PHY1016 Mathematics II

1999-2000

Code: PHY1016
Title: Mathematics II
Instructors: Dr M.E. Portnoi and Dr T.W. Preist
HE credits: 20
ECTS credits: 10
Availability: unrestricted
Level: 1
Prerequisites: Mathematics I (PHY1015) or Grade B in A-level Mathematics
Corequisites: none
Background Assumed: none
Duration: Semester I
Directed Study: 44 hours of lectures and work sessions
Private Study: 156 hours
Supports Programme Aims: 1, 5, 7 and 8
Supports Programme Objectives: none

Assessment Methods

Weekly graded homework, two 45 minute Continuous Assessment tests in weeks 5 and 9, and a three-hour examination at the end of the semester. The weighting for the final assessment is: coursework 30%, tests 30%, examination 40%.

Rationale

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 by either first or second year students depending upon the extent of their 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 courses, e.g. PHY2206 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:

• make series expansions of simple functions and determine their assymptotic 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 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.

Teaching and Learning Methods

Each week there will be three lectures and one problems class in which homework will be reviewed. Students will attempt simple exercises during the lectures.

Transferable Skills

Facility with mathematicaly formed problems and their solution. 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. Students are free to discuss homework problems with each other. Hence they have the opportunity to work cooperatively and exploit each other as a learning resource.

Assignments

Ten weekly graded homeworks.

Module Text

Spiegel M.R., Advanced Mathematics, Schaum Outline Series, McGraw-Hill (UL: 510 SPI/)

Supplementary Reading

Not applicable

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, series of sines and cosines.
   3. Linear equations and matrices: matrix multiplication,eduction of an array to reduced echelon form by elementary row operations, applications to the solution of systems of homogeneous and inhomogeneous linear equations, finding inverse matrices, and evaluating numerical determinants.
2. Differential calculus
   1. Cartesian, plane-polar, cylindrical and spherical polar coordinate systems.
   2. Partial and total derivatives.
   3. 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 parameterisation; integration of the Dirac delta function.
3. Vectors and Vector Calculus
   1. Grad, div, curl, product rules, gradient as slope.
   2. Elementary cases of Stokes' theorem and the divergence theorem.
4. Ordinary differential equations
   1. First-order-separable and integrating-factor types.
   2. Linear second-order equations with constant coefficients; damped harmonic motion.
5. 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.

Feedback to Students

Students are able to monitor their own learning by attempting the exercise sheets distributed in the lectures. Solutions are made available in the Physics library after the lecture. Weekly 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.

Feedback from Students

Feedback from students on the module is gathered via the standard student representation mechanisms.