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Modern Differential Geometry of Curves and Surfaces with Mathematica

Edition: 
3
Publisher: 
Chapman & Hall/CRC
Number of Pages: 
984
Price: 
89.95
ISBN: 
1584884487

When it comes to the use of mathematical software as a pedagogical tool, I am something of an old fogey. I find Mathematica and Maple useful, and I encourage students to use them, but I am leery of allowing such tools to occupy a very large part of the classroom or the textbook, for two reasons. First, I worry that learning to use these programs will become one more hurdle, one more difficult thing in a subject that is already quite difficult. Second, I worry about the fascination that these programs seem to exert. Some mathematicians, and some students, seem to get so fascinated with the tools that the mathematics ends up relegated to a supporting role.

Alfred Gray's Modern Differential Geometry of Curves and Surfaces was one of the first textbooks to fully integrate Mathematica into an undergraduate course on differential geometry. Doing so is clearly a good idea. Geometry should be visual, and using software allows us to make it so. Furthermore, differential geometry deals with lots of important things that are easy to define and very hard to compute. Even computing the curvature and torsion of a curve in Euclidean space can be a pain. As a result, we are either reduced to having students work on carefully selected toy examples or we tell our students to learn to use symbolic manipulation software.

In his introduction to his second edition, Gray made the first point forcefully. He pointed out that differential geometry was out of fashion in the mid-twentieth century, and said, "I attribute the decline of differential geometry, especially in the United States, to the rise of tensor analysis. Instead of drawing pictures it became fashionable to raise and lower indices." Computers made it possible once again to draw pictures. Computer graphics has, in fact, created a whole new range of applications of differential geometry. And computers make it easy to do complicated computations. To use the most obvious example, the horrible formula for the Gaussian curvature in terms of the first fundamental form that comes out of Gauss's Theorema Egregium is much too complicated to use when computing by hand, but it doesn't bother Mathematica one bit.

All of this makes me ambivalent about this new (third) edition of Gray's book, revised after his death by Elsa Abbena and Simon Salamon. It is huge, bringing to mind those scary massive software manuals that used to come with Mathematica. Opening the book reveals that much of this size is due to the decision to print the Mathematica notebooks corresponding to each chapter as a appendices to their chapters. This instantly doubles the size of the book. It is a puzzling decision. First of all, the notebooks are all available online. Second, there is very little benefit to reading a Mathematica notebook without being able to execute the code. Yes, it's nice to see the code written out, but one could see that in the electronic form too.

Seeing the size of the book and the presence of all the code, my initial reaction was to conclude that my fear had come true: the software had eaten up the mathematics. There is certainly some truth to this. In fact, someone wanting to teach a course on Mathematica and graphics could use parts of this book as a text. (Gray even says so!) But there is more to the book than that.

If one ignores those many pages of Mathematica notebooks, what is left is very good. The treatment of differential geometry is quite classical, but it is also intelligently and charmingly presented, with many more pictures than usual. For example, I had never before seen a simultaneous plot of a plane curve and its curvature function, or a plot of the Gaussian curvature of a surface. There are pictures of tangent developables, generalized cones, parallel surfaces, and other constructions that usually only show up in problem sets. There are lots of pictures of minimal surfaces.

There are also some nice theoretical touches, such as relating the existence of a signed curvature function for plane curves to the existence of a complex structure on the plane. Links to complex analysis and to quaternions are made when appropriate, which is also helpful. And the ugly equations with awful indices are also there for those who like them.

The book covers far too many topics for a one-semester course, but that is fine: one can select, and much will be left over for students to consider and study. There are little biographical footnotes and attempts at history. These are mostly inadequate and sometimes even annoying. (What's the point of pointing out that "ellipse" comes from "falling short" if one does not then explain what has fallen short of what?)

One final complaint is that the Gauss-Bonnet theorem is relegated to the very last chapter, and the proof uses (lightly) the theory of abstract surfaces developed earlier. This is unfortunate, since in an undergraduate course it is very hard to cover abstract surfaces and still get to Gauss-Bonnet… but it would be educational malpractice not to get to Gauss-Bonnet.

