Module Catalogue 2024/25

Pre Requisite Comment

A Level Mathematics or Equivalent

Co Requisite Comment

N/A

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.

Intended Knowledge Outcomes

Students will gain an understanding of complex numbers and of vector and matrix methods in algebra and geometry.

Students will
Know the basic methods for solving first and second order linear ordinary differential equations.

Understand the basic calculus for functions of many variables including differentiation and its uses and integration over 2- and 3-dimensional domains in Cartesian, polar, spherical and cylindrical coordinates.

Intended Skill Outcomes

Students will

Perform arithmetic manipulations with complex numbers.

Solve polynomial equations of small degree.

Use De Moivre’s Theorem to compute powers, determine n-th roots, and to obtain trigonometric identities.

Solve general systems of linear equations by Gauss eliminations. Use the dot product and the cross product to describe equations of lines and planes in three dimensional space.

Describe rotations and reflections in matrix form. Perform algebraic operations with matrices and apply them to solve linear equations.

Evaluate determinants and invert matrices. Determine eigenvalues and eigenvectors of matrices. Manipulate quadratic forms.

be able to solve the most basic types of ordinary differential equations

acquire a toolkit to tackle the most frequently occurring ODEs: linear first order ODEs and second order ODEs with constant coefficients.

be able to differentiate functions of many variables, classify their extrema and compute simple integrals over 2- and 3-dimensional domains using different coordinate systems.

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.

General Notes

N/A