Elementary Differential Geometry (Springer Undergraduate Mathematics Series) by A.N. Pressley (PDF)

Description

Elementary Differential Geometry presents the main results in the differential geometry of curves and surfaces suitable for a first course on the subject. Prerequisites are kept to an absolute minimum – nothing beyond first courses in linear algebra and multivariable calculus – and the most direct and straightforward approach is used throughout. New features of this revised and expanded second edition include:a chapter on non-Euclidean geometry, a subject that is of great importance in the history of mathematics and crucial in many modern developments. The main results can be reached easily and quickly by making use of the results and techniques developed earlier in the book. Coverage of topics such as: parallel transport and its applications; map colouring; holonomy and Gaussian curvature.Around 200 additional exercises, and a full solutions manual for instructors, available via www.springer.comul

User’s Reviews

Editorial Reviews: Review . Gouvêa, The Mathematical Association of America, May, 2010) From the Back Cover Curves and surfaces are objects that everyone can see, and many of the questions that can be asked about them are natural and easily understood. Differential geometry is concerned with the precise mathematical formulation of some of these questions. It is a subject that contains some of the most beautiful and profound results in mathematics yet many of these are accessible to higher-level undergraduates.Elementary Differential Geometry presents the main results in the differential geometry of curves and surfaces suitable for a first course on the subject. Prerequisites are kept to an absolute minimum – nothing beyond first courses in linear algebra and multivariable calculus – and the most direct and straightforward approach is used throughout. New features of this revised and expanded second edition include:a chapter on non-Euclidean geometry, a subject that is of great importance in the history of mathematics and crucial in many modern developments. The main results can be reached easily and quickly by making use of the results and techniques developed earlier in the book. Coverage of topics such as: parallel transport and its applications; map colouring; holonomy and Gaussian curvature.Around 200 additional exercises, and a full solutions manual for instructors, available via www.springer.comPraise for the first edition:”The text is nicely illustrated, the definitions are well-motivated and the proofs are particularly well-written and student-friendly…this book would make an excellent text for an undergraduate course, but could also well be used for a reading course, or simply read for pleasure.”Australian Mathematical Society Gazette”Excellent figures supplement a good account, sprinkled with illustrative examples.”Times Higher Education Supplement About the Author Andrew Pressley is Professor of Mathematics at King’s College London, UK. Read more

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐I must admit to being disappointed by this text. I was happy to hear the solutions to the excercises are given in the bag of the book. Unfortunately the author skips many steps in the provided solutions. Some of the problems are easy and quite straightforward. Others are quite difficult as they should be. However, the solutions should include detailed description of the techniques and formulas used. Before giving the book a try, I recommend you restudy your standard trigonometric identities, integration by substitution, linear algebra, and particularly tangent and normal vectors. Even with this preparation many of the proofs also do not make sense. The problem again is skipped steps. I have not been able to find a differential geometry text at a true beginner or elementary level. This book has a lot of beautiful math which merits the four stars, but as a self learning text it does not meet the goal. Many self learners like myself need more help. If you can find a good on-line tutor familiar with this subject and text it probably would be helpful.

⭐I am using this book for a 2 semester Differential Geometry course at my university. The school used to use Do Carmo, but apparently the book was too advanced for the undergraduate level, so this semester they decided to switch over to give this one a test and see how it worked out. This book is not bad. It is basically Do Carmo rehashed for the not so mathematically mature. In all seriousness, the book even follows almost the exact same flow as Do Carmo, the topics are just presented with less rigor. The exercises are rather tedious at the end of each chapter, and in my opinion they don’t really help to enhance the subject matter.On the other hand, if you fall in the category that most of the math majors at my university fall in (i.e. the category of people who really don’t care, they just want to get an A and graduate, and don’t care about mathematics), then you’ll love this book. Why? Because the solution to every single problem is at the end of the book. In my opinion this is a huge flaw. It would be great if everyone were honest and everybody was genuinely interested in the learning Differential Geometry, but that isn’t the case. So 90% of my class simply copies the answers out of the back of the book and hands it in to get a 100 on the homework assignments. Pretty sad if you ask me. The book is almost there. Without full solutions to every problem, this book would get 5 stars. But those students who simply turn to the back of the book 15 seconds after looking at the problem statement will learn nothing from this book, so I have to knock it down 2 stars. After all, what good is a book if it doesn’t serve it’s intended purpose. Perhaps some people would rate a book by “how easy is it to get an A in the class if this is the textbook”, in which case they would probably rate this book 5 stars.Differential Geometry is a hard subject. It’s _supposed_ to be hard. We’re not talking about taking the reciprocal of a fraction here, it’s Differential Geometry. You’re _supposed_ to think about these problems for a long time. So if you’re a professor considering this book for a course I would recommend against it. The text is good, but the students won’t learn anything from it. I’ve suggested to my professor that perhaps it would be good to not assign problems from the text, but rather get problems from other textbooks where students can’t look at the answers.In my opinion that is the only flaw with this book. Otherwise I think it’s a great introduction, and about as elementary as you can really make the subject. If another book was too hard, then this is the one for you.Also, if you’re interested in this book for self study it’s a good choice since obviously you’re genuinely interested in the subject matter and won’t be tempted to look at the answer at the first opportunity.

