Differential Geometry of Curves and Surfaces  book cover
2nd Edition

Differential Geometry of Curves and Surfaces

ISBN 9781482247343
Published September 10, 2015 by Chapman & Hall
430 Pages 89 B/W Illustrations

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Book Description

Differential Geometry of Curves and Surfaces, Second Edition 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 computer graphics applets supported by sound theory.

The book explains the reasons for various definitions while the interactive applets offer motivation for certain definitions, allow 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.

New to the Second Edition

  • Reworked presentation to make it more approachable
  • More exercises, both introductory and advanced
  • New section on the application of differential geometry to cartography
  • Additional investigative project ideas
  • Significantly reorganized material on the Gauss–Bonnet theorem
  • Two new sections dedicated to hyperbolic and spherical geometry as applications of intrinsic geometry
  • A new chapter on curves and surfaces in Rn

Suitable for an undergraduate-level course or self-study, this self-contained textbook and online software applets provide students with a rigorous yet intuitive introduction to the field of differential geometry. The text gives a detailed introduction of definitions, theorems, and proofs and includes many types of exercises appropriate for daily or weekly assignments. The applets can be used for computer labs, in-class illustrations, exploratory exercises, or self-study aids.

Table of Contents

Plane Curves: Local Properties
Position, Velocity, and Acceleration
Osculating Circles, Evolutes, and Involutes
Natural Equations

Plane Curves: Global Properties
Basic Properties
Rotation Index
Isoperimetric Inequality
Curvature, Convexity, and the Four-Vertex Theorem

Curves in Space: Local Properties
Definitions, Examples, and Differentiation
Curvature, Torsion, and the Frenet Frame
Osculating Plane and Osculating Sphere
Natural Equations

Curves in Space: Global Properties
Basic Properties
Indicatrices and Total Curvature
Knots and Links

Regular Surfaces
Parametrized Surfaces
Tangent Planes and Regular Surfaces
Change of Coordinates
The Tangent Space and the Normal Vector
Orientable Surfaces

The First and Second Fundamental Forms
The First Fundamental Form
Map Projections (Optional)
The Gauss Map
The Second Fundamental Form
Normal and Principal Curvatures
Gaussian and Mean Curvature
Developable Surfaces and Minimal Surfaces

The Fundamental Equations of Surfaces
Gauss’s Equations and the Christoffel Symbols
Codazzi Equations and the Theorema Egregium
The Fundamental Theorem of Surface Theory

The Gauss–Bonnet Theorem and Geometry of Geodesics
Curvatures and Torsion
Gauss–Bonnet Theorem, Local Form
Gauss–Bonnet Theorem, Global Form
Geodesic Coordinates
Applications to Plane, Spherical and Elliptic Geometry
Hyperbolic Geometry

Curves and Surfaces in n-Dimensional Euclidean Space
Curves in n-Dimensional Euclidean Space
Surfaces in Rn

Appendix: Tensor Notation

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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 T. 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 and Differential Geometry of Manifolds.


Praise for the First Edition:
"… 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