Introduction to Tensor Analysis and the Calculus of Moving Surfaces

by Pavel Grinfeld

Tensor calculus is a must for researchers dealing with natural phenomena as well as for highly qualified engineers working with man-made technological equipment. This textbook was conceived as a self-contained introduction to tensor calculus and grown out of the lecture notes for a course continually taught by the author at Drexel University. The text contains over 150 exercises and is divided into four parts. The first part focuses on tensors in Euclidean spaces, and includes chapters on subjects such as covariant differentiation. The second part focuses on tensors in embedded surfaces, and includes chapters on the curvature tensor and Gauss's theorem. The third part covers applications of tensor calculus and contains chapters on equations of classical mechanics, equations of continuum mechanics, and Einstein's Theory of Relativity. The fourth and final chapter explains the rules and applications of the calculus of moving surfaces. Though the book's approach is informal and avoids a formalization of the subject, it maintains a?respectable level of rigor and reflects the author's deep passion for the subject. Moreover,?it?focuses on concrete objects and appeals to the reader's intuition with regard to such fundamental concepts as the Euclidean space, surface, and length. It is intended for advanced undergraduate students?(and?first-year graduate?students)?in technical fields, and assumes a solid understanding of linear algebra and multivariable calculus.

Genres: Mathematics, Nonfiction

318 pages, Paperback

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