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Differential Geometry [Paperback]

William C. Graustein

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Book Description

April 14 2006 0486450112 978-0486450117
This first course in differential geometry presents the fundamentals of the metric differential geometry of curves and surfaces in a Euclidean space of three dimensions. Written by an outstanding teacher and mathematician, it explains the material in the most effective way, using vector notation and technique. It also provides an introduction to the study of Riemannian geometry.
Suitable for advanced undergraduates and graduate students, the text presupposes a knowledge of calculus. The first nine chapters focus on the theory, treating the basic properties of curves and surfaces, the mapping of surfaces, and the absolute geometry of a surface. The final chapter considers the applications of the theory to certain important classes of surfaces: surfaces of revolution, ruled surfaces, translation surfaces, and minimal surfaces. Nearly 200 problems appear throughout the text, offering ample reinforcement of every subject.

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Amazon.com: 2.0 out of 5 stars  2 reviews
2 of 2 people found the following review helpful
3.0 out of 5 stars Archaic and of little historical or mathematical interest Oct. 14 2013
By Alan U. Kennington - Published on Amazon.com
Format:Paperback|Verified Purchase
Regrettably, I have to report that this book "Differential Geometry" by William Caspar Graustein is of little interest to the modern reader. I had hoped that it would throw some light on the state of differential geometry in the 1930s, but the modernity of this book is somewhere between Gauss and Darboux. Nevertheless, I'm grateful to Dover for keeping it in print. It's good to keep the old books alive.

Graustein only very peripherally acknowledges the work of Riemann, Christoffel, Levi-Civita and Weyl in two pages, almost at the end of the book. He gives a very brief mention of Weyl's idea of an affine connection, which generalizes the Riemannian metric.

You will find almost no modern differential geometry in this book. Even the idea of a "tangent vector" is introduced only as a unit vector tangent to a curve. There are no tensors, no affine connection or Riemannian metric (apart from a very brief allusion), no topology, no differential forms, etc. etc. It's just the geometry of surfaces embedded in Euclidean 3-space, in the 19th century idiom of Gauss and Darboux. However, I do not regret buying it. It's good to have a book which gives me the Gaussian differential geometry in a well-presented compact format.
1.0 out of 5 stars book is 75 years old (out of date). some if the calculus notation was not reconizable. graphics was poor. Jan. 7 2014
By Brian Coughlin - Published on Amazon.com
book is 75 years old (out of date). some if the calculus notation was not recognizable. graphics was poor or non existent.

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