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.
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.Read more ›
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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.
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