Newcastle University > Undergraduate > Find a Degree > Module

• Offered for Year: 2021/22

• Module Leader(s): Dr Toby Wood
• Lecturer: Professor Anvar Shukurov

• Owning School: Mathematics, Statistics and Physics
• Teaching Location: Newcastle City Campus

Semesters

Aims

To introduce the mathematics needed to formulate and solve problems involving vector and scalar fields, and ordinary and partial differential equations.

Module Summary

Many applications of mathematics involve objects and quantities that exist in multiple dimensions, as well as their rates of change in space and/or time. This course shows how calculus can be applied to such problems, and introduces techniques for solving the resulting differential equations.

The first half of this module explains how we can mathematically define curves and surfaces in three-dimensional space, and how we can calculate their properties, such as tangent, length and area. We also introduce the concepts of scalar fields (e.g. temperature, pressure, density) and vector fields (e.g. velocity and electromagnetic fields). To describe these objects and quantities we must generalize the principles of calculus to multi-dimensions. This part of the course introduces the mathematical language and concepts that are needed to study continuous media, fluids, and electromagnetism.

The second half of this module continues the exploration of differential equation that started in Stage 1, with emphasis on methods to solve them, both exact and approximate. The essential elements in the theory of ordinary and partial differential equations, and their methods of solution, introduced in this module, provide the basis for specific studies in other modules. The methods that will be introduced, justified and practiced apply to a wide range of ordinary and partial differential equations.

Outline Of Syllabus

Semester 1:
•       Scalar and vector fields;
•       double and triple integrals;
•       parametric representations of curves and surfaces;
•       tangent vector and line integrals;
•       normal vector and surface integrals;
•       differential operators (gradient, divergence, curl, and Laplacian);
•       subscript notation and the summation convention;
•       operators in spherical and cylindrical coordinates;
•       Gauss', Stokes' and Green's theorems.

Semester 2:
•       A review of ordinary and partial differential equations;
•       Series solutions.
•       Fourier series and Fourier transforms.
•       Wave equation: simple derivation in one spatial dimension, generalisation to three dimensions, fundamental properties (D’Alembert’s solution, phase speed, plane waves, superposition, standing and travelling waves), wave packet.
•       Second-order partial differential equations. Separation of variables in Cartesian coordinates: application to the wave, heat, Laplace’s and Poisson’s equations.

Teaching Methods

Teaching Activities

Jointly Taught With

Teaching Rationale And Relationship

Lectures are used for the delivery of theory and explanation of methods, illustrated with examples, and for giving general feedback on marked work. Problem Classes are used to help develop the students’ abilities at applying the theory to solving problems.

Assessment Methods

The format of resits will be determined by the Board of Examiners

Exams

Exam Pairings

Other Assessment

Assessment Rationale And Relationship

A substantial formal unseen examination is appropriate for the assessment of the material in this module. The coursework assignments 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.

In the event of on-campus examinations not being possible, an on-line alternative assessment will be used for written examination 1.

Reading Lists

• MAS2804's Reading List

Timetable

• Timetable Website: www.ncl.ac.uk/timetable/
• MAS2804's Timetable