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Elements of Differential Geometry [Facsimile] [Paperback]

Richard S. Millman , George D. Parker
5.0 out of 5 stars  See all reviews (2 customer reviews)

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Book Description

March 29 1977 0132641437 978-0132641432 1

This text is intended for an advanced undergraduate (having taken linear algebra and multivariable calculus). It provides the necessary background for a more abstract course in differential geometry. The inclusion of diagrams is done without sacrificing the rigor of the material.

 

For all readers interested in differential geometry.


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5.0 out of 5 stars A solid introduction July 20 2002
Format:Paperback
It is hard to disagree with the idea that one must pursue the learning of mathematics in way that might be at odds with its axiomatic structure. One can pursue the study of differentiable manifolds without ever looking at a book on classical differential geometry, but it is doubtful that one could appreciate the underlying ideas if such a strategy were taken. Some background in linear algebra, topology, and vector calculus would allow one to understand the abstract definition of a differentiable manifold. However, to push forward the frontiers of the subject, or to apply it, one must have a solid understanding of its underlying intuition.
Thus a study of classical differential geometry is warranted for someone who wants to do original research in the area as well as use it in applications, which are very extensive. Differential geometry is pervasive in physics and engineering, and has made its presence known in areas such as computer graphics and robotics. In this regard, the authors of this book have given students a fine book, and they emphasize right at the beginning that an undergraduate introduction to differential geometry is necessary in today's curriculum, and that such a course can be given for students with a background in calculus and linear algebra. They also do not hesitate to use diagrams, without sacrificing mathematical rigour. Too often books in differential geometry omit the use of diagrams, holding to the opinion that to do so would be a detriment to mathematical rigour. Much is to be gained by the reading and studying of this book, and after finishing it one will be on the right track to begin a study of modern differential geometry.
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5.0 out of 5 stars Took the class and the book March 23 2004
Format:Paperback
I had the class from Prof. Parker ~20 years ago. (BS Mathematics 83 from SIU) It was a wonderful class and this is a wonderful book. I still have my signed! copy. I am now a professor of EE and a large research university and this is still a subject that I love. Credit that to the Book and Prof. Parker.
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Most Helpful Customer Reviews on Amazon.com (beta)
Amazon.com: 4.3 out of 5 stars  6 reviews
17 of 19 people found the following review helpful
5.0 out of 5 stars A solid introduction July 20 2002
By Dr. Lee D. Carlson - Published on Amazon.com
Format:Paperback
It is hard to disagree with the idea that one must pursue the learning of mathematics in way that might be at odds with its axiomatic structure. One can pursue the study of differentiable manifolds without ever looking at a book on classical differential geometry, but it is doubtful that one could appreciate the underlying ideas if such a strategy were taken. Some background in linear algebra, topology, and vector calculus would allow one to understand the abstract definition of a differentiable manifold. However, to push forward the frontiers of the subject, or to apply it, one must have a solid understanding of its underlying intuition.
Thus a study of classical differential geometry is warranted for someone who wants to do original research in the area as well as use it in applications, which are very extensive. Differential geometry is pervasive in physics and engineering, and has made its presence known in areas such as computer graphics and robotics. In this regard, the authors of this book have given students a fine book, and they emphasize right at the beginning that an undergraduate introduction to differential geometry is necessary in today's curriculum, and that such a course can be given for students with a background in calculus and linear algebra. They also do not hesitate to use diagrams, without sacrificing mathematical rigour. Too often books in differential geometry omit the use of diagrams, holding to the opinion that to do so would be a detriment to mathematical rigour. Much is to be gained by the reading and studying of this book, and after finishing it one will be on the right track to begin a study of modern differential geometry.
10 of 11 people found the following review helpful
4.0 out of 5 stars Should be required reading for undergrads Nov. 20 2011
By U of M Math Student - Published on Amazon.com
Format:Paperback|Verified Purchase
Differential Geometry is one of the toughest subjects to break into for several reasons. There is a huge jump in the level of abstraction from basic analysis and algebra courses, and the notation is formidable to say the least. An ill-prepared student can begin reading Spivak Volume I or Warner's book and get very little out of it. This is, in fact, what happened to me. Only at the advice of a professor did I take an undergraduate diff. geometry course which used this book, and am I glad that I did.

