Introduction to Tensor Analysis and the Calculus of Moving Surfaces

by Pavel Grinfeld

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 20 th 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.

Genres: Mathematics, Nonfiction

315 pages, Hardcover

First published August 17, 2013

Reviews

[WarpDrive · Rating 4 out of 5]

A quite good, conceptually rigorous introduction to tensor calculus. It is not perfect, but probably one of the most readable on the subject, requiring only knowledge of linear algebra and multivariate calculus at undergraduate level.

There are also online lectures that partially mirror the book (albeit with a slightly different sequence of subjects items, see https://www.youtube.com/channel/UCr22...).

The only issues I have with the book are:
- in relation to the number of typos (there are a few, but fortunately there is an unofficial document which contains most of the fixes: http://alexyuffa.com/Miscellaneous/er...)
- in relation to the excessive amount of derivations left as an exercise for the reader. A partial list of exercise solutions is available here: https://www.math.drexel.edu/~dws57/Te...

Also note that this book delivers a general introduction to tensor calculus, and as such it is not particularly targeted at general relativity.

A solid, readable introductory work on the subject, designed at advanced undergraduate level. 4 stars.

[David Grimmer · Rating 4 out of 5]

This book was my first real introduction to those fantastic objects from geometry called tensors, and thank god I read this before being poorly indoctrinated in college lectures. This book is not perfect, it avoids the topological setting upon which tensors are modernly understood. There is no talk of charts, overlap, diffeomorphism, ect. But you do not need any of that unnecessary topology to appreciate what you will learn from this book. If you are a physicist struggling to learn about Relativity theory, you don't need to know about smooth manifolds you need to know how to calculate. This book will teach you the foundations of index gymnastics, how to preform real computations, with just enough mathematical correctness (not exactly formal rigor/but conversationally conveyed mathematical preciseness) to give you the feeling that you know why the index gymnastics works. This book is a perfect first step, you learn mathematics by doing it and applying it to problems in mathematics or physics. A final thing this book includes, not relevant to some, is a steady collection of Mathematical history. From Gauss to Riemann, Levi-Civita to Ricci, and Grossman to Einstein tensors have come through some of the greatest thinkers in human history and solved some of, quite literally, the biggest problems in our universe. Understanding not just how to calculate but the lineage of these things has value as well.

[Curtis Penner · Rating 3 out of 5]

I had expected more. Many of the problems where explain. That is what I expected of the book, not from me. He had only a few examples and those were of little value. He was thorough, to a fault, with very little why are we here in the explanation.

I mainly got the book for the second part of the title, Calculus of Moving Surfaces. With the exception of the Lagrangian / Hamiltonian writing, this was mostly a nothing burger. This is danger of purchasing a book online. You can't flip through it to find if it is worth it. Bookstores are good for general reading, but for technical books, buying them is more of a crap-shoot. This one, not bad, came up short.

[Liquidlasagna · Rating 1 out of 5]

no wait in hell i'm gonna rate this book with nine pages of typos floating around the intertubes with this one....

So much promise, for being an easy 3-4-5 star book too!

Thank goodness Russell and Whitehead's Principia had well-paid proofreaders!

[John David Stanway · Rating 4 out of 5]

As tensor explanations go, this is about as good as it gets. But it speeds up too much at the end. I was pretty much just skimming by the time we got to moving surfaces.

[Yusuke · Rating 5 out of 5]

Great book. There are videos explaining the material