Aims

To lay the mathematical foundations for more advanced mathematics needed to describe
physical systems. Students will learn how to solve simple differential equations and how known computational tools of the calculus of functions of a single variable generalize to functions of many variables.

Outline Of Syllabus

Complex numbers, arithmetic, Argand diagram, polar form, de Moivre's theorem, powers and roots of unity.

Vectors: sums, products (scalar, dot, cross), equations of lines and planes, orthogonality, norm.

Linear algebra: row operations, solution of linear equations, Gaussian elimination, matrix operations, determinants, inverting matrices, eigenvectors, quadratic forms.

A general introduction to differential equations: Partial and ordinary; linear and non-linear; homogeneous and non- homogeneous. First-order ordinary differential equations (ODEs): direct integration; separation of variables, homogeneous equations, general linear first order ODEs. Second-order linear ODEs: Constant coefficients, inhomogeneous equations. Partial differentiation of multivariable functions: stationary points, chain rule. Integration of multivariable functions: Double and triple integration, change of variables, polar, spherical and cylindrical coordinates.

Teaching Rationale And Relationship

Non-synchronous online materials are used for the delivery of theory and explanation of methods, illustrated with examples, and for giving general feedback on assessed work. Present-in-person and synchronous online sessions are used to help develop the students’ abilities at applying the theory to solving problems and to identify and resolve specific queries raised by students, and to allow students to receive individual feedback on marked work. In addition, office hours/discussion board activity will provide an opportunity for more direct contact between individual students and the lecturer: a typical student might spend a total of one or two hours over the course of the module, either individually or as part of a group.

Assessment Rationale And Relationship

The course assessments allow the students to develop their problem solving techniques, to practise the methods learnt in the module, to assess their progress and to receive feedback; these assessments have a secondary formative purpose as well as their primary summative purpose.