Differential Geometry

Rating:
Summary: One of the best
Review: Good, thorough, self-contained. Spend a little more time on the first three chapters, and the understanding will follow. What can I say? Just rush out and get it!

Rating:
Summary: One of the best
Review: Good, thorough, self-contained. Spend a little more time on the first three chapters, and the understanding will follow. What can I say? Just rush out and get it!

Rating:
Summary: Agree -- one of the best; elegant; beautiful
Review: I had been seeking a book on differential geometry for self-study, as a preface to learning general relativity. A seasoned mathematics friend recommended Kreyszig.

So, I waded in, and patiently made my way through every page of the first six chapters, working the problems along the way, at a pace of a few pages per day. Now that the journey is behind me, I can say that I appreciated this book. It compares favorably to some other texts I had tried reading, with less success.

I realize that the author's approach is an old-style classical one, with a reliance on specific coordinate systems and transformations between coordinate systems. To work the problems requires a fair amount of paper and pencil work. Nonetheless, this approach worked well for me. On those occasions when my reading bogged down, inevitably there was a good reason. If I went back carefully, re-read and pondered, doodled on paper, and tried to visualize what Kreyszig was describing, it always worked! The light would soon go on, usually with a pleasurable sense of discovery.

I went back to re-read certain sections of the book to refresh my memory, and realized how elegant the writing is. Crystal clear, right to the heart, and always trustworthy. Everything follows in a gentle persuasive way; there are no jarring leaps or gaps.

Additionally, I had a nice sense of the different flavor brought to the field by the French geometers who made many of the key advances around the turn of the 19th-20th century.

Finally, the summary of key results and equations at the end is very smart and helpful.

Since finishing Kreysig, I did find it helpful to push on and try to grasp these same ideas from the standpoint of one-forms and the coordinate-free approach to tensors. But I'm not sorry I came at the subject this way first.

I do recommend this book, and think that a beginner needs only a moderate amount of stamina and patience here.

A postscript -- the book is also beautiful. I like that in a math book.

Rating:
Summary: Very nice, useful style
Review: I really like this book. I checked out Manfredo P. Do Carmo'sfrom the library and bought this one, and I prefer this one just oncontent. The concepts are explained in a very approachable style and in a nice order to give you an understanding of diff. geometry as well as what you might use it for.. This is not a math text for just math in my opinion. This is geared for you to use differential geometry. I thought most of the concepts are explained nicely but it doesn't hurt to read another book to get another point of view. One advantage this book has over a number of others is that every answer to the exercises is in the back of the book with a very nice solution.. If you're interested in the subject I think this book is a great deal.

Rating:
Summary: An extension of advanced Mathematics for Engineering
Review: Kreyszig conserves in this book the same style of simple explanation of his Advanced Mathematics for Engineering. Although he reserves the content for a treatment of the differential geometry in three dimensions, for that reason it doesn't exempt the generality of treating this topic in spaces of n dimensions. Excepting the Chapter 6 and 8, the rest of the book is an extension of that was missing on his book of Advanced Mathematics: a Chapter on tensors, and of course, a more general treatment on the theory of surfaces. The first Chapter tries on questions preliminary on lineal algebra, quite simple, but enough for the Chapter two that is about the theory of curves, Chapter also quite simple if we compare it with the general treatments that are usually shown the the classic books of calculus (Thomas, Leithold...). The chapters 3, 4 and 5 are the real center of the work of Kreyszig in the book. Quite clear, concise, where introduce to the reader to the riemanian geometry with such a harmony that the reader feels extremely attracted by his reading. The symbology is clear and the very coherent and easy presentation of continuing. The Chapter 7 are a small treaty on calculus differential absolute, topic well presented, with the same methodological lines that characterize to the works of Kreyszig, without complications, simple, soft, without minimizing the rigor, without seeking to end up embracing the content of the book of Levi-Civitta, but very enough and that leaves the reader prepared for future more technical readings.

Rating:
Summary: highly recommended
Review: This is a wonderfully well written book. If you have a good background in calculus and analytic geometry, you will have no problems with understanding most of the book. (If you don't, you shouldn't be studying differential geometry anyway.) The last couple of chapters are more difficult. Make sure to do the problems after each chapter; they are very well designed to enhance your understanding, and as a huge bonus, their solutions can be found at the end of the book. Forget about those books with a fancy hard cover and cost ten times as much. Buy this book and enjoy!

Rating:
Summary: Not for beginer
Review: This is the sort of math book that you pick up, get something to drink, sit on the couch and read through as you would read a novel. I dont know if its possible to write a simpler or clearer treatment on differential geomerty. But be warned that it is still only "classical". Tensros are treated as objects that tranform in a certain way, rather than studied as general multilinear functions. However, after reading this book, any book on tensors is a breeeze to go through. Well worth having, especially considering the price.