The geometry of surfaces is a classical subject which remains important today in fields as diverse as string theory and nano-materials. From a mathematical perspective it provides an excellent introduction to the ideas of contemporary Riemannian geometry.
Objectives
Understand basic topology and differentiation in Rn.
Understand and be able to apply the inverse and implicit function theorems.
Understand and be able to work with submanifolds in their various forms.
Understand and be able to calculate with the geometry of curves.
Understand and be able to calculate with the geometry of surfaces.
Understand integration on surfaces and be able to calculate such integrals.
Understand the Gauss-Bonnet theorem and be able to apply it.
Content
Introduction and review of topology on Rn (2 lectures)
Differentiable functions on Rn (5 lectures)
Inverse and implicit function theorems (3 lectures)
Submanifolds (4 lectures)
Curves (3 lectures)
Surfaces (3 lectures)
Integration on submanifolds (7 lecture)
Gauss-Bonnet theorem (3 lectures)
As this course has not been taught for 10 years there may be some minor variation
on the topic list and timings indicated here.
Linkage
Although not a pre-requisite this would be a very useful course to have done
before taking the Honours course Differential Geometry.
Pre-requisites
Prerequisite is MATHS 1007A/B Mathematics I (Pass Div I) or both MATHS 1007A/B Mathematics I (Pass Div II) and MATHS 2004 Mathematics IIM (Pass Div I). PURE MTH 2100 Real Analysis is assumed knowledge.
Recommended Texts
Manfredo de Carmo: Differential Geometry of Curves and Surfaces. 514.75 C287
John A. Thorpe: Elementary topics in differential geometry. 514.7 T519e
Peter Baxandall and Hans Liebeck: Vector calculus. 517.2 B355v
Martin Lipschutz: Schaum's outline of theory and problems of differential geometry. 513.73 L767
Alfred Gray:Modern differential geometry of curves and surfaces. 514.7 G778m
Wendell Fleming: Functions of several variables. 517.53 F598.2
Consulting time
Tuesday 11.30 to 12.30. For another time just email me and we can arrange something.
Handouts
Course plan. Will probably cover a bit more than this.
These are made available for students who have difficult keeping up or miss lectures. They are not particularly neat and I make no guarantee they agree completely with what I said in the lectures.
I am easiest to contact by email at
michael.murray@adelaide.edu.au. My office
is Room 7.29 of the Innova Building. This is behind a locked door so you need to ring me on
8313 4174 so I can come and let you in. When you come out of the lift turn right and you will see a door with a phone next to it. You can ring from there.