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Modern Differential Geometry of Curves and Surfaces with Mathematica, Second Edition, by Alfred Gray
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The Second Edition combines a traditional approach with the symbolic manipulation abilities of Mathematica to explain and develop the classical theory of curves and surfaces. You will learn to reproduce and study interesting curves and surfaces - many more than are included in typical texts - using computer methods. By plotting geometric objects and studying the printed result, teachers and students can understand concepts geometrically and see the effect of changes in parameters.
Modern Differential Geometry of Curves and Surfaces with Mathematica explains how to define and compute standard geometric functions, for example the curvature of curves, and presents a dialect of Mathematica for constructing new curves and surfaces from old. The book also explores how to apply techniques from analysis.
Although the book makes extensive use of Mathematica, readers without access to that program can perform the calculations in the text by hand. While single- and multi-variable calculus, some linear algebra, and a few concepts of point set topology are needed to understand the theory, no computer or Mathematica skills are required to understand the concepts presented in the text. In fact, it serves as an excellent introduction to Mathematica, and includes fully documented programs written for use with Mathematica.
Ideal for both classroom use and self-study, Modern Differential Geometry of Curves and Surfaces with Mathematica has been tested extensively in the classroom and used in professional short courses throughout the world.
- Sales Rank: #2980422 in Books
- Brand: Brand: CRC Press
- Published on: 1997-12-29
- Original language: English
- Number of items: 1
- Dimensions: 10.50" h x 7.50" w x 2.50" l,
- Binding: Hardcover
- 1088 pages
- Used Book in Good Condition
Review
"I wish that I would have used this book as my introduction to differential geometry instead of the here-unnamed horror-flick-of-a-book required by my early professors on the subjectIn the high-turnover of textbooks this much improved 2nd edition is truly deserving of the title and furthermore acts as an excellent tribute to its author." - The Mathematica Journal, Vol. 7 No. 2, 1998 "More than any other text I am aware of, Gray integrates computing into the materialGray's Mathematica programs offer new and fascinating ways to present material. With an appropriate selection of material the first edition and, even better, the present second edition, may provide an excellent background for a course on the subjectsecond edition provides considerable extensions of the subjectsauthor's acknowledgements show that comments from a lot of well-known geometers have lead to an improvement of the book." --Bernd Wegner, Zentralblatt MATH, Vol. 942
About the Author
Gray; Alfred University of Maryland, College Park, USA,
Most helpful customer reviews
18 of 18 people found the following review helpful.
Good introduction to differential geometry
By Dr. Lee D. Carlson
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.
19 of 22 people found the following review helpful.
Title is Misleading
By A Customer
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."
7 of 8 people found the following review helpful.
Excellent overall book
By A Customer
I strongly disagree with the reviewer at the bottom of this page. Having taken a differential geometry course last year using do Carmo's book (also excellent) I came to appreciate the intuition that this book lends to the reader. Also, this book makes greater use of elementary linear algebra than is common in some more standard texts, for example in defining the second fundamental form in terms of the Shape Operator. For students wanting to compliment their course notes or standard text with a book which will thoroughly explain both the fundamentals and isolated topics, this book is highly recommended.
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