Manifolds, Tensors, and Forms: An Introduction for Mathematicians and Physicists 1st Edition by Paul Renteln (PDF)

Description

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.

User’s Reviews

Editorial Reviews: Book Description Comprehensive treatment of the essentials of modern differential geometry and topology for graduate students in mathematics and the physical sciences. 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.

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

⭐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.

⭐This covers the math at a grad level. Do not attempt at home. The course is for professionals only. More than what you need to know unless you are a mathematician.

⭐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.

⭐The Kindle edition is severely flawed! Do not buy this edition until it is fixed. Many of the equations were not curated: they are only an unreadable smudge. Otherwise this book is very promising and the hardcopy version appears okay.Very best regards, Robert Hosken, hosken@hotmail.com

⭐I think the single best accolade I can give this book is the author’s own admission of the following dilemma: “For many mathematicians, mathematics is the art of avoiding computation. That may be so, but, more often than not, what happens is that a person works out the answer by ugly computation, and then reworks and publishes the answer in a way that hides all the gory details and makes it seem as though he or she knew the answer all along from pure abstract thought.” That being said, a flare for the abstract is highly recommended. This treatment is terse but pedagogical (and therefore intended for graduate students or advanced undergraduates). However, a solid mathematical background is necessary as such books (despite its overzealous claims) do not become easier just by virtue of the author’s particular style of writing. Seeing as it 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 (the list goes on)… I would advise one to take these remarks with a grain of salt.Several approaches are provided in this book that balance highly theoretical concepts with concrete computational tools. However, the writing is DELIBERATELY terse as part of a fast-paced, informal overview of tensor analysis and differential forms on smooth manifolds. While this book may have been written with the intention of serving as the basis for an undergrad/grad math course (i.e. one semester’s worth of study or TEN WEEKS), it is obvious that this text like so many others has suffered mission creep (lest we forget our analysis texts). A little over halfway through the book, there is an introduction to elementary homology theory (simplicial complexes, homology, the Euler characteristics etc.) that IMO you can skip altogether in order to get to the real stuff on integration on manifolds.There is a section on de Rham cohomology that leads naturally into the topology on smooth manifolds. It also contains a (brief) chapter on the homology of continuous manifolds that makes not one iota of difference to an untutored reader. Therefore, some background in point set topology and abstract algebra would certainly be useful. These sections transition nicely into integration on manifolds and, more specifically, Stokes’ theorem where the author discusses different versions of this theorem. You will be exposed to concepts such as vector bundles, connections, covariant derivatives and finally manifolds with metrics. The last chapter provides a brief foray into degree theory, which concludes with a look at the Poincaré-Hopf index theorem and the Gauss-Bonnet theorem.If you are familiar with Riemannian geometry then you will most likely appreciate chapter 8 on geometric manifolds. There is a section on holonomy, which you will remember studying in differential geometry but the author does present more interesting cases such as the holonomy of the Levi-Civita connection in Riemannian geometry that provide motivation for more rigorous study. There is a discussion on the Riemann curvature tensor, Jacobi fields and geodesic deviation, which conclude with a look at Hodge theory to study the cohomology groups of a smooth manifold.This book is surprisingly well-structured with exercises conveniently littered throughout the text and at the end of each chapter that reinforce what you were suppose to learn (you will also find these problems are sometimes revisited again in later parts of the book). Save yourself the trouble and refer to the comprehensive errata for the book that was published by Renteln in April 2017, which you can access online in pdf if you search for “List of Errata for Manifolds, Tensors, and Forms”. Cheers.

⭐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

⭐This book is just too much pick and mix for my taste. It runs through lots of stuff, but too briefly to be satisfying. It may suit some, but I regret the purchase.

⭐Good page quality

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