In the end, though I would have preferred to have the notebooks on a CD-ROM and a thinner book, I think this is a very good introductory textbook, particularly if your students are already familiar with Mathematica and can therefore concentrate on learning the geometry. I would probably not choose it as a main text, but I would certainly recommend it to appropriately inclined students and to their professors.


Fernando Q. Gouvêa just finished teaching a one-semester undergraduate introduction to differential geometry at Colby College.

Date Received: 
Thursday, October 26, 2006
Reviewable: 
Yes
Include In BLL Rating: 
No
Alfred Gray, Elsa Abbena, and Simon Salamon
Series: 
Studies in Advanced Mathematics
Publication Date: 
2006
Format: 
Hardcover
Category: 
Textbook
Fernando Q. Gouvêa
01/26/2007

 CURVES IN THE PLANE
Euclidean Spaces
Curves in Space
The Length of a Curve
Curvature of Plane Curves
Angle Functions
First Examples of Plane Curves
The Semicubical Parabola and Regularity
1.8 Exercises
Notebook 1

FAMOUS PLANE CURVES
Cycloids
Lemniscates of Bernoulli
Cardioids
The Catenary
The Cissoid of Diocles
The Tractrix
Clothoids
Pursuit Curves
Exercises
Notebook

ALTERNATIVE WAYS OF PLOTTING CURVES
Implicitly Defined Plane Curves
The Folium of Descartes
Cassinian Ovals
Plane Curves in Polar Coordinates
A Selection of Spirals
Exercises
Notebook 3

NEW CURVES FROM OLD
Evolutes
Iterated Evolutes
Involutes
Osculating Circles to Plane Curves
Parallel Curves
Pedal Curves
Exercises
Notebook 4

DETERMINING A PLANE CURVE FROM ITS CURVATURE
Euclidean Motions
Isometries of the Plane
Intrinsic Equations for Plane Curves
Examples of Curves with Assigned Curvature
Exercises
Notebook 5

GLOBAL PROPERTIES OF PLANE CURVES
Total Signed Curvature
Trochoid Curves
The Rotation Index of a Closed Curve
Convex Plane Curves
The Four Vertex Theorem
Curves of Constant Width
Reuleaux Polygons and Involutes
The Support Function of an Oval
Exercises
Notebook 6

CURVES IN SPACE
The Vector Cross Product
Curvature and Torsion of Unit-Speed Curves
The Helix and Twisted Cubic
Arbitrary-Speed Curves in R3
More Constructions of Space Curves
Tubes and Tori
Torus Knots
Exercises
Notebook 7

CONSTRUCTION OF SPACE CURVES
The Fundamental Theorem of Space Curves
Assigned Curvature and Torsion
Contact
Space Curves that Lie on a Sphere
Curves of Constant Slope
Loxodromes on Spheres
8.7 Exercises
Notebook 8

CALCULUS ON EUCLIDEAN SPACE
Tangent Vectors to Rn
Tangent Vectors as Directional Derivatives
Tangent Maps or Differentials
Vector Fields on R n
Derivatives of Vector Fields
Curves Revisited
Exercises
Notebook 9

SURFACES IN EUCLIDEAN SPACE
Patches in Rn
Patches in R3 and the Local Gauss Map
The Definition of a Regular Surface
Examples of Surfaces
Tangent Vectors and Surface Mappings
Level Surfaces in R3
Exercises
Notebook 10

NONORIENTABLE SURFACES
Orientability of Surfaces
Surfaces by Identification
The Möbius Strip
The Klein Bottle
Realizations of the Real Projective Plane
Twisted Surfaces
Exercises
Notebook 11

METRICS ON SURFACES
The Intuitive Idea of Distance
Isometries between Surfaces
Distance and Conformal Maps
The Intuitive Idea of Area
Examples of Metrics
Exercises
Notebook 12

SHAPE AND CURVATURE
The Shape Operator
Normal Curvature
Calculation of the Shape Operator
Gaussian and Mean Curvature
More Curvature Calculations
A Global Curvature Theorem
Nonparametrically Defined Surfaces
Exercises
Notebook 13

RULED SURFACES
Definitions and Examples
Curvature of a Ruled Surface
Tangent Developables
Noncylindrical Ruled Surfaces
Exercises
Notebook 14

