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Introduction to Tensor Analysis and the Calculus of Moving Surfaces, by Pavel Grinfeld
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This textbook is distinguished from other texts on the subject by the depth of the presentation and the discussion of the calculus of moving surfaces, which is an extension of tensor calculus to deforming manifolds.
Designed for advanced undergraduate and graduate students, this text invites its audience to take a fresh look at previously learned material through the prism of tensor calculus. Once the framework is mastered, the student is introduced to new material which includes differential geometry on manifolds, shape optimization, boundary perturbation and dynamic fluid film equations.
The language of tensors, originally championed by Einstein, is as fundamental as the languages of calculus and linear algebra and is one that every technical scientist ought to speak. The tensor technique, invented at the turn of the 20th century, is now considered classical. Yet, as the author shows, it remains remarkably vital and relevant. The author’s skilled lecturing capabilities are evident by the inclusion of insightful examples and a plethora of exercises. A great deal of material is devoted to the geometric fundamentals, the mechanics of change of variables, the proper use of the tensor notation and the discussion of the interplay between algebra and geometry. The early chapters have many words and few equations. The definition of a tensor comes only in Chapter 6 – when the reader is ready for it. While this text maintains a consistent level of rigor, it takes great care to avoid formalizing the subject.
The last part of the textbook is devoted to the Calculus of Moving Surfaces. It is the first textbook exposition of this important technique and is one of the gems of this text. A number of exciting applications of the calculus are presented including shape optimization, boundary perturbation of boundary value problems and dynamic fluid film equations developed by the author in recent years. Furthermore, the moving surfaces framework is used to offer new derivations of classical results such as the geodesic equation and the celebrated Gauss-Bonnet theorem.
- Sales Rank: #156956 in Books
- Published on: 2013-09-24
- Original language: English
- Number of items: 1
- Dimensions: 9.21" h x .75" w x 6.14" l, .0 pounds
- Binding: Hardcover
- 302 pages
Review
From the book reviews:
“The textbook is meant for advanced undergraduate and graduate audiences. It is a common language among scientists and can help students from technical fields see their respective fields in a new and exiting way.” (Maido Rahula, zbMATH, Vol. 1300, 2015)
“This book attempts to give careful attention to the advice of both Cartan and Weyl and to present a clear geometric picture along with an effective and elegant analytical technique … . it should be emphasized that this book deepens its readers’ understanding of vector calculus, differential geometry, and related subjects in applied mathematics. Both undergraduate and graduate students have a chance to take a fresh look at previously learned material through the prism of tensor calculus.” (Andrew Bucki, Mathematical Reviews, November, 2014) From the Back Cover
This text is meant to deepen its readers’ understanding of vector calculus, differential geometry and related subjects in applied mathematics. Designed for advanced undergraduate and graduate students, this text invites its audience to take a fresh look at previously learned material through the prism of tensor calculus. Once the framework is mastered, the student is introduced to new material which includes differential geometry on manifolds, shape optimization, boundary perturbation, and dynamic fluid film equations.
Tensor calculus is a powerful tool that combines the geometric and analytical perspectives and enables us to take full advantage of the computational utility of coordinate systems. The tensor approach can be of benefit to members of all technical sciences including mathematics and all engineering disciplines. If calculus and linear algebra are central to the reader’s scientific endeavors, tensor calculus is indispensable. The language of tensors, originally championed by Einstein, is as fundamental as the languages of calculus and linear algebra and is one that every technical scientist ought to speak. The tensor technique, invented at the turn of the 20th century, is now considered classical. Yet, as the author shows, it remains remarkably vital and relevant. The author’s skilled lecturing capabilities are evident by the inclusion of insightful examples and a plethora of exercises. A great deal of material is devoted to the geometric fundamentals, the mechanics of change of variables, the proper use of the tensor notation, and the discussion of the interplay between algebra and geometry. The early chapters have many words and few equations. The definition of a tensor comes only in Chapter 6 – when the reader is ready for it. While this text maintains a reasonable level of rigor, it takes great care to avoid formalizing the subject.
The last part of the textbook is devoted to the calculus of moving surfaces. It is the first textbook exposition of this important technique and is one of the gems of this text. A number of exciting applications of the calculus are presented including shape optimization, boundary perturbation of boundary value problems, and dynamic fluid film equations developed by the author in recent years. Furthermore, the moving surfaces framework is used to offer new derivations of classical results such as the geodesic equation and the celebrated Gauss–Bonnet theorem.
About the Author
Pavel Grinfeld is currently a professor of mathematics at Drexel University, teaching courses in linear algebra, tensor analysis, numerical computation, and financial mathematics. Drexel is interested in recording Grinfeld's lectures on tensor calculus and his course is becoming increasingly popular. Visit Professor Grinfeld's series of lectures on tensor calculus on YouTube's playlist: http://bit.ly/1lc2JiY http://bit.ly/1lc2JiY
Also view the author's Forum/Errata/Solution Manual (Coming soon): http://bit.ly/1nerfEf
The author has published in a number of journals including 'Journal of Geometry and Symmetry in Physics' and 'Numerical Functional Analysis and Optimization'. Grinfeld received his PhD from MIT under Gilbert Strang.
