Coordinator: Michael Pounds

Weighting: 10 CATS

This is a composite module made of 2 components: physics problems (5 credits) and five worksheets (5 credits). Problem solving forms a vital part of the learning process and therefore each lecturer issues a set of problems for their module which you are expected to make serious attempts to solve. A subset of these problems is marked for credit. These problems are discussed in the weekly Examples Classes.

Aims:
To cover background mathematical material assumed by other modules, to give experience of learning by self-study and to develop the habit of keeping up with the problem sheets handed out in physics modules

Objectives:

By the end of the module, students should be able to:

• Demonstrate a facility with complex numbers, curve sketching, the mathematics used to model waves, integration and Fourier methods
• Tackle problems associated with the physics covered by year 1 modules

Syllabus:

Techniques
Revision of mathematics from A-level - mainly algebra, differentiation, integration and trigonometry

Worksheets

• Complex Numbers Their construction from the reals; norm, argument, real and imaginary parts; addition, subtraction, multiplication and division; the Argand diagram and geometric view of complex numbers. de Moivre's theorem, exponential representation of a complex number in terms of its norm and its argument.

• Curve Sketching Basic Functions: trigonometric, exponential, modulus. Odd/even functions. Limiting values, continuity, differentiability. L'Hopital's rule. Asymptotes and strategies for graph sketching.

• Maths for Waves Notation for partial derivatives. Examples of equations admitting wave-like solutions: wave equation, advection equation, traffic flow. Linear operators, principle of superposition. Boundary conditions, reflection and transmission coefficients. Plane waves, exponential form. Energy in waves. Wave groups, group velocity.

• Integration along Lines, Surfaces and Volumes Notation for integration of both scalar and vector quantities over lines, surfaces and volumes. Integration along lines using parameterised curves, circulation around a contour. Infinitesimal surface element as a vector in 3D, use to compute flux across a surface. Volume integrals and revision of the Jacobian.

• Fourier Series Revision of lectured material: definition of Fourier series, the coefficients, periodic extensions, Gibbs phenomenon. The complex form, Parseval's theorem. Functions on intervals of length 2L. Introduction to Fourier transforms as the limit of L -> infinity.

You should answer the questions on each of the worksheets and hand in your answers to your personal tutors as directed.

Weekly Problem Sheets
You will be asked to hand in written answers to designated problems from the problem sheets or attempt designated problems from the Mastering Physics package.

Operation

Mathematics
All students will take a diagnostic mathematics test in the 1st week of the year. This test has 4 sections - algebra, trigonometry, differentiation and integration. Students failing any section will be required to be retested in that section (a total of 2 retests will be available and the best mark obtained will count). Students failing a section will be given work sheets and there will be surgeries where students can obtain individual assistance.

Mathematics Worksheets
Students will receive copies of each worksheet. These are self-study with problems which should be attempted and submitted to personal tutors for marking.

Physics Problems
For each physics module a number of problems will be set and a number of these will be identified as 'for credit'. Following submission of their attempts at these problems each student will attend a weekly small group seminar in which solutions to the problems will be presented by a tutor. Some of the assessments will be paper based and others will be web based.

Assessment: Worksheets - 50%; Physics Problems - 50%.