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Manifolds, Tensors, and Forms: An Introduction for Mathematicians and Physicists, by Paul Renteln
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Providing a succinct yet comprehensive treatment of the essentials of modern differential geometry and topology, this book's clear prose and informal style make it accessible to advanced undergraduate and graduate students in mathematics and the physical sciences. The text covers the basics of multilinear algebra, differentiation and integration on manifolds, Lie groups and Lie algebras, homotopy and de Rham cohomology, homology, vector bundles, Riemannian and pseudo-Riemannian geometry, and degree theory. It also features over 250 detailed exercises, and a variety of applications revealing fundamental connections to classical mechanics, electromagnetism (including circuit theory), general relativity and gauge theory. Solutions to the problems are available for instructors at www.cambridge.org/9781107042193.
- Sales Rank: #908550 in Books
- Published on: 2013-12-23
- Original language: English
- Number of items: 1
- Dimensions: 9.69" h x .83" w x 7.44" l, 1.80 pounds
- Binding: Hardcover
- 340 pages
About the Author
Paul Renteln is Professor of Physics in the Department of Physics, California State University, San Bernardino, where he has taught a wide range of courses in physics. He is also Visiting Associate in the Department of Mathematics, California Institute of Technology, where he conducts research into combinatorics.
Most helpful customer reviews
9 of 9 people found the following review helpful.
Great intuition, efficient notation, a joy to read.
By SAHM
On the face of things you might agree with the other reviewers that this covers the same topics as other similar texts... however, to compare this book with another just on the basis of a table of contents is absurd. This book is efficient. The author's definitions and notation are superior to many other texts. The notation and typesetting is modern, crisp, a joy to read. This book is like the text of Flander's in it's ambition to exhibit the power of differential form calculation. But, having spent some time calculating in Flanders, I can assure you this text is far clearer.
Here you find the modern concept of an abstract vector space as well as quotient vector spaces used throughout. The linear algebra shown is a good amount, not overly tedious, not overly terse. He gives two proofs of Stokes' Theorem and clarifies their connection. Both Homology of a smooth manifold and Homotopy are nicely introduced. It's not meant as a reference on these topics, but, it is quite complete and always with references where proofs are omitted.
When he introduces tensors he does so formally, but, without needless digression into universal principles (those can be discussed elsewhere). Then, he follows up by connecting the formal view to that of concrete multilinear maps. Likewise, the wedge product is discussed both from a formal axiomatic perspective, and as it connects to the exterior power of a map. Many similarly ambitious texts have little to offer in their exercise sets. In contrast, Rentlen shines with exercise after exercise which are as lucid as the body of the text. I'm using these to supplement an advanced calculus course I teach this semester. My goal in that course is to bring differential forms to life, this text gives me hope to present topics which occupy the entirety of courses. You can gain much intuition by the efficient introductions to topological topics in this text.
There are too many hidden gems in this text for me to account for in this review. Do not dismiss this text as just another of this type, it is not a fair characterization.
5 of 5 people found the following review helpful.
A strong background in abstract and linear algebra is recommended but not required
By Gabe Jurado
Straight to the point textbook for the mathematically inclined physicist. With respect to the first three chapters, you will encounter more of the theory which builds the foundation for application in the succeeding chapters. A strong background in abstract and linear algebra is recommended but not required. This book provides experienced insight into the relationship between physics and mathematical theory through numerous exercises, necessary examples, and thought provoking hints.
6 of 8 people found the following review helpful.
Looks like just another terse differential geometry book
By A. Ali
Looks like just another terse differential geometry book. I don't see anything novel in either ideas or presentation. For price I would stick with Bishop and Goldberg as it's equally terse but far cheaper. For exposition I would go with Suhubi's "Exterior Analysis" as the author discusses the ideas carefully, with worked examples.
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