Differential Forms and Applications (Universitext) by Manfredo P. Do Carmo (PDF)

Description

An application of differential forms for the study of some local and global aspects of the differential geometry of surfaces. Differential forms are introduced in a simple way that will make them attractive to “users” of mathematics. A brief and elementary introduction to differentiable manifolds is given so that the main theorem, namely Stokes’ theorem, can be presented in its natural setting. The applications consist in developing the method of moving frames expounded by E. Cartan to study the local differential geometry of immersed surfaces in R3 as well as the intrinsic geometry of surfaces. This is then collated in the last chapter to present Chern’s proof of the Gauss-Bonnet theorem for compact surfaces.

User’s Reviews

Editorial Reviews: Review M.P. Do CarmoDifferential Forms and Applications”This book treats differential forms and uses them to study some local and global aspects of differential geometry of surfaces. Each chapter is followed by interesting exercises. Thus, this is an ideal book for a one-semester course.”―ACTA SCIENTIARUM MATHEMATICARUM

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

⭐Mathematicians have a penchant for underestimating the amount of background necessary to understand something. The text, like Spivak’s, claims that elementary calculus is all that is needed, but the level of sophistication is far above that.This is a good, clean development of the topics in the last two chapters of Calculus on Manifolds by Spivak, plus elementary applications in differential geometry (which is why I bought the book). The writing and the choice of notation are clearer than Spivak, although even less motivated, if that were possible. Someone reading this for self-study without an previous knowledge of differential forms would not find it useful for developing an intuitive understanding of the subject. However, it may be a good text if there is a talented instructor. It is also good if you have some background (e.g., Hubbard and Hubbard’s text) and want to get a more rigorous understanding of differential forms.

⭐This book is short, which could have been good. But it fails to be clear and even lacks explanation of notations, which is in fact a formidable hurdle for beginners. Compared to Loring Tu’s ‘An Introduction to Manifolds’, this book doesn’t seem nearly as friendly.

⭐Helpful

⭐This is a short book suitable for people who have little background in differential forms, but need a quick and concise development of it.

⭐On August 1, 2011, Congress amended the CPSIA to exclude ordinary books and ordinary paper-based printed materials from the third party testing and certification requirements of the CPSIA. An “ordinary book” as defined by the amendment, means: “a book printed on paper or cardboard, printed with inks or toners, and bound and finished using a conventional method, and that is intended to be read or has educational value.” H.R.2715.The manufacturer of the book containing this web address only manufactures ordinary books bound and finished in a conventional manner; thus, the books produced by this manufacturer are not subject to the testing and certification requirements of the CPSIA.Do Carmo es el autor de un clásico de la geometria diferencial para novatos. Es un libro a tener en cuenta pero yo empezaria por “Differential forms: a complement to vector calculus” de Weintraub y “Advanced calculus: a differential forms approach” H.M. Edwards.

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