# AMA Level 2 Modules

**AMA2001 - Classical Mechanics (1**^{st}semester)

*Pre-requisites:* MTH1001 and MTH1002

*Lecturer:* Dr C Ramsbottom

**Introduction**

Mechanics has always been a source of inspiration for mathematics and mathematicians. Calculus, as well as the famous Newton's laws, were “invented” by Newton largely because he wanted to solve a single but very important mechanical problem, the problem of planetary motion. Mechanics was at the beginning of such branches of mathematics as theory of functions, calculus of variations, differential equations and more recently, theory of chaos. Mechanics and its mathematical methods are important in optics, electromagnetic theory, statistical mechanics, quantum mechanics, theory of relativity, quantum field theory and many ‘non-physical’ applications such as theory of optimisation and control.

For students, besides giving a comprehensive picture of mechanical phenomena and teaching how to solve a wide variety of problems, Classical Mechanics offers a unique opportunity to see the mathematical methods they have learned at work and to practice their mathematical skills. Vectors, partial derivatives, single and multiple integrals, differential equations, stationary points, complex numbers, as well as matrices and determinants are all among the tools used in the course.

**Contents**

*Introduction:*Basic revision of Newtonian mechanics: conservation of mechanical energy and angular momentum. Revision of key concepts in linear algebra (matrices): eigenvalues and eigenvectors. Revision of basic multivariable calculus (partial differentiation and multiple integration).*Motion of a single particle in a central potential:*Attractive forces: planetary motion and transfer orbits. Repulsive forces: Rutherford scattering.*Rotating frames of reference:*Angular velocity, effect of the Earth's rotation; apparent gravity, cyclones and anticyclones, Foucault's pendulum, Larmor precession.*Conservation laws for a system of particles:*internal and external forces, conservation of energy, centre of mass as the origin.*Rigid body motion:*Moments and products of inertia, parallel axis theorem, Euler's equations of motion, Euler angles and rotation matrices.*Lagrange's equations:*Constraints, equations for holonomic constraints; examples.*Lagrange's equations for impulses:*Applications to systems of rods.*Small oscillations:*Normal modes of oscillation.

**Assessment**

Exam 60%, Class Test 20%, Project 15% and Assignments 5%.

**MTH2002 Introduction to Partial Differential Equations (1**^{st}semester)

*Pre-requisite:* MTH1001 and MTH1002

*Lecturer:* Dr A Brown

**Introduction**

Partial differential equations are differential equations whose solution is a function of more than one variable. Many real phenomena in the sciences, technology and finance are modelled with partial differential equations. For instance, if you switch on a radiator in a cold room, the temperature varies depending on where in the room you stand, and how long the heat has been on. Similarly, the prices of commodities vary depending on how far you are from the source, and the time of year.

In this module we will address two common partial differential equations- the diffusion equation and the wave equation. In order to assemble all of the mathematical tools that go in to solving these equations, we will first have to address ideas of orthogonality, basis functions, ordinary differential equations and Fourier analysis. With all of these pieces in place we will then be able to solve the diffusion and wave equations using the method of separation of variables.

Also in this module you will use computer programming to solve partial differential equations numerically. This will allow you to explore and analyse the solutions of partial differential equations and gain a greater understanding both of the mathematical models, and the systems they describe.

**Contents**

*Revision**functions of a single variable**even and odd functions**limits**differentiation**integration**Taylor series expansion**functions of two variables**partial differentiation**total differentials,**Taylor series expansion for two variables and stationary points**Ordinary differential equations*

- Orthogonality
- Basis expansions
- Fourier series
*The method of separation of variables**The diffusion equation**The wave equation*

**Assessment**

Class test 50%

Team project 50%

**AMA2004 - Numerical Analysis (2**^{nd}semester)

*Pre-requisite:* MTH1001 and MTH1002

*Lecturer:* Dr M Gruening

**Introduction**

Numerical Analysis is concerned with devising methods for finding approximate, numerical solutions to mathematically expressed problems. The methods are analysed for their accuracy, efficiency and robustness. As a simple example, consider solving the equation *f*(*x*) = 0, where *f*(*x*) is a specified function. If *f*(*x*) is of even moderate complexity, we will not be able to solve this equation analytically. In Numerical Analysis we develop different procedures, or algorithms, to solve this problem. Finding the most suitable requires an appreciation of the methods. For example, some will guarantee convergence to a solution, but may require much effort, while other methods may converge quickly with less effort, but may also diverge, depending on the function and initial guess. Faced with such differing behaviour of the methods, what is the ‘best’ strategy to adopt? In AMA2004 we cover the basic introductory material of Numerical Analysis. We investigate the solution of equations, interpolation, function approximation, differentiation, integration and the solution of ordinary differential equations.

An essential element of this module are the practical classes, in which algorithms developed during the lectures are implemented using Python. The practical work consists of several numerical projects, in which a Python program is written to implement a particular algorithm and to investigate the behaviour of a method.

