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Outline:

This module aims to provide students with the mathematical foundations required for all the first term and some of the second term modules in the first year of our physics-related degree programmes, and to give students practice in mathematical manipulation and problem-solving. Topics include: complex numbers, vectors, (partial) differentiation, integration, series and limits.Ìý

Aims:

• To provide the mathematical foundations required for all term 1 modules and some of the Term 2 modules in the first year of the Physics and Astronomy programmes; To prepare students for the term 2 follow-on mathematics course PHAS0009 Mathematical Methods II; To give students practice in mathematical manipulation and problem solving.

Intended Learning Outcomes:

After completing this course, the student should be able to:

• understand the relation between the hyperbolic and exponential functions;
• differentiate simple functions and apply the product and chain rules to evaluate the differentials of more complicated functions;
• find the positions of the stationary points of a function of a single variable and determine their nature;
• understand integration as the reverse of differentiation;
• evaluate integrals by using substitutions, integration by parts, and partial fractions;
• understand a definite integral as an area under a curve and make simple numerical approximations;
• differentiate up to second order a function of 2 or 3 variables and test when an expression is a perfect differential;
• change the independent variables by using the chain rule and work with polar coordinates;
• find the stationary points of a function of two independent variables and show whether these correspond to maxima, minima or saddle points;
• evaluate line integrals along simple curves in three-dimensional space;
• manipulate real three-dimensional vectors, evaluate scalar and vector products, find the angle between two vectors in terms of components;
• construct vector equations for lines and planes and find the angles between them, understand frames of reference and direction for interception using vectors;
• express vectors, including velocity and acceleration, in terms of basis vectors in polar coordinate systems; understand the concept of convergence for an infinite series and apply simple tests to investigate it; expand an arbitrary function of a single variable as a power series (Maclaurin and Taylor), make numerical estimates and apply l’Hôpital’s rule to evaluate the ratio of two singular expressions; represent complex numbers in Cartesian and polar form on an Argand diagram;
• perform algebraic manipulations with complex numbers, including finding powers and roots;
• apply de Moivre’s theorem to derive trigonometric identities and understand the relation between trigonometric and hyperbolic functions using complex arguments.

Teaching and Learning Methodology:

This module is delivered via weekly lectures supplemented by a series of workshops and additional discussion. In addition to timetabled lecture hours, it is expected that students engage in self-study in order to master the material. This can take the form, for example, of practicing example questions and further reading in textbooks and online.

Indicative Topics:

• Complex Numbers;
• Vectors;
• Differentiation;
• Integration;
• Partial Differentiation;
• Series & LimitsÌý