Lectures on Classical Differential Geometry: Second Edition

by Dirk J. Struik

Overview

Elementary, yet authoritative and scholarly, this book offers an excellent brief introduction to the classical theory of differential geometry. It is aimed at advanced undergraduate and graduate students who will find it not only highly readable but replete with illustrations carefully selected to help stimulate the student's visual understanding of geometry. The text features an abundance of problems, most of which are simple enough for class use, and often convey an interesting geometrical fact. A selection of more difficult problems has been included to challenge the ambitious student.
Written by a noted mathematician and historian of mathematics, this volume presents the fundamental conceptions of the theory of curves and surfaces and applies them to a number of examples. Dr. Struik has enhanced the treatment with copious historical, biographical, and bibliographical references that place the theory in context and encourage the student to consult original sources and discover additional important ideas there.
For this second edition, Professor Struik made some corrections and added an appendix with a sketch of the application of Cartan's method of Pfaffians to curve and surface theory. The result was to further increase the merit of this stimulating, thought-provoking text — ideal for classroom use, but also perfectly suited for self-study. In this attractive, inexpensive paperback edition, it belongs in the library of any mathematician or student of mathematics interested in differential geometry.

Product Details

ISBN-13:	9780486138183
Publisher:	Dover Publications
Publication date:	04/26/2012
Series:	Dover Books on Mathematics
Sold by:	Barnes & Noble
Format:	eBook
Pages:	240
File size:	16 MB
Note:	This product may take a few minutes to download.

Table of Contents

PREFACE
BIBLIOGRAPHY
CHAPTER 1. CURVES
1-1 Analytic representation
1-2 "Arc length, tangent "
1-3 Osculating plane
1-4 Curvature
1-5 Torsion
1-6 Formulas of Frenet
1-7 Contact
1-8 Natural equations
1-9 Helices
1-10 General solution of the natural equations
1-11 Evolutes and involutes
1-12 Imaginary curves
1-13 Ovals
1-14 Monge
CHAPTER 2. ELEMENTARY THEORY OF SURFACES
2-1 Analytical representation
2-2 First fundamental form
2-3 "Normal, tangent plane"
2-4 Developable surfaces
2-5 Second fundamental form
2-6 Euler's theorem
2-7 Dupin's indicatrix
2-8 Some surfaces
2-9 A geometrical interpretation of asymptotic and curvature lines
2-10 Conjugate directions
2-11 Triply orthogonal systems of surfaces
CHAPTER 3. THE FUNDAMENTAL EQUATIONS
3-1 Gauss
3-2 The equations of Gauss-Weingarten
3-3 The theorem of Gauss and the equations of Codazzi
3-4 Curvilinear coordinates in space
3-5 Some applications of the Gauss and the Codazzi equations
3-6 The fundamental theorem of surface theory
CHAPTER 4. GEOMETRY ON A SURFACE.
4-1 Geodesic (tangential) curvature
4-2 Geodesics
4-3 Geodesic coordinates
4-4 Geodesics as extremals of a variational problem
4-5 Surfaces of constant curvature
4-6 Rotation surfaces of constant curvature
4-7 Non-Euclidean geometry
4-8 The Gauss-Bonnet theorem
CHAPTER 5. SOME SPECIAL SUBJECTS
5-1 Envelopes
5-2 Conformal mapping
5-3 Isometric and geodesic mapping
5-4 Minimal surfaces
5-5 Ruled surfaces
5-6 lmaginaries in surface theory
SOME PROBLEMS AND PROPOSITIONS
APPENDIX: The method of Pfaffians in the theory of curves and surfaces
ANSWERS TO PROBLEMS
INDEX