Differential Geometry of Curves and Surfaces

ISBN-13: 9781032281094 / Angielski / Twarda / 2022 / 384 str.

Thomas F. Banchoff; Stephen T. Lovett

Opis

The book explains the reasons for various definitions. The interactive applets offer motivation for definitions, allowing students to explore examples, and give a visual explanation of complicated theorems. More elementary exercises are added and some challenging problems are moved later in exercise sets to assure more graduated progress.

Recenzje

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

Spis treści

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