- Capa comum: 512 páginas
- Editora: Dover Publications Inc.; Edição: Updated, Revised (27 de janeiro de 2017)
- Idioma: Inglês
- ISBN-10: 0486806995
- ISBN-13: 978-0486806990
- Dimensões do produto: 15,2 x 3,2 x 22,9 cm
- Peso de envio: 721 g
- Avaliação média: 2 avaliações de clientes
- Lista de mais vendidos da Amazon: no. 62,668 em Livros (Conheça o Top 100 na categoria Livros)
Differential Geometry of Curves and Surfaces: Second Edition (Inglês) Capa Comum – 27 jan 2017
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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.
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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.
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A resourceful reader will then discover that tracing out the historical path of development from Euclidean to non-Euclidean Geometries is important to understanding DG: specifically that at the intersection of Affine and Metric Geometry, one finds non-Euclidean Geometry arising when either the metric requirement is relaxed, or the parallel postulate is replaced by a suitable alternative. Either case leads directly to the study of curves and surfaces — the proper province of DG.
In 1854, Bernard Riemann, following Gauss et. al., picked up on these ideas, and added a few of his own, like “manifolds,” “the Riemannian metric,” and “curvature,” and used them to invent an infinite family of non-Euclidean geometries. By formulating them in terms of the differential calculus of curvature tensors, Riemann allowed non-Euclidean geometry to be applied to curves and surfaces at higher dimensions, and thus allowed DG to take its proper place in the kit of tools for investigating the mechanical sciences.
Being able to see how this mid-18th Century “veering away from” the complications of Euclid’s Fifth Postulate led to studying curves and surfaces using differential calculus and curvature tensors, and without having to rely entirely on coordinate geometry, seems to me essential background information for basic understanding of DG itself. Otherwise, one could leave DG as I did 55 years ago, thinking that it was little more than a field of endless theorem-proving headed nowhere.
Even though I managed to pass a first course in DG at Washington University 55 years ago, I had no idea how the substance would ever be used? Now, fifty-five years on, while struggling to understand how Einstein’s field equations describe General Relativity, I have finally found the first practical use for DG.
From those readings, I had roughly figured out on my own that one of the most important uses of DG was describing the curvature of surfaces in n-space without the need to rely on coordinates or vector components. What remaining doubts I had, became crystal clear with this book and “Modern Classical Physics,” by Thorne and Blandford. In an early discussion of “Tensor Algebra without a Coordinate System,” the utility of DG finally came home.
Because, this book too states clearly and early what the goals for DG are — to use differential calculus in the neighborhood of a point to study the local properties of curves and surfaces — I am very pleased with the layout of its content. Later, the local properties whose behavior affect an entire curve or surface, are added and also studied.
Thus, in the clearest of expositions, including annotated diagrams, the necessary theorem proofs, and a graduated set of problems, the author then proceeds to show us just how DG is done. Five Stars