Outline:

This module aims to provide students with the mathematical foundations required for all the first term and some of the second term courses 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. Although parts of the module may seem to be a review for some students, it is essential in bringing all students up to the same benchmark.

Aims:

To provide the mathematical foundations required for all the first semester and some of the second semester courses in the first year of the Physics and Astronomy programmes;
	To prepare students for the second semester follow-on mathematics course PHAS0009;
	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:

Video lectures are provided weekly on each of the main topics of the syllabus. These are reinforced by quizzes and a frequent drop-in session. Examples are used throughout to illustrate concepts. In addition, there are "virtual" online revision sessions in term 3. Though mathematical formalism is developed throughout the course, the emphasis in the homework sheets and final questions is very much on problem solving rather than demonstrations of bookwork.

Since it is important that students become fluent in the application of Mathematics to Physics and Astronomy, the lectures are supplemented by around 8 hours of problem-solving tutorials (PSTs), which are an integral part of the course.

In addition to timetabled lecture and PST 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:

In total, there are six sections to the course. Each section has one or two videos which present the material; each section also has a quiz associated with it. Each week, there is a 2-hour lecture that recaps the relevant material and goes through appropriate examples. There are 4 problem-solving tutorials (2 hours each), and a drop-in session once a week in those weeks which have no tutorials. The following gives a loose idea of the distribution:

Complex Numbers [4 hrs]
	Vectors [4 hrs]
	Differentiation (mainly revision) [4 hrs]
	Integration (mainly revision) [4 hrs]
	Partial Differentiation [8 hrs]
	Series & Limits [8 hrs]