⭐For a stand-alone course in differential geometry at the undergraduate level, this is one of the clearest and most accessible texts around, with perhaps its only rival being McCleary’s “Geometry from a Differentiable Viewpoint.” It’s written in the spirit of Struik’s classic, “Lectures on Classical Differential Geometry,” and explains the classical material developed largely by Gauss. The caveat is that a student planning to later study Riemannian geometry will perhaps not be best served by such a book — e.g., there’s no mention of covariant derivatives, which — as Riemannian connections on a differentiable manifold — serve as the bedrock of Riemannian geometry. Such a student would be better served either by O’Neill’s “Elementary Differential Geometry” or Oprea’s “Differential Geometry and its Applications” (see in particular the last chapter). This is the difference between classical and modern treatments of differential geometry.

⭐I would not recommend this book for math majors. I used this book for a course in differential geometry which was not intended for math majors (so yes, it is sort of my fault), but thought I would mention this here just in case there are math students who are considering it. Like the author mentions, some of the methods he uses don’t generalize and so they keep the requirements to a minimum and parts of the book cover topics that a math major would already know and not as rigorously. A math student should be able to tackle the classic in the genre by do Carmo. My professor often times used the proofs of do Carmo instead of the ones from Pressley. However, I suppose for non-math majors this is probably a very good book as it also includes solutions to every problem at the back. So its excellent for self-study!

⭐The proofs in the book cover all of the necessary material. The one problem with this book is “obvious.” I am a physics major with a minor in mathematics and electronics, perhaps this book is geared entirely toward math majors, however it would have been nice to see explanations instead of solutions followed by the author pointing out the fact that his conclusion was obvious. That said, it is not a bad book. If you are buying the book you are either a grad student or a capable undergrad, and either way you have likely taken stridulous courses in the past. Well, this is another of those courses. Unless you are required to purchase this specific book buy another. There is no reason to make an already difficult class more difficult.

⭐This is one of the clearest and easiest to follow textbooks I have had the pleasure to use in the last year. It happened to be one of the recommended texts for our 2nd year “Geometry of Surfaces Course”.It is well written and fairly well structured, providing a consistently flowing and logical exposition of the material. There are a plenty of diagrams and worked out examples, so one can easily see whether he’s on the ball or not. Apart from worked out exaples directly in the text, there is a wealth of exercies, with solutions (or major hints) at the back. The exercies are almost worth the price of the book by themself, starting from basic ones, checking that one understands definitions, followed by more difficult ones outlining the subtler points of the subject and a couple of rather involved ones; ones which it is easy to spend a whole afternoon with.The author does a good job at pointing the diffucult parts of proofs and constructions. The style is very enthusiastic, which might help motivating the reader. The proofs are sometimes a bit too fast paced for someone who might not be as quick witted as the author when it comes to differetial calculus. I did find certain steps not obvious the first time round. A second or third reading (plus trying to work out the steps on a paper) helps a lot.What is rather important to know, before buying, is that the book is only concerned with curves and surfaces in 3 dimensional euclidean space. The approaches taken here would not, very often, generalise to higher dimensional cases. This means that the material is easily accesible to begginers. At the same time, it is not for people who are after an introduction to coordinate free geometry and manifolds.Topics covered here include: curvature, smooth surfaces, tangents normals & orientability, first & second fundametal forms, curvatures of surfaces, gaussian curvature, geodesics, and the amazing Gauss-Bonet theorem. By the time Gauus-Bonnet theorems are discussed, I had the feeling that the proofs are not as detailed and rigorous as they could be. On the other hand this makes them easier to follow and one is not overwhelmed by techicalities.So why not give it 5? I don’t know, maybe there are other better books out there, maybe I’m just not too keen on differential geometry in general. I guess I might have given it 5, but then maybe Prof. Pressley won’t feel like improving this great book and that would be a shame.

⭐This book of “elementary differential geometry ” is not a self explanatary but it needs to be study with Do Carmo,Differential Geometry of curves and surfaces.No doubt Author A.N. Pressley has tried to write this one with modern point of view,but this book failes to explain many more theories of curves and surfaces.I want summarize to tell thatA.N. Pressley has explained less in his book.

⭐Just a superficial initial look through the contents made me very interested.Elementary means different things to different people. Sometimes it means simple. abut here it means ” the elements. “Here we are greeted with a great detailed exposition. There is no rush for Abstract Formalism to provide a quick road to ” higher ” results.Highly recommended book, especially to those that are working thru a text of ” Mathematical Physics. ”

⭐An alright book for advanced undergraduates. Keep in mind, though, that this is about the classical differential geometry of curves and surfaces. Those who want to learn modern differential geometry, for instance because they want to explore the general theory of relativity, this book does not have what you need.

⭐I like this textbook.It is well written, with only a few minor errors (typos). The notation is clear and fairly easy to understand, considering the level of complexity. Printing quality was ideal, economic without sacrificing readability.This is less expensive than the average textbook, and easier to follow than any other math textbook I have come across.

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