Tensor Analysis Basics for Non-Experts

[quasar_4]

Okay, second question - does anyone know of a good text that covers the basics of manipulating tensors without assuming a very extensive math background? I am reasonably good at linear algebra and taking a differential geometry class, but that's about the extent of my math background. I am in a multilinear algebra class but we aren't using a textbook which makes it very hard to study (my notes tend to become kind of hazy when not in class). Any good recommendations?

 

[robphy]

My favorites [in classical tensor notation] are
Synge & Schild "Tensor calculus"
and Schouten "Tensor Analysis for Physicists",
both are available from Dover.

 

[Daverz]

Tensor analysis books emphasize analysis rather than algebra, so they won't help much at all with the multilinear algebra course.

For the diff. geom course, it depends on how advanced it gets. The Schaum's Outline of Diff. Geom. is pretty useful. Though it's completely classical in approach, that approach is very concrete. There's also a Schaum's Outline of Tensor Analysis.

 

[robphy]

The texts I mention develop the algebra and the analysis...
and they are intended to be introductions to tensors... for physicists.

In particular, there are some discussions of counting independent components when the tensor has symmetric or antisymmetric indices.
https://www.amazon.com/gp/product/0486636127/?tag=pfamazon01-20
https://www.amazon.com/gp/product/0486655822/?tag=pfamazon01-20

While these are useful guides, it's still up to you to become familiar with techniques for doing the calculations.

 

[Daverz]

The texts I mention develop the algebra and the analysis...
and they are intended to be introductions to tensors... for physicists.

Yeah, but multilinear algebra is really a different kettle of fish. Take a look at the TOC of this book. More like the material physicists learn in advanced QM courses.

 

[robphy]

Yeah, but multilinear algebra is really a different kettle of fish. Take a look at the TOC of this book. More like the material physicists learn in advanced QM courses.

Ok.
I was going on the request for "the basics of manipulating tensors" and the fact that this forum is "Tensor Analysis & Differential Geometry". So, maybe the question belongs elsewhere.

In any case, from a google search for Greub's book, these came up:
http://www.math.niu.edu/~rusin/known-math/98/multilin_alg
https://www.physicsforums.com/showthread.php?t=161630

 

[MathematicalPhysicist]

on the same issue anyone knows anything about the book on holors, i read on wiki that's the only book on this issue, is it worth the money, i mean if it's a generalization of tensors it should be worthwile for everyone, but only one book of it makes me suspicious about its genuity and if it's really worth the money, i mean only one book on holors sounds not good for me, i.e perhaps this generalization isn't useful in neither of maths nor physics.

 

[robphy]

on the same issue anyone knows anything about the book on holors, i read on wiki that's the only book on this issue, is it worth the money, i mean if it's a generalization of tensors it should be worthwile for everyone, but only one book of it makes me suspicious about its genuity and if it's really worth the money, i mean only one book on holors sounds not good for me, i.e perhaps this generalization isn't useful in neither of maths nor physics.

I remember peeking into this book a while back... but decided that this would have to take a backseat to other approaches of more interest to me (e.g. differential forms, then maybe geometric algebra). Given the finite amount of time and effort that one can devote to something, one should ask the question... of what use is it [to me, a physicist]? Is there some "killer-application" [borrowing from the software-tech world] of this formalism?

You can look inside:
https://www.amazon.com/gp/product/0521019001/?tag=pfamazon01-20
and read some reviews:
https://www.amazon.com/dp/0521019001/?tag=pfamazon01-20

As you suggested earlier ( https://www.physicsforums.com/showthread.php?t=25661 ), you might try to find it in a library first. Try http://www.worldcatlibraries.org/wcpa/top3mset/11916192
I see you have already seen the item at Amazon.

It seems there are some familiar applications:
http://www.csa.com/partners/viewrecord.php?requester=gs&collection=TRD&recid=A8220504AH
http://staff.um.edu.mt/ccam1/ (follow "Tensor Analysis" link)
and not-so-familiar applications:
http://jn.nutrition.org/cgi/content/abstract/104/12/1535

 

[quasar_4]

hmm. The multilinear class covers tensors, but we're eventually moving towards representation theory. I'm in the class mostly for the tensor analysis though, so I can eventually do general relativity. I don't exactly know how representation theory fits into physics, though I'm sure it has some place... any way, we have a text already for that part of the course. But our instructor doesn't have any text for the tensor part or the bilinear forms. That's where I am the most confused right now!

 

[Chris Hillman]

Suggest some textbooks

does anyone know of a good text that covers the basics of manipulating tensors without assuming a very extensive math background?

Tensors, or tensor fields?

I am reasonably good at linear algebra and taking a differential geometry class, but that's about the extent of my math background. I am in a multilinear algebra class but we aren't using a textbook which makes it very hard to study (my notes tend to become kind of hazy when not in class). Any good recommendations?

D. G. Northcutt, Multilinear Algebra, Cambridge University Press, 1984, or Werner Greub, Multilinear Algebra, 2nd edition, Springer, 1987, should be more than enough. For multilinear algegra applied to representations of groups, try R. Shaw, Linear algebra and group representations, Academic Press, 1983.

 

[quasar_4]

thanks. :)

we haven't done anything with tensor fields. Seems to me that it's good to get a good understanding of tensors before moving on to tensor fields.

 

[Chris Hillman]

D'accord.

 

[MathematicalPhysicist]

I remember peeking into this book a while back... but decided that this would have to take a backseat to other approaches of more interest to me (e.g. differential forms, then maybe geometric algebra). Given the finite amount of time and effort that one can devote to something, one should ask the question... of what use is it [to me, a physicist]? Is there some "killer-application" [borrowing from the software-tech world] of this formalism?

You can look inside:
https://www.amazon.com/gp/product/0521019001/?tag=pfamazon01-20
and read some reviews:
https://www.amazon.com/dp/0521019001/?tag=pfamazon01-20

As you suggested earlier ( https://www.physicsforums.com/showthread.php?t=25661 ), you might try to find it in a library first. Try http://www.worldcatlibraries.org/wcpa/top3mset/11916192
I see you have already seen the item at Amazon.

It seems there are some familiar applications:
http://www.csa.com/partners/viewrecord.php?requester=gs&collection=TRD&recid=A8220504AH
http://staff.um.edu.mt/ccam1/ (follow "Tensor Analysis" link)
and not-so-familiar applications:
http://jn.nutrition.org/cgi/content/abstract/104/12/1535

well i myself don't have time to read it, but i wondered how come there's only one book on this issue, which as i said makes me think this is unpopular approach.

p.s
thanks for linking the post from 2004, i remember we had this conversation, but still only one person commented about reading the book.

 

[matthewbanar]

I recently had a good introduction to tensors (primarily cartesian proofs and general tensor transformation laws) in my Math Methods for Engineers course, and the detailed course reading can be found here:

http://comp.uark.edu/~icjong/4703/in&ct06.pdf

Hopefully that will help.

-mtm