Introductory description

The tutor's mark is made up from marks for answers to the assessed weekly problems (50%) and from work associated with five worksheets (50%). The worksheets cover some background mathematical material assumed by other modules.

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.

Worksheets

Vectors:
Vectors have magnitude and direction. Addition and subtraction, the null vector. Geometry of simple figures written in vector notation, equation of lines and planes, equation for centroid of a triangle. The dot product, the normal to a plane and alternative form for equations of planes, perpendiculars from points of a triangle to opposite sides meet at a point. Cross-product and the notion of an area in three dimensions as a vector. Equation of line of intersection of two planes. Triple scalar product, associative law, relation to volume of parallelopiped. Triple vector product

Matrices:
Motivation and definition. The 2 x 2 case: operations on vectors. Eigenvalues and eigenvectors. Diagonalizing matrices. Exponential of a diagonalizable matrix. Mention of the 3 x 3 and N x N cases.

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.

Probability:
Definition of probability spaces and axioms. Discrete and continuous probability spaces. Common probability distributions, including binomial, Poisson, normal and Boltzmann distributions. Expectation and variance. Joint, conditional and marginal probabilities. Bayes' theorem.

Statistics:
Random variables and central limit theorem. Visualising and quantifying data. Uncertainty, errors and confidence intervals. Fitting, including least squares and maximum likelihood. Entropy and Shannon information.

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:

• Work with vectors, wave-functions, probability theory and statistics at a level necessary to cope with all first year physics modules and some second year maths modules.
• Analyse a simple problem and decide on an approach to its solution

Subject specific skills

Mathematical techniques, physics problem-solving

Transferable skills

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