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.
Author(s): Thomas F. Banchoff, Stephen Lovett
Edition: 3
Publisher: CRC Press/Chapman & Hall
Year: 2022
Language: English
Commentary: Improvements with respect to [MD5] EE9D8DC315DDE1D979D23853FC261E00 :pagenated and added detailed bookmarks.
Pages: 385
Contents
Preface
Acknowledgments
Authors
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
4.3.1 Knots
4.3.2 Links
5. Regular Surfaces
5.1 Parametrized Surfaces
5.2 Tangent Planes; The Differential
5.2.1 Tangent Planes
5.2.2 the Differential
5.3 Regular Surfaces
5.4 Change of Coordinates; Orientability
5.4.1 Change of Coordinates
5.4.2 Change of Coordinates and the Tangent Plane
5.4.3 Orientability
6. First and Second Fundamental Forms
6.1 The First Fundamental Form
6.2 Map Projections (Optional)
6.2.1 Metric Properties of Maps of the Earth
6.2.2 Azimuthal Projections
6.2.3 Cylindrical Projections
6.2.4 Coordinate Changes on the Sphere
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
6.7.1 Developable Surfaces
6.7.2 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
8.1.1 Natural Frames
8.1.2 Normal Curvature
8.1.3 Geodesic Curvature
8.1.4 Geodesic Torsion
8.2 Gauss-Bonnet Theorem, Local Form
8.3 Gauss-Bonnet Theorem, Global Form
8.4 Geodesics
8.5 Geodesic Coordinates
8.5.1 General Geodesic Coordinates
8.5.2 Geodesic Polar Coordinates
8.6 Applications to Plane, Spherical, and Elliptic Geometry
8.6.1 Plane Geometry
8.6.2 Spherical Geometry
8.6.3 Elliptic Geometry
8.7 Hyperbolic Geometry
8.7.1 Synthetic Hyperbolic Geometry
8.7.2 The Poincaré Upper Half-plane
8.7.3 The Poincaré Disk
8.7.4 The Pseudosphere Revisited
9. Curves and Surfaces in n-Dimensional Space
9.1 Curves in n-Dimensional Euclidean Space
9.1.1 Curvatures and the Frenet Frame
9.1.2 Osculating Planes, Circles, and Spheres
9.1.3 the Fundamental Theorem of Curves in ℝⁿ
9.2 Surfaces in Euclidean n-Space
9.2.1 Regular Surfaces in ℝⁿ
9.2.2 Intrinsic Geometry for Surfaces
9.2.3 Orientability
9.2.4 The Gauss-Bonnet Theorem
A. Tensor Notation
A.1 Tensor Notation
A.1.1 Curvilinear Coordinate Systems
A.1.2 Tensors: Definitions and Notation
A.1.3 Operations on Tensors
A.1.4 Examples
A.1.5 Symmetries
A.1.6 Numerical Tensors
Bibliography
[13]
[30]
Index
abc
def
ghijklm
nopqr
stu
vw
