UCR MATH 138A - Classical Differential Geometry of Curves

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I . Classical Differential Geometry of CurvesThis is a first course on the differential geometry of curves and surfaces. It begins withtopics mentioned briefly in ordinary and multivariable calculus courses, and two major goals areto formulate the mathematical concept(s) of curvature for a surface and to interpret curvature forseveral basic examples of surfaces that arise in multivariable calculus.Basic references for the courseWe shall begin by citing the official text for the course:B. O’Neill. Elementary Differential Geometry. (Second Edition), Harcourt/AcademicPress, San Diego CA, 1997, ISBN 0–112–526745–2.There is also a Revised Second Edition (published in 2006; ISBN-10: 0–12–088735–5) which is closebut not identical to the Second Edition; the latter (not the more recent version) will be the officialtext for the course.This document is intended to provide a fairly complete set of notes that will reflect the contentof the lectures; the approach is similar but not identical to that of O’Neill. At various points weshall also refer to the following alternate sources. The first two of these are texts at a slightly higherlevel, and the third is the Schaum’s Outline Series review book on differential geometry, which iscontains a great deal of information on the classical approach, brief outlines of the underlyingtheory, and many worked out examples.M. P. do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall, SaddleRiver NJ, 1976, ISBN 0–132–12589–7.J. A. Thorpe, Elementary Topics in Differential Geometry, Springer-Verlag, New York,1979, ISBN 0–387–90357–7.M. Lipschutz, Schaum’s Outlines – Differential Geometry, Schaum’s/McGraw-Hill, 1969,ISBN 0–07–037985–8.At many points we assume material covered in previous mathematics courses, so we shall includea few words on such background material. This course explicitly assumes prior experience withthe elements of linear algebra (including matrices, dot products and determinants), the portions ofmultivariable calculus involving partial differentiation, and some familiarity with the a few basicideas from set theory such as unions and intersections. At a few points in later units we shallalso assume some familiarity with multiple integration. but we shall not be using results likeGreen’s Theorem, Stokes’ Theorem or the Divergence Theorem. For the sake of completeness, filesdescribing the background material (with references to standard texts that have been used in theDepartment’s courses) are included in the course directory and can be found in the files calledbackground∗.pdf, where n = 1, 2 or 3.The name “differential geometry” suggests a subject which uses ideas from calculus to obtaingeometrical information about curves and surfaces; since vector algebra plays a crucial role inmodern work on geometry, the subject also makes extensive use of material from linear algebra. Atmany points it will be necessary to work with topics from the prerequisites in a more sophisticatedmanner, and it is also necessary to be more careful in our logic to make sure that our formulas1and conclusions are accurate. Also, at numerous steps it might be necessary to go back and reviewthings from earlier courses, and in some cases it will be important to understand things in moredepth than one needs to get through ordinary calculus, multivariable calculus or matrix algebra.Frequently one of the benefits of a mathematics course is that it sharpens one’s understanding andmastery of earlier material, and differential geometry certainly provides many opportunities of thissort.The origins of differential geometryThe paragraph below gives a very brief summary of the developments which led to the emer-gence of differential geometry as a subject in its own right by the beginning of the 19thcentury.Further information may be found in any of several books on the history of mathematics.Straight lines and circles have been central objects in geometry ever since its beginnings.During the 5thcentury B.C.E., Greek geometers began to study more general curves, most notablythe ellipse, hyperbola and parabola but also other examples (for example, the Quadratrix of Hippias,which allows one to solve classical Greek construction problems that cannot be answered by means ofstraightedge and compass, and the Spiral of Archimedes, which is given in polar coordinates by theequation r = θ). In the following centuries Greek mathematicians discovered a large number of othercurves and investigated the properties of such curves in considerable detail for a variety of reasons.By the end of the Middle Ages in the 15thcentury, scientists and mathematicians had discoveredfurther examples of curves that arise in various natural contexts, and still further examples andresults were discovered during the 16thcentury. Problems involving curves played an importantrole in the development of analytic geometry and calculus during the 17thand 18thcenturies, andthese subjects in turn yielded powerful new techniques for analyzing curves and analyzing theirproperties. In particular, these advances created a unified framework for understanding the workof the Greek geometers and a setting for studying new classes of curves and problems beyond thereach of classical Greek geometry. Interactions with physics played a major role in the mathematicalstudy of curves beginning in the 15thcentury, largely because curves provided a means for analyzingthe motion of physical objects. By the beginning of the 19thcentury, the differential geometry ofcurves and surfaces had begun to emerge as a subject in its own right.This unit describes the classical nineteenth century theory of curves in the plane and 3-dimensional space.References for examplesHere are some web links to sites with pictures and written discussions of many curves thatmathematicians have studied during the past 2500 years, including the examples mentioned above:http://www-gap.dcs.st-and.ac.uk/∼history/Curves/Curves.htmlhttp://www.xahlee.org/SpecialPlaneCurves dir/specialPlaneCurves.htmlhttp://facstaff.bloomu.edu/skokoska/curves.pdfClickable links to these sites — and others mentioned in these notes — are in the course directoryfile dg2010links.pdf.2REFERENCES FOR RESULTS ON CURVES FROM CLASSICAL GREEK GEOMETRY. A survey ofcurves in classical Greek geometry is beyond the scope of these notes, but here are references forArchimedes’ paper on the spiral named after him and a


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UCR MATH 138A - Classical Differential Geometry of Curves

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