SURFACES OF REVOLUTION AND CONSTANT CURVATURE
Surfaces of Revolution
Principal Curves
Curvature of a Surface of Revolution
Generalized Helicoids
Surfaces of Constant Positive Curvature
Surfaces of Constant Negative Curvature
More Examples of Constant Curvature
Exercises
Notebook 15

A SELECTION OF MINIMAL SURFACES
Normal Variation
Deformation from the Helicoid to the Catenoid
Minimal Surfaces of
More Examples of Minimal Surfaces
Monge Patches and Scherk's Minimal Surface
The Gauss Map of a Minimal Surface
Isothermal Coordinates
Exercises
Notebook 16

INTRINSIC SURFACE GEOMETRY
Intrinsic Formulas for the Gaussian Curvature
Gauss's Theorema Egregium
Christoffel Symbols
Geodesic Curvature of Curves on Surfaces
Geodesic Torsion and Frenet Formulas
Exercises
Notebook 17

ASYMPTOTIC CURVES AND GEODESICS ON SURFACES
Asymptotic Curves
Examples of Asymptotic Curves and Patches
The Geodesic Equations
First Examples of Geodesics
Clairaut Patches
Use of Clairaut Patches
Exercises
Notebook 18

PRINCIPAL CURVES AND UMBILIC POINTS
The Differential Equation for Principal Curves
Umbilic Points
The Peterson-Mainardi-Codazzi Equations
Hilbert's Lemma and Liebmann's Theorem
Triply Orthogonal Systems of Surfaces
Elliptic Coordinates
Parabolic Coordinates and a General Construction
Parallel Surfaces
The Shape Operator of a Parallel Surface
Exercises
Notebook 19

CANAL SURFACES AND CYCLIDES OF DUPIN
Surfaces Whose Focal Sets are 2-Dimensional
Canal Surfaces
Cyclides of Dupin via Focal Sets
The Definition of Inversion
Inversion of Surfaces
Exercises
Notebook 20

THE THEORY OF SURFACES OF CONSTANT NEGATIVE CURVATURE
Intrinsic Tchebyshef Patches
Patches on Surfaces of Constant Negative Curvature
The Sine-Gordon Equation
Tchebyshef Patches on Surfaces of Revolution
The Bianchi Transform
Moving Frames on Surfaces in R3
Kuen's Surface as Bianchi Transform of the Pseudosphere
The B¨ acklund Transform
Exercises
Notebook 21

MINIMAL SURFACES VIA COMPLEX VARIABLES
Isometric Deformations of Minimal Surfaces
Complex Derivatives
Minimal Curves
Finding Conjugate Minimal Surfaces
The Weierstrass Representation
Minimal Surfaces via Björling's Formula
Costa's Minimal Surface
Exercises
Notebook 22

ROTATION AND ANIMATION USING QUATERNIONS
Orthogonal Matrices
Quaternion Algebra
Unit Quaternions and Rotations
Imaginary Quaternions and Rotations
Rotation Curves
Euler Angles
Further Topics
Exercises
Notebook 23

DIFFERENTIABLE MANIFOLDS
The Definition of a Differentiable Manifold
Differentiable Functions on Manifolds
Tangent Vectors on Manifolds
Induced Maps
Vector Fields on Manifolds
Tensor Fields
Exercises
Notebook 24

RIEMANNIAN MANIFOLDS
Covariant Derivatives
Pseudo-Riemannian Metrics
The Classical Treatment of Metrics
The Christoffel Symbols in Riemannian Geometry
The Riemann Curvature Tensor
Exercises
Notebook 25

ABSTRACT SURFACES AND THEIR GEODESICS
Christoffel Symbols on Abstract Surfaces
Examples of Abstract Metrics
The Abstract Definition of Geodesic Curvature
Geodesics on Abstract Surfaces
The Exponential Map and the Gauss Lemma
Length Minimizing Properties of Geodesics
Exercises
Notebook 26

THE GAUSS-BONNET THEOREM
Turning Angles and Liouville's Theorem
The Local Gauss-Bonnet Theorem
An Area Bound
A Generalization to More Complicated Regions
The Topology of Surfaces
The Global Gauss-Bonnet Theorem .
Applications of the Gauss-Bonnet Theorem
Exercises
Notebook

Bibliography
Name Index
Subject Index
Notebook Index

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Friday, January 26, 2007

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