Where should I start with tensors and their applications?

[cmcraes]

I'm not really sure where I should post this forum in particular so I guess I'll just put it here haha.

My questions: What are tensors in general? What are they used for? What Mathematics do I need to understand well, before I begin to learn tensor mathematics? Also does anyone know a good online resource I could use to begin learning? Thanks a ton guys!

 

[WannabeNewton]

This might get more responses if posted in the academic guidance section. Tensors aren't really related to calculus in their bare form (they are algebraic objects). Tensors have an insanely wide range of use; they are ubiquitous in fundamental physics for example (general relativity, QFT, QG etc.). It would actually help if you stated for what use you want to learn them i.e. is this for physics, for pure math, or for some other application? Different texts exist on tensors/tensor analysis for different uses.

At the bare minimum, you would need to have a very good handle on linear algebra to start learning the proper theory of tensors. You will usually see tensors being developed in intermediate to advanced linear algebra texts (Roman comes to mind on the advanced side).

 

[cmcraes]

Okay thank you Ill go post it there! And I'm planning to learn it so that I can eventually truly understand Einstein's Field Equations; but for now I just want to know the basics, so I can work my way up from there.

 

[micromass]

Okay thank you Ill go post it there! And I'm planning to learn it so that I can eventually truly understand Einstein's Field Equations; but for now I just want to know the basics, so I can work my way up from there.

Why don't you get a good GR book, such as Schutz? They teach everything you need to know.

 

[WannabeNewton]

No need to repost, it's already been moved there. What is your background in mathematics i.e. what's the highest level of math you have gotten up to?

 

[cmcraes]

My highest level I think is In the middle of second year calculus; So far I've mastered Everything from limits to Line integral and now I'm beginning to learn Gradient, Divergence and Curl.

 

[micromass]

My highest level I think is In the middle of second year calculus; So far I've mastered Everything from limits to Line integral and now I'm beginning to learn Gradient, Divergence and Curl.

Be sure to master multivariable calculus and linear algebra first. Then you can start with tensors and GR.

 

[ahsanxr]

Depends on what approach you want to take. I think for a first approach learning it without the linear algebra, (i.e description as multilinear maps) as objects that "transform" in a certain way is probably fine. You'll learn how to do basic computations and the important formulas, which should give you enough to understand the basics of General Relativity. For such an approach I think you should be quite comfortable with multi-variable calculus and working with several variables and coordinate systems. I didn't learn tensors like this, but people who do think of them this way seem to do fine when it comes to the practical, physics-y computations, but are usually in the conceptual dark and find tensors as mysterious and confusing objects.

The approach where you learn everything "properly" will require you to know some topics in abstract linear algebra, such as multilinear maps, dual spaces etc, and differential geometry because when physicists usually say "tensors" they actually mean "tensor fields". Such an approach requires some background in math, but it'll give you a good conceptual understanding of tensors and will also shed light on the mysterious transformation properties that they obey. This is the approach I would recommend.

 

[cmcraes]

Okay thanks a ton ahsanxr! :)

 

[vanhees71]

I only wonder, how tensors should make any sense without linear algebra. After all they are multilinear forms in a vector space. If anything is linear algebra then it's tensors.

In the physics context it's of course very important to deal with their behavior under invertible linear transformations which form the group GL(n,R) or GL(n,C) (general linear group in the n-dimensional real or complex vector space) or one of its subgroups. The most important ones are the rotation group SO(3) in the 3D Euclidean space and the Minkowski group SO(1,3) [or SO(3,1), depending on the sign convention used] in 4D pseudo-Euclidean spacetime. This is sufficient for Newtonian and special relativistic physics.

If you want to learn general relativity you need differential geometry on top, i.e., some idea what a differentiable manifold and the special case of Riemannian and pseudo-Riemannian manifolds are.

 

[WannabeNewton]

Depends on what approach you want to take. I think for a first approach learning it without the linear algebra, (i.e description as multilinear maps) as objects that "transform" in a certain way is probably fine.

I disagree completely. This is a terribly outdated way to learn it. Linear algebra is a must.

 

[romsofia]

I think a book you should pick up is Mathematical Methods in the Physical Sciences by Mary Boas. This will allow you to get some LA, and then move onto tensors if you so please. It's also a good book for physics majors anyway.

