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Rating: 
Summary: Great Book!
Review: Gray does not intend for you to buy his book if you don't haveaccess to Mathematica and simply want to learn about differentialgeometry from an axiomatic standpoint. Of course if you don't have access to Mathematica, this isn't for you, and even if you do have Mathematica, you will probably want to have a good "standard" text to go along with your learning. Having said this, the book and Mathematica make an excellent addition to anyone's diferential geometry course.
Rating: 
Summary: Excellent overall book
Review: 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.
Rating:
Summary: Impressive both in size and content
Review: 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.
Rating: 
Summary: No CD-Rom!
Review: The main purpose of this textbook is supposed to be the combination of the visualization of curves and surfaces along with the traditional differential geometry materials. This book makes a great effort to realize this lofty goal with some success as a reference book. As a main textbook, however, it fails to deliver what it promised by overlooking one small point: the lack of CD-Rom. The readers are expected to type in the sample programs manually. This can be very time-consuming especially for inexperienced students. Inevitably, a lot of valuable class time has be consumed helping students looking for and correcting errors, many of which are small typographic errors completely unrelated to either their mathematical understanding or their computer skill. Instructors can attempt writing their own Mathematica class-notes, but copyright issue will come up if they don't be careful distributing the notebook files to the students. All such hassle can easily be eliminated with the CD-Rom(s). Hopefully this issue is resolved when the next edition is printed. Until then, I cannot recommend this book as a main textbook.
Rating:
Summary: No CD-Rom!
Review: The main purpose of this textbook is supposed to be the combination of the visualization of curves and surfaces along with the traditional differential geometry materials. This book makes a great effort to realize this lofty goal with some success as a reference book. As a main textbook, however, it fails to deliver what it promised by overlooking one small point: the lack of CD-Rom. The readers are expected to type in the sample programs manually. This can be very time-consuming especially for inexperienced students. Inevitably, a lot of valuable class time has be consumed helping students looking for and correcting errors, many of which are small typographic errors completely unrelated to either their mathematical understanding or their computer skill. Instructors can attempt writing their own Mathematica class-notes, but copyright issue will come up if they don't be careful distributing the notebook files to the students. All such hassle can easily be eliminated with the CD-Rom(s). Hopefully this issue is resolved when the next edition is printed. Until then, I cannot recommend this book as a main textbook.
Rating:
Summary: Good introduction to differential geometry
Review: 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.
Rating:
Summary: Title is Misleading
Review: 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."
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