Most helpful customer reviews
65 of 66 people found the following review helpful.
Awesome Tensor Analysis Book
By Alex J. Yuffa
This review applies to the Hardcover edition.
I definitely wish this book was available when I was doing my PhD in physics. Having used tensor notation in grad. school in various courses I was still hopelessly confused. There are number of Tensor Analysis books available (some of which, unfortunately, I have purchased) but they all, roughly, fall into one of these categories:
1. Too mathy, i.e., even if you can follow the proof of some theorem you still have no idea what was just proven and why you should care. These books typically also have something related to Differential Geometry in the title.
2. Very applied books, e.g., books on General Relativity will typically devote a good portion of the book to tensor analysis. Needless to say, they view Tensor Analysis as a tool and thus, don't painstakingly explain it.
I tried learning tensor analysis from the above two categories but, for the most part, failed, i.e., learned the rules of moving indices around but had no real idea as to what I was actually doing. This brings me to Pavel Grinfeld's "Introduction to Tensor Analysis and the Calculus of Moving Surfaces" book, which is simply the best. Currently, I'm working through Chapter 11 and almost everything has been crystal clear to me thus far. I did have to work through the exercises, which are strategically placed throughout the text. The book is fairly rigorous and may be understood with minimal background (Calculus and some Linear Algebra).
Here is what I like the best about the book:
1. Clear explanation of the notation and why this or that notation works so well.
2. NO awful phrases like "as one can clearly see" or "as it can be shown."
3. Derivations are carried out in great detail with a verbal explanation between each major step. No need to guess, why did the author do step 2 before step 3. The derivations are first outlined and then carried out in a pedantic fashion. This is really good, trust me!
4. Exercises are well chosen to solidify the newly acquired knowledge.
5. Examples and not generalizing things for the sake of generality. I think it is much easier to learn things in 2D and 3D and then see how it can be analogously extended into n-D.
Some remarks to previous reviewers:
1. Yes, the book has some typos; no big deal. Most of them are obvious if you read the book with paper and pencil. If you want to read this book as a novel or use it to simply look up formulas then look elsewhere. This book is NOT an encyclopedia. Here is a link to the typos I have found [...]
2. There is no solution manual as far as I know. But, you don't need one. Most of exercises are of type "show A = B." For other exercises, simply skip forward a section or two and you will find the solution and the reason for the exercise.
Other FREE Goodies:
1. Youtube Lecture videos by Pavel Grinfeld himself. His lectures a very good. I watch them when I need a hint or have a question. He answers a lot of questions in his lectures. He is a link to the videos: [...]
2. Forum devoted to Tensor Calculus [...] Doesn't appear to be very active but has some useful info. and a place to ask questions.
24 of 24 people found the following review helpful.
Clear introduction to tensor analysis
By B. DeZonia
This book is the third book I've read that touches on the subject of tensor analysis. Previously I had read Schaum's Outline on Vector Analysis which had a chapter on tensor analysis that piqued my curiosity. I then delved more deeply by reading most of Borisenko's Vector and Tensor Analysis with Applications. I found Grinfeld's book in my search for a more clear and thorough explanation of tensor analysis.
Within Grinfeld there is a good mix of exposition and derivation. One thing Grinfeld's book is good at is thoroughly explaining the various notations presented in the literature. This was a great aid in clearing up questions I had after reading the other two works. Grinfeld is also good at building up concepts methodically and completely. And the conversational tone of the text allowed me to survey the subject without getting bogged down. There are about 350 exercises (that I did not tackle) but I did notice there are no solutions given.
In my copy there are numerous typos throughout. However the typos do not appear in any formulas and are in the expository sections of the text. They are actually just poor edits and it is usually easy to see what the author meant to say.
Overall I enjoyed this work, learned a lot, and would recommend it to others. I'm docking it one star because of the typos.
22 of 24 people found the following review helpful.
If you'd like learn differential geometry, this is your book
By Alex Benjamin
This textbook focuses on tensors, but it is really a textbook on elementary differential geometry. It starts with a super-intuitive description of Euclidean spaces and goes on to cover the most critical concepts in differential geometry and vector calculus: directional derivative, gradient, general coordinate systems, Laplacians, and an outstanding discussion of curvature of mean and Gaussian curvature.
The study of moving surfaces is an unexpected bonus. I was excited to learn that moving surfaces can be described by essentially the same framework as stationary geometry. I discovered a number of unexpected applications of moving surfaces: a proof of the Gauss Bonnet theorem, and shape optimization. The textbook also includes a discussion of fluid films, which are apparently the author's own research. I must admit I didn't understand much of that.
The book's style takes a while to get used to. There are no theorems or proofs, just a continuous narrative mixed with exercises. The lack of a solution manual makes the open ended exercises difficult.
In summary, this textbook was an eye-opener for me regarding the tensor technique and gave me an entirely new (and better) view on differential geometry and vector calculus.
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