Aims

To introduce the mathematics needed to formulate and describe problems involving vector and scalar fields in 3D space.

Module Summary

Many applications of mathematics involve objects and quantities that exist in multiple dimensions, as well as their rates of change in space. This module shows how calculus can be applied to such problems, which is essential in many branches of applied mathematics.

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.

Outline Of Syllabus

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);

A brief introduction to subscript notation and the summation convention;

operators in spherical and cylindrical coordinates;

Gauss', Stokes' and Green's theorems.

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. The teaching methods are appropriate to allow students to develop a wide range of skills, from understanding basic concepts and facts to higher-order thinking.

Problem Classes are used to help develop the students’ abilities at applying the theory to solving problems. Tutorials are used to identify and resolve specific queries raised by students and to allow students to receive individual feedback on marked work. In addition, office hours (two per week) 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

A substantial formal unseen examination is appropriate for the assessment of the material in this module. The format of the examination will enable students to reliably demonstrate their own knowledge, understanding and application of learning outcomes. The assurance of academic integrity forms a necessary part of the programme accreditation.

Examination problems may require a synthesis of concepts and strategies from different sections, while they may have more than one ways for solution. The examination time allows the students to test different strategies, work out examples and gather evidence for deciding on an effective strategy, while carefully articulating their ideas and explicitly citing the theory they are using.

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.