Modern Differential Geometry of Curves and Surfaces with Mathematica, Second Edition [Hardcover]

Alfred Gray (Author)

Book Description

ISBN-10: 0849371643 | ISBN-13: 978-0849371646 | Publication Date: December 16, 1997 | Edition: 2nd

This book not only explains and develops the classical theory of curves, but also allows the reader to reproduce and study curves and surfaces using computer methods. This second edition contains eight new chapters on global curve theory, space curves, minimal surfaces, inversions, cyclides, the Gauss-Bonnet Theorem and global surface theory.

Editorial Reviews

About the Author

Gray; Alfred University of Maryland, College Park, USA,

Product Details

• Hardcover: 1088 pages
• Publisher: CRC-Press; 2nd edition (December 16, 1997)
• Language: English
• ISBN-10: 0849371643
• ISBN-13: 978-0849371646
• Product Dimensions: 10.3 x 7.3 x 2.5 inches
• Shipping Weight: 4.6 pounds
• Average Customer Review: 3.6 out of 5 stars  See all reviews (7 customer reviews)
• Amazon Best Sellers Rank: #1,922,809 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews

16 of 16 people found the following review helpful:

4.0 out of 5 stars Good introduction to differential geometry, May 7, 2002

By 

Dr. Lee D. Carlson (Baltimore, Maryland USA) - See all my reviews
(VINE VOICE)    (HALL OF FAME REVIEWER)    (REAL NAME)   

This review is from: Modern Differential Geometry of Curves and Surfaces with Mathematica, Second Edition (Hardcover)

The visualization of complicated geometrical objects
is now routine thanks to the excellent software that
has been developed over the past two decades. Now
students and professionals can have a better
appreciation of the geometrical properties of these
objects thanks to these software packages. In this
book the author has done a great job of doing this,
having chosen one of the best tools for this purpose:
Mathematica. The book is a hefty one, totaling almost
1100 pages, but its perusal is worth the effort for
those who want a more intuitive appreciation behind
the concepts of differential geometry. Physicists in
particular, who usually need a pictorial approach to
complement the learning of a subject, should really
enjoy this book. It could definitely be used as a
textbook in a beginning course in differential
geometry since there are problems at the end of each
chapter and most of the results in the book are proven
with the required mathematical rigor, I.e. this book
is not just code and pictures, and a substantial
portion of it is devoted to definitions and rigorous
proofs. This is especially true for the discussion on
differentiable manifolds and Riemannian geometry. The
author also includes a brief biography of the
mathematicians who have been involved in differential
geometry at various places in the book. The
Mathematica code in the book though can be revised to
make it look more like standard mathematical notation,
thanks to the new features of Mathematica that have
appeared since this book was published (1997). The use
of color shading is not done in the book, except for a
short insert with pictures of several surfaces, but
the reader can easily experiment with the color
functions available in Mathematica if needed. A very
lengthy appendix that lists the functions and code
used in the book is included.
Some of the concepts that are usually
difficult to grasp intuitively for those approaching
differential geometry for the first time but are here
illustrated nicely include: 1. The computation of the
curvature of plane curves and the plotting of this
curvature. The curvature of the famous Lissajous
curves, very familiar from oscilloscope traces, is
computed. The author might have spent a little more
time explaining why the curvature plots have the shape
they do however. 2. The treatment of osculating curves
to plane curves. 3. The finding of curves whose
curvature is equal to the arc length times a Bessel
function. The resulting plots are very entertaining.
4. The computation of the torsion of a curve in space.
The discussion on torus knots is particularly well-
done. 5. The author's discussion on surfaces in
Euclidean space motivates well the concept of a
differentiable manifold. He plots a few surfaces with
coordinate patches that have a singularity, and shows
how to plot surfaces that defined nonparametrically.

Kummer's surface, of particular importance in
algebraic geometry, is plotted here. Even more useful
is the author's treatment of nonorientable surfaces,
wherein he shows the reader how to plot the Moebius
strip, the Klein bottle, and two realizations of the
projective plane using Mathematica. Several examples
of the Gaussian curvature of surfaces are plotted. The
Gauss map, one of the most important tools for the
physicist, is given detailed treatment. 6. Rare in
textbooks at this level of differential geometry is a
discussion of minimal surfaces, but the author gives a
very nice treatment in this book. The Enneper's,
Scherk's Henneberg's and Catalan's minimal surfaces
are plotted along with the Gauss map of Enneper's
surface. Minimal surfaces are extremely important in
theoretical physics, such as superstring and membrane
theories, and are also very important in optimization
theory, so it was nice to see a discussion of them
included in the book. In recent years galleries of
minimal surfaces have appeared on the Web, and this
book allows one to plot these without too much effort.
The author even introduces the use of complex analysis
in the study of minimal surfaces. Readers interested
in understanding the mathematics of string theory will
appreciate this discussion. In addition, the
Weierstrass representation, which allows generation of
new minimal surfaces, is introduced. Readers familiar
with the Weierstrass function for elliptic curves will
see it used here for this generation.

15 of 17 people found the following review helpful:

1.0 out of 5 stars Title is Misleading, December 8, 1999

By A Customer

This review is from: Modern Differential Geometry of Curves and Surfaces with Mathematica, Second Edition (Hardcover)

Using this book might help you learn Mathematica, but if your goal is to learn differential geometry, please try something else. The Mathematica learner can benefit from the numerous useful examples and exercises in the book. In most programming books, the examples provided are trivial and usually do nothing useful, and this can make reading the book extremely boring. By using "Modern Differential Geometry of Curves and Surfaces with Mathematica"as a companion to a standard book on Mathematica, you can find a way out of this boredom. But the book essentially lacks the qualities of a text on pure mathematics. In some cases it lacks mathematical rigor, and even sometimes the definitions and their usage are inconsistent (see pages 159-161 for instance.) The proofs are also usually nothing but simple manipulations of formulae, the type that we have encountered mostly in high school. And when complicated mathematical reasoning becomes necessary, the author simply tries to avoid it. To summarize, if I were to pick a title for this book, I would chose "Fun with Mathematica through Differential Geometry."

5 of 5 people found the following review helpful:

5.0 out of 5 stars Impressive both in size and content, December 10, 1999

By 

angelos Vorvolakos - See all my reviews

This review is from: Modern Differential Geometry of Curves and Surfaces with Mathematica, Second Edition (Hardcover)

I would recommend this book to anyone that needs an intuitive introduction to the subject that is complete in many ways and that provides visualization and examples using mathematica when needed. Before purchasing this I was expecting the treatment to be 'informal' and using mathematica rather than mathematical rigour to introduce concepts and results. This is not the case however. In my opinion the author has struck the right balance between a formal maths treatment and the abilities provided by mathematica to make the book easier to read and coprehend.