures, 1 tutorial and 1 computer lab per week.

Assessment: One 2 hour exam (65%), assignments (10%), quizzes (5%), computer practical reports (10%), and a one-hour computer practical class assessment task (10%).

The unit provides an introduction to the standard methods of statistical analysis of data: Tests of hypotheses and confidence intervals, including t-tests, analysis of variance, regression least squares and robust methods, power of tests, nonparametric tests, nonparametric smoothing, tests for count data goodness of fit, contingency tables. Graphical methods and diagnostics are used throughout with all analyses discussed in the context of computation with real data using an interactive statistical package.