by





Free Shipping on all Orders Over $35!*
*excludes Marketplace items.
We're Sorry
Sold Out
This item is being sold by an Individual Seller and will not ship from the Online Bookstore's warehouse. The Seller must confirm the order within two business days. If the Seller refuses to sell or fails to confirm within this time frame, then the order is cancelled.
Please be sure to read the Description offered by the Seller.
Preface | |
Contents | |
Curves in the plane and in space | |
What is a curve? | p. 1 |
Arc-length | p. 9 |
Reparametrization | p. 13 |
Closed curves | p. 19 |
Level curves versus parametrized curves | p. 23 |
How much does a curve curve? | |
Curvature | p. 29 |
Plane curves | p. 34 |
Space curves | p. 46 |
Global properties of curves | |
Simple closed curves | p. 55 |
The isoperimetric inequality | p. 58 |
The four vertex theorem | p. 62 |
Surfaces in three dimensions | |
What is a surface? | p. 67 |
Smooth surfaces | p. 76 |
Smooth maps | p. 82 |
Tangents and derivatives | p. 85 |
Normals and orientability | p. 89 |
Examples of surfaces | |
Level surfaces | p. 95 |
Quadric surfaces | p. 97 |
Ruled surfaces and surfaces of revolution | p. 104 |
Compact surfaces | p. 109 |
Triply orthogonal systems | p. 111 |
Applications of the inverse function theorem | p. 116 |
The first fundamental form | |
Lengths of curves on surfaces | p. 121 |
Isometries of surfaces | p. 126 |
Conformal mappings of surfaces | p. 133 |
Equiareal maps mid a theorem of Archimedes | p. 139 |
Spherical geometry | p. 148 |
Curvature of surfaces | |
The second fundamental form | p. 159 |
The Gauss and Weingarten maps | p. 162 |
Normal and geodesic curvatures | p. 165 |
Parallel transport and covariant derivative | p. 170 |
Gaussian, mean and principal curvatures | |
Gaussian and mean curvatures | p. 179 |
Principal curvatures of a surface | p. 187 |
Surfaces of constant Gaussian curvature | p. 196 |
Flat surfaces | p. 201 |
Surfaces of constant mean curvature | p. 206 |
Gaussian curvature of compact surfaces | p. 212 |
Geodesics | |
Definition and basic properties | p. 215 |
Geodesic equations | p. 220 |
Geodesics on surfaces of revolution | p. 227 |
Geodesics as shortest paths | p. 235 |
Geodesic coordinates | p. 242 |
Gauss' Theorema Egregium | |
The Gauss and Codazzi-Mainardi equations | p. 247 |
Gauss' remarkable theorem | p. 252 |
Surfaces of constant Gaussian curvature | p. 257 |
Geodesic mappings | p. 263 |
Hyperbolic geometry | |
Upper half-plane model | p. 270 |
Isometries of H | p. 277 |
Poincaré disc model | p. 283 |
Hyperbolic parallels | p. 290 |
Beltrami-Klein model | p. 295 |
Minimal surfaces | |
Plateau's problem | p. 305 |
Examples of minimal surfaces | p. 312 |
Gauss map of a minimal surface | p. 320 |
Conformal parametrization of minimal surfaces | p. 322 |
Minimal surfaces and holomorphic functions | p. 325 |
The Gauss-Bonnet theorem | |
Gauss-Bonnet for simple closed curves | p. 335 |
Gauss-Bonnet for curvilinear polygons | p. 342 |
Integration on compact surfaces | p. 346 |
Gauss-Bonnet for compact surfaces | p. 349 |
Map colouring | p. 357 |
Holonomy and Gaussian curvature | p. 362 |
Singularities of vector fields | p. 365 |
Critical points | p. 372 |
Inner product spaces and self-adjoint linear maps | |
Isometries of Euclidean spaces | |
Möbius transformations | |
Hints to selected exercises | |
Solutions | |
Index | |
Table of Contents provided by Ingram. All Rights Reserved. |
An electronic version of this book is available through VitalSource.
This book is viewable on PC, Mac, iPhone, iPad, iPod Touch, and most smartphones.
By purchasing, you will be able to view this book online, as well as download it, for the chosen number of days.
A downloadable version of this book is available through the eCampus Reader or compatible Adobe readers.
Applications are available on iOS, Android, PC, Mac, and Windows Mobile platforms.
Please view the compatibility matrix prior to purchase.