Tensors Introductory Textbook

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Tensors Introductory Textbook

#1  Postby AlanF » Apr 10, 2021 6:52 pm

Any suggestions for a good introductory textbook on tensors?
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Re: Tensors Introductory Textbook

#2  Postby Cito di Pense » Apr 12, 2021 3:41 pm

AlanF wrote:Any suggestions for a good introductory textbook on tensors?


I hate to see a post like this languishing unanswered. Maybe the short answer is that tensor analysis is not an introductory topic. Most treatments are going to assume you've already been exposed to vector spaces and linear transformations. Harold Jeffreys published a rather thin volume in 1931, entitled Cartesian Tensors, but after the first couple of chapters, the material seems to assume you've mastered multivariable calculus and the applications are in dynamics. Jeffreys was a geophysicist, so his research interests were in topics like elasticity theory. The relations between stress and strain in continuous media is the way many students first approach the topic. The physics of space-time is probably not where you want to dig in for the first time.

If that doesn't seem like what you want, there are any number of basic introductory videos at youtube on the topic of how tensors are defined and (intuitively) how they are applied.
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Re: Tensors Introductory Textbook

#3  Postby AlanF » Apr 12, 2021 6:28 pm

Cito di Pense wrote:
AlanF wrote:Any suggestions for a good introductory textbook on tensors?


I hate to see a post like this languishing unanswered. Maybe the short answer is that tensor analysis is not an introductory topic. Most treatments are going to assume you've already been exposed to vector spaces and linear transformations. Harold Jeffreys published a rather thin volume in 1931, entitled Cartesian Tensors, but after the first couple of chapters, the material seems to assume you've mastered multivariable calculus and the applications are in dynamics. Jeffreys was a geophysicist, so his research interests were in topics like elasticity theory. The relations between stress and strain in continuous media is the way many students first approach the topic. The physics of space-time is probably not where you want to dig in for the first time.

If that doesn't seem like what you want, there are any number of basic introductory videos at youtube on the topic of how tensors are defined and (intuitively) how they are applied.


Thanks for responding! Your points are well taken. I'm a retired Microchip designer with an MSEE, and have taken a bunch of graduate math/physics courses, but never had a need to learn much about tensors. Now that I have some time, I'm trying to learn stuff that I never got around to before.

I found a YouTube series on tensors by Pavel Grinfeld and am working my way through it. Just ordered his book from Amazon, since it got good reviews.
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Re: Tensors Introductory Textbook

#4  Postby Calilasseia » Nov 13, 2021 2:14 pm

If you're looking for an elementary course on tensors, then the book:

Vector Analysis, With an Introduction To Tensor analysis by Murray R. Spiegel, Schaum's Outline Series, McGraw-Hill Publishing, ISBN-10 07 084378 3, is one I would recommend. Might be somewhat on the terse side, but it contains numerous solved problems illustrating the thinking underlying tensors.

However, there is one BIG issue to raise here, namely that this book concentrates upon the coordinate based approach to tensors. If you want a book that covers the coordinate free treatment of tensors, then you need to look elsewhere.

Likewise, if you're thinking of pursuing tensors all the way to the Ricci Calculus, you'll need to look for other texts.

However, the good news about the Spiegel book, is that it teaches you quickly that a tensor representation isn't just valid in one choice of coordinate system, but in all possible coordinate systems, which is what makes tensors so massively powerful.

Quite simply, you can choose whichever coordinate system is most convenient for you, the moment you have a tensor form of an equation, because that tensor form represents the quantities in question in all possible coordinate systems, in one compact, dense notation. Though it's precisely because that notation is compact and dense, that it'll take some time learning to understand said notation.

Head off to Chapter 8 in the Spiegel book, and have fun. It'll take you to the stage where you can, at the end of the chapter, begin to understand how general relativity works, though it's a fairly steep learning curve if you don't already have a fair amount of vector analysis under your belt first.
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Re: Tensors Introductory Textbook

#5  Postby hackenslash » Nov 15, 2021 8:40 am

This is a good potted introduction from a course on General Relativity, with more in subsequent videos.



Worth noting that the prior video in the series - the first - is a more general geometric approach to spacetime.
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