Differential Geometry of Curves and Surfaces

by Thomas F. Banchoff, Stephen T. Lovett

Overview

Through two previous editions, the third edition of this popular and intriguing text takes both an analytical/theoretical approach and a visual/intuitive approach to the local and global properties of curves and surfaces.

Requiring only multivariable calculus and linear algebra, it develops students’ geometric intuition through interactive graphics applets. Applets are presented in Maple workbook format, which readers can access using the free Maple Player.

The book explains the reasons for various definitions while the interactive applets offer motivation for definitions, allowing students to explore examples further, and give a visual explanation of complicated theorems. The ability to change parametric curves and parametrized surfaces in an applet lets students probe the concepts far beyond what static text permits. Investigative project ideas promote student research.

At users of the previous editions' request, this third edition offers a broader list of exercises. More elementary exercises are added and some challenging problems are moved later in exercise sets to assure more graduated progress. The authors also add hints to motivate students grappling with the more difficult exercises.

This student-friendly and readable approach offers additional examples, well-placed to assist student comprehension. In the presentation of the Gauss-Bonnet Theorem, the authors provide more intuition and stepping-stones to help students grasp phenomena behind it. Also, the concept of a homeomorphism is new to students even though it is a key theoretical component of the definition of a regular surface. Providing more examples show students how to prove certain functions are homeomorphisms.

Product Details

ISBN-13:	9781568814568
Publisher:	Taylor & Francis
Publication date:	03/01/2010
Pages:	352
Product dimensions:	7.50(w) x 9.30(h) x 0.90(d)

About the Author

Thomas F. Banchoff is a geometer and a professor at Brown University. Dr. Banchoff was president of the Mathematical Association of America (MAA) from 1999 to 2000. He has published numerous papers in a variety of journals and has been the recipient of many honors, including the MAA’s Deborah and Franklin Tepper Haimo Award and Brown’s Teaching with Technology Award. He is the author of several books, including　Linear Algebra Through Geometry with John Wermer and Beyond the Third Dimension.

Stephen Lovett is an associate professor of mathematics at Wheaton College. Dr. Lovett has taught introductory courses on differential geometry for many years, including at Eastern Nazarene College. He has given many talks over the past several years on differential and algebraic geometry as well as cryptography. In 2015, he was awarded Wheaton’s Senior Scholarship Faculty Award. He is the author of Abstract Algebra: Structures and Applications, Differential Geometry of Manifolds, Second Edition, A Transition to Advanced Mathematics with Danilo Dedrichs (forthcoming), all published by CRC Press.

Table of Contents

Preface
1 Plane Curves: Local Properties
1.1 Parametrizations
1.2 Position, Velocity, and Acceleration
1.3 Curvature
1.4 Osculating Circles, Evolutes, Involutes
1.5 Natural Equations
2 Plane Curves: Global Properties
2.1 Basic Properties
2.2 Rotation Index
2.3 Isoperimetric Inequality
2.4 Curvature, Convexity, and the Four-Vertex Theorem
3 Curves in Space: Local Properties
3.1 Definitions, Examples, and Differentiation
3.2 Curvature, Torsion, and the Frenet Frame
3.3 Osculating Plane and Osculating Sphere
3.4 Natural Equations
4 Curves in Space: Global Properties
4.1 Basic Properties
4.2 Indicatrices and Total Curvature
4.3 Knots and Links
5 Regular Surfaces
5.1 Parametrized Surfaces
5.2 Tangent Planes; The Differential
5.3 Regular Surfaces
5.4 Change of Coordinates; Orientability
6 First and Second Fundamental Forms
6.1 The First Fundamental Form
6.2 Map Projections (Optional)
6.3 The Gauss Map
6.4 The Second Fundamental Form
6.5 Normal and Principal Curvatures
6.6 Gaussian and Mean Curvatures
6.7 Developable Surfaces; Minimal Surfaces
7 Fundamental Equations of Surfaces
7.1 Gauss’s Equations; Christoffel Symbols
7.2 Codazzi Equations; Theorema Egregium
7.3 Fundamental Theorem of Surface Theory
8 Gauss-Bonnet Theorem; Geodesics
8.1 Curvatures and Torsion
8.2 Gauss-Bonnet Theorem, Local Form
8.3 Gauss-Bonnet Theorem, Global Form
8.4 Geodesics
8.5 Geodesic Coordinates
8.6 Applications to Plane, Spherical, and Elliptic Geometry
8.7 Hyperbolic Geometry
9 Curves and Surfaces in n-dimensional Space
9.1 Curves in n-dimensional Euclidean Space
9.2 Surfaces in Euclidean n-Space
Appendix A: Tensor Notation
Index

What People are Saying About This

From the Publisher

… a complete guide for the study of classical theory of curves and surfaces and is intended as a textbook for a one-semester course for undergraduates … The main advantages of the book are the careful introduction of the concepts, the good choice of the exercises, and the interactive computer graphics, which make the text well-suited for self-study. …The access to online computer graphics applets that illustrate many concepts and theorems presented in the text provides the readers with an interesting and visually stimulating study of classical differential geometry. … I strongly recommend [this book and Differential Geometry of Manifolds] to anyone wishing to enter into the beautiful world of the differential geometry.
—Velichka Milousheva, Journal of Geometry and Symmetry in Physics, 2012

Students and professors of an undergraduate course in differential geometry will appreciate the clear exposition and comprehensive exercises in this book … Some of the more interesting theorems explore relationships between local and global properties. A special feature is the availability of accompanying online interactive java applets coordinated with each section. The applets allow students to investigate and manipulate curves and surfaces to develop intuition and to help analyze geometric phenomena.
—L’Enseignement Mathématique (2) 57 (2011)

… an intuitive and visual introduction to the subject is beneficial in an undergraduate course. This attitude is reflected in the text. The authors spent quite some time on motivating particular concepts and discuss simple but instructive examples. At the same time, they do not neglect rigour and precision. … As a distinguishing feature to other textbooks, there is an accompanying web page containing numerous interactive Java applets. … The applets are well-suited for use in classroom teaching or as an aid to self-study.
—Hans-Peter Schröcker, Zentralblatt MATH 1200

Coming from intuitive considerations to precise definitions the authors have written a very readable book. Every section contains many examples, problems and figures visualizing geometric properties. The understanding of geometric phenomena is supported by a number of available Java applets. This special feature distinguishes the textbook from others and makes it recommendable for self studies too. … highly recommendable …
—F. Manhart, International Mathematical News, August 2011

… the authors succeeded in making this modern view of differential geometry of curves and surfaces an approachable subject for advanced undergraduates.
—Andrew Bucki, Mathematical Reviews, Issue 2011h

… an essential addition to academic library Mathematical Studies instructional reference collections, as well as an ideal classroom textbook.
—Midwest Book Review, May 2011

Editorial Reviews

I endorse this third edition with enthusiasm. Moving away from Java based applets for the previous edition to Maplesoft based applets is significant. There are many new examples, and these are more clearly identified. There are many significant changes to produce a more useful and attractive teaching too. Figures are well-placed to be easily matched to the text. Adding historical vignettes help students appreciate how the topic developed.--Ken Rosen

From the Publisher