Banchoff / Lovett Differential Geometry of Curves and Surfaces, Second Edition

2. Auflage 2015
ISBN: 978-1-4822-4737-4
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 430 Seiten

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.

Autoren/Hrsg.

Banchoff, Thomas F.

Lovett, Stephen T.

Fachgebiete

• Mathematik | Informatik Mathematik Geometrie

Inhaltsverzeichnis

Plane Curves: Local Properties
Parametrizations
Position, Velocity, and Acceleration
Curvature
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
Geodesics
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

Über Autor(innen)

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.