Description
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
Module deliveries for 2024/25 academic year
Last updated
This module description was last updated on 19th August 2024.
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