Elementary Differential Geometry, Revised 2nd Edition, Second Edition (Hardcover)

~ (Author) "This book presupposes a reasonable knowledge of elementary calculus and linear algebra..." (more)
Key Phrases: cylindrical frame field, principal frame field, geodesic polar parametrization, Definition Let, Lemma Let, Theorem Let (more...)

Editorial Reviews

Book Description

Includes fully updated computer commands in line with the latest software

Product Description

Written primarily for students who have completed the standard first courses in calculus and linear algebra, ELEMENTARY DIFFERENTIAL GEOMETRY, REVISED SECOND EDITION, provides an introduction to the geometry of curves and surfaces.

The Second Edition maintained the accessibility of the first, while providing an introduction to the use of computers and expanding discussion on certain topics. Further emphasis was placed on topological properties, properties of geodesics, singularities of vector fields, and the theorems of Bonnet and Hadamard.

This revision of the Second Edition provides a thorough update of commands for the symbolic computation programs Mathematica or Maple, as well as additional computer exercises. As with the Second Edition, this material supplements the content but no computer skill is necessary to take full advantage of this comprehensive text.

*Fortieth anniversary of publication! Over 36,000 copies sold worldwide
*Accessible, practical yet rigorous approach to a complex topic--also suitable for self-study
*Extensive update of appendices on Mathematica and Maple software packages
*Thorough streamlining of second edition's numbering system
*Fuller information on solutions to odd-numbered problems
*Additional exercises and hints guide students in using the latest computer modeling tools

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Product Details

• Hardcover: 520 pages
• Publisher: Academic Press; 2 edition (April 10, 2006)
• Language: English
• ISBN-10: 0120887355
• ISBN-13: 978-0120887354
• Product Dimensions: 9.1 x 6 x 1.4 inches
• Shipping Weight: 2 pounds (View shipping rates and policies)
• Average Customer Review: 4.5 out of 5 stars  See all reviews (10 customer reviews)
• Amazon.com Sales Rank: #258,785 in Books (See Bestsellers in Books)

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Customer Reviews

Most Helpful Customer Reviews

34 of 35 people found the following review helpful:

4.0 out of 5 stars Solid and Modern Introduction, October 17, 2000

By	A. Ali "Harkonnen" (Minneapolis, MN USA) - See all my reviews (REAL NAME)

This review is from: Elementary Differential Geometry, Second Edition (Hardcover)

I worked through the first edition of this book some years back. After finishing this book I was ready for more abstract treatments of Riemannian Geometry. For example, having seen covariant derivatives on 2-surfaces embedded in R^3 motivates the abstract definition of connections on manifolds.

Chapter 1 is a decent introduction to pullbacks and pushforwards of differntial forms and tangent vectors respectively. In fact, all the subsequent geometry is based on pullbacks and pushforwards.This itself motivates the more abstract definition of a differentiable manifold with its coordinate charts. True,tangent vectors are not described in the most abstract fashion (e.g. as derivations on the algebra of functions) but this is not appropriate for a first course.

Chapter 2 describes the language of frame field and connection forms and derives the Frenet-Serret equations in terms of moving frames and structure equations. We associate this with the methods of Elie Cartan, who used moving frames in a systematic manner.

Chapter 3 deals with isometries; frankly speaking I never understood the raison d'etre for such a long chapter on such a topic.

Chapter 4 discusses coordinate patches. Again, this is thoroughly modern, and you won't find this in Struik or Kreyszig. The idea of piecing together coordinate patches to get geometric or topological information is a twentieth-century conception.

Chapter 5 introduces the Shape Operator, which is subsequently used in Chapter 6 to derive the equations of surface theory. This is really moving frames again, in another guise.

Chapter 7 finally tries to put this in a more abstract setting by defining abstract surfaces with an intrinsically defined covariant derivative.Holonomy and the Gauss-Bonnet theorem are discussed.

After reading this book, one would be equipped to handle do Carmo's book on Riemannian geometry, or O'Neill's book on Semi-Riemanninan geometry, or the more recent book by Lee, again on Riemannian geometry.

