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This is a book that the author wishes had been available to him when he was student. It reflects his interest in knowing (like expert mathematicians) the most relevant mathematics for theoretical physics, but in the style of physicists. This means that one is not facing the study of a collection of definitions, remarks, theorems, corollaries, lemmas, etc. but a narrative — almost like a story being told — that does not impede sophistication and deep results.
It covers differential geometry far beyond what general relativists perceive they need to know. And it introduces readers to other areas of mathematics that are of interest to physicists and mathematicians, but are largely overlooked. Among these is Clifford Algebra and its uses in conjunction with differential forms and moving frames. It opens new research vistas that expand the subject matter.
In an appendix on the classical theory of curves and surfaces, the author slashes not only the main proofs of the traditional approach, which uses vector calculus, but even existing treatments that also use differential forms for the same purpose.
Readership: Physicists and mathematicians working on differential geometry.
- Sales Rank: #2387505 in Books
- Published on: 2014-04-18
- Format: Abridged
- Original language: English
- Number of items: 1
- Dimensions: 9.02" h x .75" w x 5.98" l, 1.25 pounds
- Binding: Hardcover
- 312 pages
From the Inside Flap
This is a book that the author wishes had been available to him when he was student. It reflects his interest in knowing (like expert mathematicians) the most relevant mathematics for theoretical physics, but in the style of physicists. This means that one is not facing the study of a collection of definitions, remarks, theorems, corollaries, lemmas, etc. but a narrative almost like a story being told that does not impede sophistication and deep results.
It covers differential geometry far beyond what general relativists perceive they need to know. And it introduces readers to other areas of mathematics that are of interest to physicists and mathematicians, but are largely overlooked. Among these is Clifford Algebra and its uses in conjunction with differential forms and moving frames. It opens new research vistas that expand the subject matter.
An appendix on the classic theory of curves and surfaces slashes is included. It does not only contain the traditional approach that uses vector calculus, but also the treatments of the subject by those who have already used differential forms for the same purpose.
Most helpful customer reviews
4 of 4 people found the following review helpful.
Interesting Book!
By Jessie Hearle
I'm not done reading and studying this book but I have perused it and read the introductory chapter. I like the author's conversational approach and he has some ideas that he presents that I've not seen in print before, my background being a BS Degree in Applied Mathematics and Physics, with some additional study in Civil and Electrical Engineering. I have studied math through the level of advanced calculus and some advanced 400/500 level courses in differential equations, partial differential equations, special functions of mathematical physics, complex analysis, and vector analysis. I share the author's view that much energy is wasted and the real reason for the math is obscured in making formal proofs and not getting down to the business of applications early on. In short, anyone new to the study of any branch of mathematics would prefer being shown what it can be used for first, then gradually introduced to the proofs later in their studies.
The book contains no problem sets, only a number of illustrative examples scattered throughout the text. It might be better titled "Lectures on Differential Geometry for Physicists and Mathematicians." The author even cites Feynman's "Lectures on Physics" and references a passage from one of Feynman's lectures in the first chapter. In many ways, it is very similar in its approach to Feynman's Lectures, though the material it covers is certainly of much different character and at a somewhat more advanced level, while it remains accessible to perhaps an advanced undergraduate student with the mathematical maturity that would be expected of perhaps an honors student in intermediate calculus or differential equations. By this I mean to say it isn't the easiest read but with a bit of patience, it can be understood.
I have a few criticisms intended for the author. There are some glaring flaws in spelling and English grammar stemming from his mother language obviously being Spanish rather than English. However, the manner of writing, the numerous misspellings, and grammatical errors should in no way be viewed as detrimental to the reading of the book. I quickly made up my mind that the author views getting the material into print far more important than satisfying academia's stuffy approach to making proper sentences. This is mathematics, not a course in English literature after all! Perhaps my biggest criticism is in his overeducated way of throwing complicated words into his sentences where a simpler word would have been a better choice. But having been guilty of a lot of that myself until late in my university education, I know how difficult it can be to trim the fat out of the language as well as sentence structure. Some might consider it arrogance but I'd say that view would be wrong. He is, after all, trying to help in the best way he knows to get some very valuable views into the hands of those who are best equipped to use it.
In short, I find the author very real and down to earth, though rightfully proud of his work. And why shouldn't he be? It is, after all, rather unique!
My suggestions for improvement in a future edition would be the following:
1.) Have someone proof read the manuscript and correct the English, eliminating overly pompous writing in favor of a direct, modern, and very clear style while keeping the tone informal. Simpler is always better when it comes to writing.
2.) Add some graded problem sets at the end of each chapter that contain numerous application examples and brief, detailed solutions to every problem in an appendix at the back of the book. Have the solutions thoroughly checked by a couple of colleagues. Don't try to add proofs. Proofs will only make the book less accessible to the intended audience of physicists and applied mathematicians. If there is a desire to add proof problems, collect all of them in an appendix at the back of the book and include detailed steps to each proof in the solutions appendix..
4 of 4 people found the following review helpful.
Potentially very interesting, but an extremely poorly written book.
By Tullius Agrippa
This book covers many interesting topics and could have been extremely interesting. Unfortunately, it is very poorly written and contains many misleading typographical errors and inconsistencies. More seriously, the author freely adorns his text with the jargon of fibre bundle and frame bundle theory without ever explaining in detail what that jargon means. He also keeps referring to pullbacks and pushforwards without defining what functions are doing the pulling back or the pushing forward, or what are the domains of these functions. This is most unfortunate and extremely frustrating for someone interested in getting a detailed understanding of this fascinating and important theory. Some of the sections, which ought to have been the most interesting, degenerate into a garbled string of indecipherable assertions. It is almost as if the author has been "baptized by the spirit" and feels compelled to speak to us in unknown tongues. The nett effect of these episodic breakdowns is that this book teaches us how to sound like experts without providing us with any expertise of substance.
That said, if one first learns the theory of frame bundles from a superb book like Lawrence Conlon's Differentiable Manifolds (get the superior 2nd edition), one may be able to derive some benefit from Vargas' insights as to how Cartan generalized the Klein geometries to manifolds and developed his own unique (but generally ignored) approach to geometry.
2 of 2 people found the following review helpful.
An excellent textbook - highly recommended!
By High Fly
This is an excellent book for the beginning graduate student or advanced undergraduate student who wants a brief introduction to modern differential geometry. All other textbooks will be way too advanced for the young mathematician. An easy-to-follow ABC of modern differential geometry, differential forms, tensor geometry, Cartan theory, Clifford theory, Kahler calculus, and many related applications. Examples are basic and simple enough. I have used Do Carmo's book on Differential forms (Spring yellow cover thin book) which covers Cartan theory .... as well as "Cartan for beginners" by Ivey and Landsberg -- in the classroom. Those are rigorous, substantial textbooks, but only the very advanced PhD students were able to read them. This introductory book would be a great resource for the talented undergraduate senior student (while studying Mathematics, Physics or Engineering) as well as the beginning graduate student in his/her first 2 years. A good incentive and good starting point for future researchers in the field. The author knows his audience and cares about it.
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