Differential Geometry of Curves and Surfaces: Revised and Updated Second Edition (Dover Books on Mathematics) by Manfredo P. do Carmo (PDF)

Description

One of the most widely used texts in its field, this volume introduces the differential geometry of curves and surfaces in both local and global aspects. The presentation departs from the traditional approach with its more extensive use of elementary linear algebra and its emphasis on basic geometrical facts rather than machinery or random details. Many examples and exercises enhance the clear, well-written exposition, along with hints and answers to some of the problems.The treatment begins with a chapter on curves, followed by explorations of regular surfaces, the geometry of the Gauss map, the intrinsic geometry of surfaces, and global differential geometry. Suitable for advanced undergraduates and graduate students of mathematics, this text’s prerequisites include an undergraduate course in linear algebra and some familiarity with the calculus of several variables. For this second edition, the author has corrected, revised, and updated the entire volume.

User’s Reviews

Editorial Reviews: From the Back Cover One of the most widely used texts in its field, this volume introduces the differential geometry of curves and surfaces in both local and global aspects. The presentation departs from the traditional approach with its more extensive use of elementary linear algebra and its emphasis on basic geometrical facts rather than machinery or random details. Many examples and exercises enhance the clear, well-written exposition, along with hints and answers to some of the problems.The treatment begins with a chapter on curves, followed by explorations of regular surfaces, the geometry of the Gauss map, the intrinsic geometry of surfaces, and global differential geometry. Suitable for advanced undergraduates and graduate students of mathematics, this text’s prerequisites include an undergraduate course in linear algebra and some familiarity with the calculus of several variables. For this second edition, the author has corrected, revised, and updated the entire volume.Dover revised and updated republication of the edition originally published by Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1976.www.doverpublications.com About the Author Manfredo P. do Carmo is a Brazilian mathematician and authority in the very active field of differential geometry. He is an emeritus researcher at Rio’s National Institute for Pure and Applied Mathematics and the author of Differential Forms and Applications.

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

⭐very nice and clear book, but be aware that it does not talk about metrics etc it uses a much simpler notation to explain common ideas of diff geo instead, without even introducing metric tensor, wedge product etc..

⭐Docarmo seems to be used in a lot of differential geometry classes, and this second edition cleans up errors in the earlier edition (and adds a few) and expands on some of the material and problems.The book is clearly for classroom use and less for self-study. A lot (really, a lot) of the material “happens” in the problems, of which there are many. That’s all well and good if you’re in a class, but not if you want to learn on your own.The book lacks a bit of balance. Some things get covered in more detail than others, and it’s not necessarily based on importance. But on the other hand, I found the explanations to be pretty lucid and easy enough to follow. And the big plus is that, at least by the outrageous standard of math texts, the book is inexpensive.Supplement this with on-line material (there’s a lot, and some excellent video lectures) and/or another book. I picked up a copy of J.J. Stoker’s book on Differential Geometry (a little expensive for a used edition but worth it) and found it to be more rigorous and complete, and good for reference and filling in some gaps.I’m probably swimming against the tide but I give Docarmo 4 stars. I don’t think the book rises to the 5 star level.

⭐I brought this book for self learnig. it is indeed a very good material. Another nice thing about it is that it is so popular, you can find tones of supplimental material online, including solution to ex. and dicussion on forum. Its more likely your doubts have been already asked, answered and posted online.Some downside of this book, it is old. Some topic can be presented better