 

[ahsanxr]

I disagree completely. This is a terribly outdated way to learn it. Linear algebra is a must.

Of course you say this as a math-physics double major, but not all physics students are pure math majors as well. Most physics students don't learn about bi and multilinear maps, dual spaces, and the behavior of these objects under linear transformations and it would simply be too distracting to learn all these when it's quite time-consuming (especially if your audience isn't mathematically mature) and at the end, it would most likely be used in a non-essential way, at least in an elementary treatment of General Relativity. All I'm saying is that if you're really eager to learn about the basics of GR, a first approach like this would be OK as long as you go back and relearn everything rigorously.

As an example, introducing the metric tensor as a symmetric matrix that allows you to measure "distance" on a "space" sounds a lot less scarier (from the perspective of a physics student) than saying it's a non-degenerate symmetric tensor field of rank (0,2) acting on the tangent space of a semi-riemannian manifold, when in most physics contexts, it does no harm to think of it as the former.

Of course I recommend the linear algebra approach to everyone, because I learned it that way and thus as a result never understood the ghost stories about tensors that physics students tell each other. But I mean to learn about the basics of curved spaces, Einstein's equations, the Schwarzschild solution and FRW cosmology, does one really need all that mathematical machinery?

 

[WannabeNewton]

After seeing many threads on the topic of "covariant" and "contravariant" vectors, it is quite obvious that linear algebra is essential in order to properly learn tensors at any decent level. Theoretical linear algebra is an extremely important subject in any regards, not just in the study of tensors. Sugar coating the math leads to more misconceptions than it does benefits.

 

[micromass]

After seeing many threads on the topic of "covariant" and "contravariant" vectors, it is quite obvious that linear algebra is essential in order to properly learn tensors at any decent level. Theoretical linear algebra is an extremely important subject in any regards, not just in the study of tensors. Sugar coating the math leads to more misconceptions than it does benefits.

And really, things like multivariate linear maps and dual spaces are not that complicated. You can learn it rather quickly, and it will save you time since later misconceptions won't arise. Furthermore, those are topics that are also useful in other domains of physics and math.

It's not like we're expecting him to read a math book on differential geometry or something.

 

[Aero51]

For a preliminary review, "Functional and Structured Tensor Analysis For Engineers" is a good free PDF you can find on the web. The tensor generally has different definitions depending on who you talk too. From an engineering standpoint a tensor is essentially a matrix that obeys special transformation rules.

 

[lurflurf]

After seeing many threads on the topic of "covariant" and "contravariant" vectors, it is quite obvious that linear algebra is essential in order to properly learn tensors at any decent level. Theoretical linear algebra is an extremely important subject in any regards, not just in the study of tensors. Sugar coating the math leads to more misconceptions than it does benefits.

Yes Wald mentions this problem here.
http://arxiv.org/abs/gr-qc/0511073
Students actually learn something in linear algebra, who knew. Unfortunately what they learn is a special case that causes much confusion later when they refuse to abandon it. Another case where being a bit more general from the start is advantageous. I wish I could remember which blog I read that made a case for introducing dual space as early as possible (as I have many times). Many popular linear algebra books were compared and put into order by introduction of dual space. The books that introduced dual space earliest were most effective.

 

[WannabeNewton]

Haha I was thinking of that exact same article by Robert Wald! It certainly puts things in perspective. Also, learning theoretical LA helps out with QM so why not kill two birds with one stone. Wald makes a great point though: compared to the level of mathematics required to rigorously understand QM, the math required to rigorously understand GR pretty much pales in comparison.

 

[vanhees71]

Of course you say this as a math-physics double major, but not all physics students are pure math majors as well. Most physics students don't learn about bi and multilinear maps, dual spaces, and the behavior of these objects under linear transformations and it would simply be too distracting to learn all these when it's quite time-consuming (especially if your audience isn't mathematically mature) and at the end, it would most likely be used in a non-essential way, at least in an elementary treatment of General Relativity. All I'm saying is that if you're really eager to learn about the basics of GR, a first approach like this would be OK as long as you go back and relearn everything rigorously.

