Introductory description

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 on 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.

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

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 and attempt designated problems from the Mastering Physics package.

Learning outcomes

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
• Prepare and submit pieces of work on a weekly basis
• Discuss questions arising out of their modules in small groups

Subject specific skills

Mathematical techniques, physics problem-solving

Transferable skills

Communication, group working, problem-solving, self-study