⭐Very good intro to differential geometry. The prerequisites are: a basic course in linear algebra, calculus, and an intro course in real analysis. Anything more than this (such as a course on differential equations) would be helpful but isn’t necessary. With the said prerequisites, you should be able to follow most, if not all, the proofs.Hard cover books obviously tend to be more durable and longer lasting than paperback ones. However, I’m actually pleased with this book. The paper quality and printing is good and the binding is standard quality for a paperback book. If you treat the book reasonably, then the book should definitely last you long enough to read the book inside and out.There are still typos (ranging from grammatical errors to typos such as h_{uu} instead of h_{vu} or a missing “<") but there aren't a lot of them, so confusion due to typos should be minimal.Overall, the math content of the book is perfect for a first introduction to differential geometry. The quality of the physical paperback book is also pretty good, considering you can buy it for a good price of $20. If you have $20 and are trying to learn differential geometry for the very first time, I'd recommend buying this book. ⭐The book is very modern and surely useful for undertanding the most important facts of this theory. For example the concepts of area of a surface, the Gauss map, the geodesic are into an approach which has great attention for the lector. In fact the proofs and the definitions are introduced with good clearness. ⭐Muy buena oferta por su precio y la baja disponibilidad local. Ojalá se amplíe la posibilidad de acceder a textos similares.The price is reasonable. However, the problem sets are a lot harder than expected. You definitely need to know Cal 3 well to do those problems ⭐so much better than Kreyszig's from Dover. This book is clear and goes in depth on the analysis of functions. I like this becuase it helps me review continuous mappings, parametrized surfaces and curves, vector functions, and Real Analysis in general. This book doesnt really use any Tensors (good because I haven't studied Tensors yet). It does use Christoffel Symbols later in the book. ⭐This is a beautiful book, certainly one of my favourites. It talks about the differential geometry of curves and surfaces in real 3-space. If you want a book on manifolds, then this isn't what you're looking for (though it does say something about manifolds at the end); but it is a good book for a course just below that level, or to gain interest and motivation in preparation for a course on manifolds. Top-class exposition, a healthy number of good exercises and lots of very, very nice illustrations. Make no mistake, the book requires some comfort with so-called "advanced calculus" - the reader should be familiar with differentiation of real functions of several variables - but it's certainly a contender for "best book" for a properly prepared student.If you're buying this book to read it in your own time, i.e., it's not for some class you're attending, then you might want to check out the set of lectures on YouTube by Claudio Arezzo, posted on the ICTP channel. Just search "ICTP differential geometry" on YouTube. It's a nice set of recorded lectures based following this book. It's certainly not necessary to read and understand the book, but it will help if you have the time. ⭐Este livro (ou sua tradução para o português) é praticamente obrigatório para qualquer estudante de matemática no Brasil, além de ser largamente adotado no exterior. É um ótimo primeiro livro de geometria diferencial para quem já tem noções sobre álgebra linear e diferenciação de funções em R^n (em particular, o famoso teorema da função inversa). Em alguns momentos, noções de topologia dos espaços métricos são assumidas como conhecidas. Ensina as noções fundamentais sobre curvas e superfícies de maneira fácil de entender, supondo que o leitor tenha os pré-requisitos básicos e serve como preparação para o livro de geometria riemanniana do mesmo autor, Não cabe dar mais detalhes sobre o conteúdo, visto que o público alvo (estudantes de matemática e áreas afins) conhece muito bem este trabalho.Se ler em inglês não for problema, pode ser vantajoso comprar esta edição da Dover no lugar da tradução brasileira. Quando comprei, o preço era quase o mesmo, mas o frete da SBM é muito caro. Por fim, gosto mais da tipografia da versão em inglês, achei visualmente mais agradável.Es un libro clásico para la enseñanza de la geometría diferencial, es claro, aunque deja muchos paso intermedios sin resolver en muchas de las demostraciones, y que no son tan obvia a la hora de ser estudiante. También es importante tener otros libros de referencia, como el O’Neill, ya que en el do Carmo usan otra notación; notación que llega a diferir con la enseñanza de la materia, pero al final es algo equivalente. Cómprenlo y diviértanse con la geometría diferencial. ❤️Livro contempla muito bem a proposta do autor. Tanto que é utilizado como referencia em vários centros universitários ao redor do mundo. A segunda edição mantém a excelente qualidade da primeira edição, acrescentando as correções necessárias.Just like any other text on Differential Geometry, this book as well has "holes". I would recommend to supplement this book with Introduction to Differential Geometry by Andrew Presley and Multivariate Calculus and Geometry by Sean Dineen. Both as well are incomplete. But a combination of all three would help for Differential Geometry in 3D.

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