This is a big mistake in physics didactics! In my undergrad studies I had a very hard time in the beginning, because the physics professors used hand-waving pseudomath to handle vector calculus (div, grad, curl, line, area, volume integrals, and all that). Only when I heard the linear-algebra lectures (2 semesters!) and vector calculus in the analysis lectures from the mathematicians, I had a revelation about the meaning of all that. In my university (Technical University Darmstadt) physicists had to take the math course together with the mathematicians, and there was no course "Mathematics for Physicists" available, and that's great luck (at least for theoretical physicists, who need some refined understanding of the math used in physics).

 

[WannabeNewton]

Anyways, once you have your LA and Calc 3 down (very important!) and you have a basic understanding of some advanced math, there are a wealth of texts/notes you can use to start dealing with tensor calculus/algebra in the context of GR.

At an elementary level (as far as the math goes), you might like: https://www.amazon.com/dp/0805386629/?tag=pfamazon01-20 and/or https://www.amazon.com/dp/0521887054/?tag=pfamazon01-20 although these focus more on the direct physics of GR (which is good of course).

There's also Carroll's book (more advanced than the aforementioned texts): https://www.amazon.com/dp/0805387323/?tag=pfamazon01-20 It's a pretty awesome text and will get you up to speed on tensor calculus/algebra in GR.

At a more advanced level, I very much like the following notes because they have a thorough introduction to tensor calculus/algebra as used in GR: http://www.socsci.uci.edu/~dmalamen/bio/GR.pdf
and also Robert Geroch's notes on GR as well as differential geometry: http://home.uchicago.edu/~geroch/Links_to_Notes.html (1st and 2nd links).

Keep in mind that if your goal is to eventually understand GR (as you said earlier) then you can't just know the math required for it. You also need to know the pre-requisite physics i.e. classical mechanics and electrodynamics. I don't know if you already know these things or not but make sure to keep that in mind.

 

[micromass]

Anyways, once you have your LA and Calc 3 down (very important!) and you have a basic understanding of some advanced math, there are a wealth of texts/notes you can use to start dealing with tensor calculus/algebra in the context of GR.

At an elementary level (as far as the math goes), you might like: https://www.amazon.com/dp/0805386629/?tag=pfamazon01-20 and/or https://www.amazon.com/dp/0521887054/?tag=pfamazon01-20 although these focus more on the direct physics of GR (which is good of course).

There's also Carroll's book (more advanced than the aforementioned texts): https://www.amazon.com/dp/0805387323/?tag=pfamazon01-20 It's a pretty awesome text and will get you up to speed on tensor calculus/algebra in GR.

At a more advanced level, I very much like the following notes because they have a thorough introduction to tensor calculus/algebra as used in GR: http://www.socsci.uci.edu/~dmalamen/bio/GR.pdf
and also Robert Geroch's notes on GR as well as differential geometry: http://home.uchicago.edu/~geroch/Links_to_Notes.html (1st and 2nd links).

Keep in mind that if your goal is to eventually understand GR (as you said earlier) then you can't just know the math required for it. You also need to know the pre-requisite physics i.e. classical mechanics and electrodynamics. I don't know if you already know these things or not but make sure to keep that in mind.

I can't shake off the feeling that you forgot a book...

 

[WannabeNewton]

I can't shake off the feeling that you forgot a book...

Haha, well as much as I love Wald's text I can't recommend it for learning tensor calculus because it is just way too terse and dry. Malament's notes (albeit being just as dry as Wald's text) and Geroch's notes do things at a similar level but are much more thorough and I think Geroch offers much more geometrical insight anyways; he has a knack for geometrical insight.

 

[micromass]

 

[cmcraes]

So would it be better to learn about LA ambiguously or Read books on it applying to GR first? I'm guessing the former would be better (Because I intend to do a joint Math-Physics major) But I'm just making sure.

 

[micromass]

So would it be better to learn about LA ambiguously or Read books on it applying to GR first? I'm guessing the former would be better (Because I intend to do a joint Math-Physics major) But I'm just making sure.

Yes, studying rigorous LA will be incredibly helpful to you. Not only for GR, but also for other kinds of physics like QM.

 

[ahsanxr]

For the Linear Algebra, I would strongly recommend that you either take a proof-based LA class or go through chapters 1,2 (especially 2.6), 4,5,6 of "Linear Algebra" by Friedberg, Insel and Spence. All these would be very useful for QM and GR.