Aims

To give students a grounding in the theory of curves and surfaces in 3 dimensions. Students will learn about parameterised curves and the use of local surface patches to describe surfaces. Students will learn about the precise definition of curvature and compute this for many famous examples. Students will see applications to the isoperimetric inequality and minimal surfaces (soap bubbles).

Module Summary

The theory of geometry in the three dimensional space is important to mathematicians as it is the theory of the world as we find it. It has been studied since time out of mind and continues to this day as an active area of research. This module focuses on the mathematical description of the shape of curves and surfaces. With these notions in hand, we can describe many interesting phenomena such as why folding a pizza slice is the best way of eating it or why soap bubbles take the shape that they do. In the early years of the 20th century, these ideas became the cornerstone for the mathematics that underpinned general relativity, variational methods, Lagrangian dynamics and geometry in higher dimensions. This module will focus on computing examples so would suit a student that likes to apply results to specific cases.

Outline Of Syllabus

Parameterised curves: examples, arc length, curvature torsion and the Frenet--Serret formulae. The isoperimetric problem. Parameterising surfaces: examples, tangents to surfaces. The Ist fundamental form and isometries. The IInd fundamental form and curvature. Selected topics: minimal surfaces and soap bubbles, Weierstrass parameterisation of minimal surfaces.

Teaching Rationale And Relationship

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.

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 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.