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# Differential Geometry of Curves and Surfaces

ISBN-10: 1568814569

ISBN-13: 9781568814568

Pub. Date: 03/01/2010

Publisher: Taylor & Francis

Students and professors of an undergraduate course in differential geometry will appreciate the clear exposition and comprehensive exercises in this book that focuses on the geometric properties of curves and surfaces, one- and two-dimensional objects in Euclidean space. The problems generally relate to questions of local properties (the properties observed at a

## Overview

Students and professors of an undergraduate course in differential geometry will appreciate the clear exposition and comprehensive exercises in this book that focuses on the geometric properties of curves and surfaces, one- and two-dimensional objects in Euclidean space. The problems generally relate to questions of local properties (the properties observed at a point on the curve or surface) or global properties (the properties of the object as a whole). 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.

## 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)

## Related Subjects

Preface

Acknowledgements

1. Plane Curves: Local Properties
2. Parameterizations

Position, Velocity, and Acceleration

Curvature

Osculating Circles, Evolutes, and Involutes

Natural Equations

3. Plane Curves: Global Properties
4. Basic Properties

Rotation Index

Isoperimetric Inequality

Curvature, Convexity, and the Four-Vertex Theorem

5. Curves in Space: Local Properties
6. Definitions, Examples, and Differentiation

Curvature, Torsion, and the Frenet Frame

Osculating Plane and Osculating Sphere

Natural Equations

7. Curves in Space: Global Properties
8. Basic Properties

Indicatrices and Total Curvature

9. Regular Surfaces
10. Parametrized Surfaces

Tangent Planes and Regular Surfaces

Change of Coordinates

The Tangent Space and the Normal Vector

Orientable Surfaces

11. The First and Second Fundamental Forms
12. The First Fundamental Form

The Gauss Map

The Second Fundamental Form

Normal and Principal Curvatures

Gaussian and Mean Curvature

Ruled Surfaces and Minimal Surfaces

13. The Fundamental Equations of Surfaces
14. Tensor Notation

Gauss’s Equations and the Christoffel Symbols

Codazzi Equations and the Theorema Egregium

The Fundamental Theorem of Surface Theory

15. Curves on Surfaces

Curvatures and Torsion

Geodesics

Geodesic Coordinates

Gauss-Bonnet Theorem and Applications

Intrinsic Geometry

Bibliography

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