In short, here is a book which takes the key aspects of classical and modern differential geometry, and teaches them in the concrete setting of R^3. This has several advantages:

(1) The student isn't lost in the abstraction immediately. When I took my first diff. geometry course, we spent the entire time taking derivatives in n-dimensional projective space and other equally abstract spaces. This book keeps it concrete, and supplements each idea with several worked out examples to help ground the student's intuition.

(2) The book uses modern techniques when applicable. Just because this book teaches the material in a concrete/classical setting does not mean that its methods are outdated. The student will become very used to modern techniques, but applied here in easier settings than what you would find in a standard graduate leveled book. Hence, when the student eventually takes graduate leveled courses, he or she will come to see the definitions and techniques as natural extensions of those learned previously.

(3) The student learns the classical theory first, which entirely motivates the modern theory. Going back to (2), this allows students to view the modern theory as a natural extension of the classical theory, and hence their intuition learned from this book is still applicable in the more abstract settings. Also, for arbitrary manifolds, our intuition comes entirely from surfaces in R^3. The fact that this book has a 100+ page chapter on surfaces alone makes it worth reading.

The material covered in this book is expansive, and i think students will find that even in more advanced Riemannian geometry courses they will see material from this book still being taught. Hence, it will be a long time before a student completely outgrows this book.

I've said only good things, but there is one thing that annoys me. This book drops the ball in providing intuition in several topics. For example, the book defines Christofell symbols as abstract sums of inner products. However, the point is that they provide the scalars for the coordinates of mixed partial derivatives in a tangent space. There is a theorem that implies this, but this intuition is never explicitly stated. In a book aimed at undergraduates, I feel that everything, including intuition, should be stated and highlighted. This has not been too bad of a hinderance for me, because a quick trip to the professor's office will often sort things out for me. However, any self-learners may get the wrong idea about how to thing about certain topics. This is not too bad of a problem, and for the most part this book does a good job at explaining and motivating topics, so I do not feel that any more than one star should be taken off.

Prerequisites for reading:

(1) Strong Linear Algebra background. And by this I do not mean computations in Lay, I mean theory in Axler or something comparable. Often times, change of bases are applied without being explicitly stated in the proofs, and you should be able to pick up on this immediately.

(2) Analysis-both single variable and multiple variable. You do not need anything past the basics of multivariable differentiation and integration, i.e. no Stoke's theorem or differential forms are needed.

(3) While not completely necessary, there are a few proofs which use the existence and uniqueness theorem of ordinary differential equations, so knowing this exhausts all possible prerequisites. I do no know ODE theory, and I am not having trouble understanding the book as a whole, so this prerequisite may be relaxed/forgotten.
7 of 8 people found the following review helpful
4.0 out of 5 stars understandable, clear differential geometry book Sept. 29 2005
By math monkey - Published on Amazon.com
Format:Paperback
There are many differential geometry books out there. Some are very rigorous others not. This book walks the road in the middle. Intuition is developed in the first few chapters by discussing familiar surfaces in R^n, and then a discussion on more abstract manifolds follow.

The book requires some very basic knowledge of linear algebra and some multivariate calculus knowledge. So basically every undergrad in the sciences should find this book easy to understand, and a good introduction to differential geometry.
8 of 10 people found the following review helpful
5.0 out of 5 stars A Perfect Introduction Feb. 22 2001
By josh kaplan - Published on Amazon.com
Format:Paperback
A Must !!! After reviewing a few dozen books in the subject, this is without any doubt one of the best. It it written with rare clarity, and gives enough motivation and examples to understand the more abstract and difficult aspects of the field. The book is intended for advanced undergraduate (with good understanding of linear algebra and calculus III) and should be read prior to an abstract course in differential geometry (such as is covered in the books of Warner and Hicks).
14 of 21 people found the following review helpful
3.0 out of 5 stars Another Differential Geometry Book - So So Dec 10 2000
By None - Published on Amazon.com
Format:Paperback
This book I also purchased as a resource for studying differential geometry. It's a little bit better than the one by Thorpe, but not by much. The text is dedicated to the 'hard-core' mathematical, and even they would have to have some experience/guidance in this subject. I'm a self-learning type of guy, with an MS in physics. Too many questions arise to justify this book for the self-learner. There are problems, and a FEW examples.
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