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Differential Geometry Curves - Surfaces - Manifolds

ISBN-10: 0821839888

ISBN-13: 9780821839881

Edition: 2nd 2005 (Revised)

Authors: Wolfgang K�hnel

List price: $54.00
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Description:

Our first knowledge of differential geometry usually comes from the study of the curves and surfaces in $I\!\!R3$ that arise in calculus. Here we learn about line and surface integrals, divergence and curl, and the various forms of Stokes' Theorem. If we are fortunate, we may encounter curvature and such things as the Serret-Frenet formulas. With just the basic tools from multivariable calculus, plus a little knowledge of linear algebra, it is possible to begin a much richer andrewarding study of differential geometry, which is what is presented in this book. It starts with an introduction to the classical differential geometry of curves and surfaces in Euclidean space, then leads to an introduction to the Riemannian geometry of more general manifolds, including a look atEinstein spaces. An important bridge from the low-dimensional theory to the general case is provided by a chapter on the intrinsic geometry of surfaces. The first half of the book, covering the geometry of curves and surfaces, would be suitable for a one-semester undergraduate course. The local and global theories of curves and surfaces are presented, including detailed discussions of surfaces of rotation, ruled surfaces, and minimal surfaces. The second half of the book, which could be usedfor a more advanced course, begins with an introduction to differentiable manifolds, Riemannian structures, and the curvature tensor. Two special topics are treated in detail: spaces of constant curvature and Einstein spaces. The main goal of the book is to get started in a fairly elementary way, thento guide the reader toward more sophisticated concepts and more advanced topics. There are many examples and exercises to help along the way. Numerous figures help the reader visualize key concepts and examples, especially in lower dimensions. For the second edition, a number of errors were corrected and some text and a number of figures have been added.
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Book details

List price: $54.00
Edition: 2nd
Copyright year: 2005
Publisher: American Mathematical Society
Publication date: 12/13/2005
Binding: Paperback
Pages: 380
Size: 5.50" wide x 8.25" long x 1.00" tall
Weight: 1.034

Notations and prerequisites from analysis
Curves in $I\!\!R^n$
The local theory of surfaces
The intrinsic geometry of surfaces
Riemannian manifolds
The curvature tensor
Spaces of constant curvature
Einstein spaces
Bibliography
List of notation
Index