Manifolds, Tensors, and Forms: An Introduction for Mathematicians and Physicists

by Paul Renteln

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.

Genres: Mathematics, Textbooks

340 pages, Hardcover

First published November 21, 2013

Reviews

[Liquidlasagna · Rating 5 out of 5]

amazon reviews

One of the best of its kind

Why do publishers publish such useful books - I've hardly put this one down. It has all that one needs at first year graduate level and beyond. Added to which it is written in a simple and brief, yet not academically/dry/terse style - consequently the reading flows easily, making it easier to absorb the material. The book will also be a useful quick-reference book for me, because of its style.

Ronnie

---

Great intuition, efficient notation, a joy to read.

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 its 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.

SAHM

[Valeh Farzaliyev · Rating 5 out of 5]

Though I had not enough nathematical background fir this book, I enjoyed it. Exercises were challenging and examples were helpful

[Sabrina · Rating 4 out of 5]

I found this book to be an absolutely fascinating read! this was the book that made me love manifolds and tensors!!!