PHY1116 Mathematics for Physicists

2007-2008

Code: PHY1116
Title: Mathematics for Physicists
Instructors: Dr A. Usher
CATS credits: 20
ECTS credits: 10
Availability: Physics programmes only
Level: 1
Pre-requisites: Mathematical Skills (PHY1115) or Grade B in A-level Mathematics
Co-requisites: N/A
Background Assumed: N/A
Duration: Semesters I and II
Directed Study Time: 66 hours of lectures and work sessions
Private Study Time: 55 hours
Assessment Tasks Time: 79 hours
Observation report: 2004/05 ASP (MEP)

Aims

This module aims to introduce students to some of the mathematical techniques that are most frequently used in physics, and to give students experience in their use and application. The module may be taken at either Stage 1 or Stage 2, depending upon the extent of the student's existing knowledge. The module is offered in Semester I so that second year students will have had opportunity to develop all the mathematical skills required for core physics modules that occur in the second semester of the second year. Emphasis is placed on the use of mathematical techniques rather than their rigorous proof.

Intended Learning Outcomes

Students should be able to:

Module Specific Skills

• make series expansions of simple functions and determine their asymptotic behaviour;
• perform basic arithmetic and algebra with complex numbers;
• manipulate vectors and matrices and solve systems of simultaneous linear equations;
• calculate partial and total derivatives of functions of more than one variable;
• evaluate single, double and triple integrals using commonly occuring coordinate systems;
• apply differential operators to vector functions;
• apply Stokes's and Gauss's theorems;
• solve simple first-order differential equations and second-order differential equations with constant coefficients;
• recognise the Dirac delta function and be aware of its properties;
• make a Fourier-series expansion of a simple periodic function;
• obtain the Fourier transform of a simple function;

Discipline Specific Skills

• tackle, with facility, mathematically formed problems and their solution;

Personal and Key Skills

• Time Management: students are required to work to weekly deadlines for the completion of homework and must therefore develop appropriate coping strategies. In particular, it will be necessary for them to work consistently through the week and manage their time carefully.
• Work Co-operatively: students are free to discuss homework problems with each other. Hence they have the opportunity to work co-operatively and exploit each other as a learning resource.

Learning and Teaching Methods

Each week there will be two lectures and a problems class in which homework will be reviewed. Students will also attempt simple exercises during the lectures. e-learning resources.

Assignments

Ten graded homeworks. Ungraded exercises support each lecture.

Assessment

Coursework comprising ten pieces of graded homework, two 45 minute Continuous Assessment tests in weeks 10 and 21, and a three-hour examination at the end of the year. The weighting for the final assessment is: coursework 30%, tests 30%, examination 40%.

Penalties for Late Submission of Work: Coursework submitted after the corresponding problems class will receive a mark of zero.

Syllabus Plan and Content

1. Basic Algebra and Calculus
   1. Series: Taylor and Maclaurin series, expansions of standard functions.
   2. Complex numbers: Argand diagram, modulus-argument form, de Moivre's theorem, trigonometric functions, hyperbolic functions, series of sines and cosines.
   3. Linear equations and matrices: matrix multiplication, applications to the solution of systems of homogeneous and inhomogeneous linear equations, finding inverse matrices, evaluating numerical determinants, and an introduction to eigenvalues and eigenvectors.
2. Coordinate Systems in 2- and 3-Dimensional Geometries
   1. Cartesian, plane-polar, cylindrical and spherical polar coordinate systems.
3. Differential calculus
   1. Partial and total derivatives.
   2. Multiple integrals: line and surface integrals; application of integration to arc lengths, surface areas, volumes and masses; evaluation of multiple integrals in different coordinate systems and using parametrisation; integration of the Dirac delta function.
4. Vectors and Vector Calculus
   1. Grad, div, curl, product rules, gradient as slope.
   2. Elementary cases of Stokes's theorem and the divergence theorem.
5. Ordinary differential equations
   1. First-order separable and integrating-factor types.
   2. Linear second-order equations with constant coefficients; damped harmonic motion.
6. Fourier analysis
   1. Fourier series: the concept of orthogonal functions, examples of Fourier series, Fourier series in exponential notation.
   2. Fourier transforms: derivation; examples of Fourier transforms, including exponential, 'top hat', the Dirac delta function, and the Gaussian function; the convolution integral and theorem.

Core Text

Stroud K.A. and Booth D.J. (2003), Advanced Engineering Mathematics (4^th edition), Paulgrave, ISBN 1-4039-0312-3 (UL: 510.2462 STR)

Supplementary Text(s)

Arfken G.B. and Weber H.J. (2001), Mathematical methods for physicists (5^th edition), Academic Press, ISBN 0-120-59826-4 (UL: 510 ARF)
Spiegel M.R. (1971), Advanced Mathematics for Engineers and Scientists, Schaum Outline Series, McGraw-Hill, ISBN 0-070-60216-6 (UL: 510 SPI)
Stroud K.A. (2001), Engineering Mathematics (5^th edition), Paulgrave, ISBN 0-333-91939-4 (UL: 510.2462 STR)

Formative Mechanisms

Students are able to monitor their own learning by attempting the exercise sheets distributed in the lectures. Solutions are available in the School Office Reception and on the Web. Homework assignments are graded and given back to students in the problems class at which points of difficulty may be discussed with the lecturers and postgraduate demonstrators. Continuous Assessment tests allow students to gauge their level of progress: the graded test scripts are shown to the students at a problems class.

Evaluation Mechanisms

The module will be evaluated using information gathered via the student representation mechanisms, the staff peer appraisal scheme, and measures of student attainment based on summative assessment.