Introducing Curves

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

Date:

These notes supplement the lectures and provide practise exercises. We begin with some material you will have met before, perhaps in other forms, to set some terminology and notation. Further details on unfamiliar topics may be found in, for example Cohn [3] for algebra, Dodson and Poston [5] for linear algebra, topology and differential geometry, Gray [6] for curves, surfaces and calculations using the computer algebra package Mathematica, and Wolfram [10] for Mathematica itself. Several on-line hypertext documents are available to support this course [1].

Introduction
This document briefly summarizes definitions and hints at proofs of principal results for a first course on curves. 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 here are: elementary knowledge of Euclidean geometry and the definition of $\mathbb{R} ^n,$ 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 , Surfaces , 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.

The document you are reading was first created with L^ATEX 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.

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Comments Please!
Please send me your comments on these course materials, and let me know of errors by email at: dodson@umist.ac.uk