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# Introducing Surfaces

C.T.J. Dodson, Department of Mathematics, UMIST

Date:

Introduction
This document briefly summarizes definitions and hints at proofs of principal results for a first course on surfaces. It is intended as an aide memoire--a companion to lectures, tutorials and computer lab classes, with exercises and proofs to be completed by the student. Exercises include the statements to be verified--mathematics needs to be done, not just read!

The prereqisites are: elementary knowledge of Euclidean geometry and the definition of familiarity with vector and scalar product, norms and basic linear algebra, fundamental theorem of calculus, inverse function theorem, implicit function theorem and a little vector calculus.

This is one of several companion electronic study aids, other topics include Using Computers , Curves , Knots . For an overview of curves and surfaces, see the course information Differential Geometry and Knot Theory .

Where possible, we encourage use of computer algebra software to experiment with the mathematics, to perform tedious analytic calculations and to plot graphs of functions that arise in the studies. For this purpose, we shall make use of Gray's book [6]--which contains all of the theory we need for curves and surfaces--and we use the computational packages he provides free in the form of Mathematica NoteBooks Mathematica for curves and surfaces. For general information about the Mathematica software, see Wolfram's book [10] and the website Mathematica. For further study of more general differential geometry and its applications to relativity and spacetime geometry, see Dodson and Poston [5]. For an introduction to algebraic topology see Armstrong [2] and for more advanced topics and their applications in analysis, geometry and physics, see Dodson and Parker [4]. The abovementioned books contain substantial bibliography lists for further reference.

Click Möbius strip movie for an animation showing a strip, with its faces of different colours, twisted into a Möbius band. As you probably know, this yields a surface with one face and one edge.

The document you are reading was first created with LATEX and the hypertext version on the World Wide Web was converted using a free program called latex2html and there is information about this in the last section of this document--see About This Document.