**Contents**

*Introduction and basic properties of errors:*Introduction; Review of basic calculus; Taylor's theorem and truncation error; Storage of non-integers; Round-off error; Machine accuracy; Absolute and relative errors; Richardson's extrapolation.*Solution of equations in one variable:*Bisection method; False-position method; Secant method; Newton-Raphson method; Fixed point and one-point iteration; Aitken's δ^{2}process; Roots of polynomials.*Solution of linear equations:*Gaussian elimination; Pivoting strategies; Calculating the inverse; LU decomposition; Norms; Condition number; Ill-conditioned linear equations; Iterative refinement; Iterative methods.*Interpolation and polynomial approximation:*Why use polynomials? Lagrangian interpolation; Neville's algorithm; Other methods.*Approximation theory:*Norms; Least-squares approximation; Linear least-squares; Orthogonal polynomials; Error term; Discrete least-squares; Generating orthogonal polynomials.*Numerical quadrature:*Newton-Cotes formulae; Composite quadrature; Romberg integration; Adaptive quadrature.*Numerical solution of ordinary differential equations:*Boundary-value problems; Finite-difference formulae for first and second derivatives; Initial-value problems; Errors; Taylor-series methods; Runge-Kutta methods.

**Assessment**

Exam 50% Portfolio (20%) Project (30%)

All assessment components are compulsory

**AMA2005 - Fluid Mechanics (2**^{nd}semester)

*Pre-requisite:* MTH1001 and MTH1002

*Lecturer:* Dr D Dundas

**Introduction**

This course provides an elementary introduction to the mechanics of non-viscous and viscous fluids. The course uses the methods of vector field theory that have been introduced at level 1. It also provides a useful link to several level-3 courses, such as AMA3001 Electromagnetic Theory and AMA3003 Tensor Field Theory.

The course starts with a revision of the main aspects of vector field theory that are needed later in the course. The main body of the course is concerned with fluid mechanics. The fundamental equations are derived. The theory is then applied to situations in which the equations simplify to the extent that analytic solutions are possible. This course does not consider computational solutions. It is hoped that, in addition to some reasonably challenging mathematics, the course contains enough simple treatments of every-day phenomena that all level-2 students will find manageable, stimulating and rewarding.

**Contents**

I. Review of Vector Field Theory

Vector algebra: scalar product; vector product; triple products. Co-ordinates: Cartesian; general orthogonal curvilinear coordinates; spherical polar; cylindrical polar. Scalar Fields: gradient; time-dependent fields. Vector fields: flux and divergence; Gauss divergence theorem; line-integrals and curl; Stokes theorem. Miscellaneous topics: ∇ notation; ∇^{2}; identities; Green's Theorem.

II. Fluid Mechanics

*1) General:* Basic definitions and properties: materials; solids, plastics and fluids; density; pressure; equation of state; body forces; surface forces; viscosity. Velocity: pathlines; streamlines; boundary conditions. Continuity equation; incompressible fluids; streamtubes. Euler's Momentum Equation.

*2) Applications:* Hydrostatics; Sound waves.

*3) Vorticity and circulation:* Definitions; Vortex lines and tubes; Vorticity equation; Kelvin's Circulation Theorem.

*4) Bernoulli's equation:* Derivation and conditions; Simple examples; Class demonstrations; Open channel flows.

*5) Two-dimensional flow of incompressible Fluid:* Stream function: properties; flux; vorticity; solid boundaries. Some model flows. Steady flow of an inviscid liquid past a cylinder.

*6) Irrotational flow in two dimensions:* Review of complex variables. Complex potential: definition; some model flows; Shifts and rotations; examples. Image theorems for walls and circles. Conformal mapping: general theory; particular mappings; the Joukowski transformation; the cambered aerofoil.

*7) Irrotational flow in three dimensions:* Method of separation of variables. Spherical polar solution; Cylindrical polar solution; Cartesian solution. Gravity surface-waves.

*8) Viscous fluids:* The stress tensor; surface forces revisited; deformations and the strain tensor; Newtonian fluids. Navier-Stokes equation: general derivation; incompressible fluids; boundary conditions. Examples. Laminar and turbulent flow; Reynolds and Froude numbers; dimensional analysis.

**Assessment**

Exam 70% Class test 10% Project 20%

**AMA2014 - Numerical Methods for Finance (1st semester)**

*Pre-requisite:* MTH1001 and MTH1002 Enrolment on B.Sc. Mathematics with Finance

*Lecturer*: Prof H van der Hart

**Contents**

An overview of essential numerical methods, including error analysis, root-finding, LU decomposition, function approximation, numerical differentiation and integration, and ordinary differential equations. Lectures are accompanied by practicals, where these techniques will be developed within MATLAB, and applied to a wide range of problems with an emphasis on applications within computational finance.

**Assessment**

Reports x3 70% Presentation 20% Attendance/Participation 10%

**MTH2010 Employability for Mathematics (0 CAT Points) (1st semester)**

Pre-Requisites: None

Lecturers: Dr D Dundas

Introduction:

This is a 0 CAT point module that is compulsory for students planning to take a placement year. The module consists of 8 lectures and 2 workshops, and is assessed 100% by attendance.

Course Content:

Introduction to placement for mathematics and physics students, CV building, international options, interview skills, assessment centres, placement approval, health and safety and wellbeing. Workshops on CV building and interview skills. The module is delivered in-house with the support of the QUB Careers Service and external experts.

Compulsory Element: Attend at least five of the six lectures and attend all two workshops.

Learning Outcomes: Identify gaps in personal employability skills. Plan a programme of work to result in a successful work placement application.

Skills Plan self-learning and improve performance, as the foundation for lifelong learning/CPD. Decide on action plans and implement them effectively. Clearly identify criteria for success and evaluate their own performance against them.

Assessment: Attend at least five of the six lectures and attend and participate in the two workshops.