15 of 16 people found the following review helpful:

5.0 out of 5 stars Introductory level text with empasis on intuition examples and exercise., July 10, 2005

By	Rehan Dost (Canada) - See all my reviews (REAL NAME)

This review is from: Elementary Differential Geometry, Second Edition (Hardcover)

If you are looking for abstraction with little in the way of intuition I suggest Conlan " differential manifolds"

If you are an applied mathematician or physicist this book is for you.

I have always beleived that to truly grasp mathematics one must be provided with a reason for WHY things are the way they are and WHAT IDEAS the expression must express. This is best done with examples and exercises.

I digress.

The book restricts is exposition to two and three dimensions. Some of the topics can readily be bootstrapped to higher dimensions.

The book starts with basic ideas of curve, directional derivative and tangent vector in Euclidean space with a sprinkling of differential forms to wet the appetite.

It then moves into the notion of frame fields along curves resulting in the Frenet formulas which express how the frame fields change along the curve. These are expressed in terms of the frame field themselves giving ideas of curvature and torsion.

The book then abstracts these concepts to show how we can talk about change of frame fields along arbritrary directions not just along the curve. The tools used to do this are the covariant derivative and connection forms which can then be used to develop connection equations ( abstracted analogue of frenet formulas ) and then the cartan structural equations.

The book talks about isometries and defines euclidean geometry as those properties preserved by isometries. It then abstracts once again to surfaces in R3 using patches and appropriate conditions on the overlap without introducing manifolds although these are briefly mentioned later.

We then see how calculus in euclidean space can be adapted to surfaces using these patches. The corresponding concepts of function, differentiability and tangent vectors on these objects is introduced. Forms on these surfaces are introduced and their application to integration theory on these surfaces is developed showing how forms on the surface are " pulled back" to euclidean space using the idea of differential of a map and integrated there. The integration gives the volume ( area ) of that surface. Stokes theorem is introduced.

We now move into the idea of shape operators on the surface and show how these describe how the normal vector on the surface move in various directions giving ideas of mean and gaussian curvature . We see a very nice interplay of algebraic analysis leading to a geometric analysis.

The book then deals with studying geometrical properties on surfaces using the Cartan methods described earlier.

We then see how to define intrinsic geometry of any surface. Namely those properties of the surface that are preserved by isometries. From the definition of isometry we see that these rely on on the concepts of tangent vector and inner products. Shape operators and mean curvature are not intrinsic.

We now study the geometry of surfaces specifically the intrinsic geometry without reference to an imbedding space ( R3). An abstract "surface" is endowed with an inner product. A different inner product gives a different geometry. We talk about gaussian curvature and covariant derivative which are intrinsic.

Geodesics are introduced as is the gauss bonnet theorem which relates a geometric property to a topological one.

The book concludes with a chapter on global properties ( 2 d surfaces ) especially how gaussian curvature influences geodesics and how the two completely determine the geometry of the surface.

9 of 11 people found the following review helpful:

4.0 out of 5 stars Good low dimensional calculation, July 28, 2003

By	"joshfeng" (Madison, WI United States) - See all my reviews

This review is from: Elementary Differential Geometry, Second Edition (Hardcover)

It's easy to read with enough examples. Suitable for self study after your advanced calculus (inverse function thm/implicit function thm should be covered here) and linear algebra classes. Tons of exercises will help you familiarize yourself with the calculation in low dimension. (Do I love the exercises on minimal surfaces and surfaces of revolution in chapter 5 and 6!) Most of them are workable. This is the strength of the book. Since the author limits the material to low dimensions, some definitions are a bit misleading, such as the definition of exterior derivative of 1-form in chapter 4, where another term to be added happens to be zero. I think there is a big gap in style and level of difficulty between this book and author's "Semi-Riemannian Geometry: With Applications To Relativity".

After this book, probably you want to read Hicks' "Notes on differential geometry", if you can find a copy in some lib. Darling's "Differential Forms and Connections" is also highly recommended. It is modern but not much topological stuff.
Company it with Warner's "Foundations of differentiable manifolds and Lie groups" for topology also much higher algebra.