Intro to tensor material - If you have time please

[pmb_phy]

Hi folks

I am reworking my website to (1) so that each page will have same font, style etc of any other page and (2) look for mistakes that I missed the first time around. The purpose of the web pages is to help others to learn the math and physics that the come to the web to learn.

I fear that some of the pages might be unclear in some places. I would like to ask for your help in this respect. If you have the time and inclination then please take a look at either or both of the following newly reconstructed web pages

http://www.geocities.com/physics_world/ma/geodesics.htm
http://www.geocities.com/physics_world/ma/tensors_via_analytic.htm
http://www.geocities.com/physics_world/ma/tensor_via_geometric.htm

All constructive criticism is welcome. Thank you.

 

[cliowa]

My overall impression of your sites is okay, but there are a few things I'd like to point out:
1) You give 2 different definitions of tensors. Wouldn't it be nice to show that they actually amount to the same thing? I feel that's quite important.
2) Concerning the "analytic" approach: I find that rather unsatisfying and unconcinving. The little jump you execute when you talk about the infinitesimal arc length (around Eq.14) and its approximations to suddenly having a completely well-behaved nice formula makes me personally feel like I'm missing the point. Frankly speaking, I don't think that in the context of the analytic description you would naturally introduce the concept of tensors. It seems almost completely useless.
3) Concerning the geometric approach: At the very beginning you present the definition of a manifold, although that isn't used later on. On the contrary: You make heavy use of specific properties of the reals, for example when defining what a vector is. Furthermore I find your definition of tensors (Eq. 32 and so on) too restrictive. The vectors do not have to be 4-vectors.

I have one more question: Are you familiar with the more abstract stuff like vector spaces? If you are, I can only suggest presenting tensors in that setting, as it greatly helped me. I found it very useful to first see the definition of a tensor without any special underlying vector space. That helps the reader distinguish what part really is tensors and what comes in from the environment.
I suggest you look over these things and your site will be well worth reading.

 

[uiulic]

pmb_phy,

First great thanks to the linked websites by your efforts. I hope that that your websites will eventually help everyone understand tensor completely.

I have some pieces of advice (which may improve in your websites) as follows

1 What is prequisite knowledge in order to learn tensor. I think for different levels, the prequisite will differ, e.g. one can learn the tensor by the coordinate transformation with calculus and basic matrix/linear algebra...

2 As Cliowa pointed out, it is important to show what is difference(or relation) of the definitions of tensors. Are they describing the same thing?

3 The different definitions of tensors may include e.g. under coordinate transformations (you call it analytical), vector spaces (multilinear algebra object, you call it geometric),in the expressions of component (sometimes matrix) and component-free (such as dyadic) description

4 As to that under coordinate transformations used by many physics and engineering applications, the word "admissible transformation" seems often omitted often.Therefore, such a definition is "conditional"? Or the word "transformation" also indicated "admissible" ?

5 Tensor bundle (Chris Hillman, a good guy, long time no see, often used this word) seems an appropriate item for "tensor geometry". An element of the bundle is a "tensor field" (I guess so)? Should this be included in your "geometrical definition" and be further explained?

6 I want to point out that the English word tensor seems related to elasticity (but Love did not call it this way), the usage of which was later generally adopted in continuum mechanics (e.g. Prager , Truesdell ,Malvern, Eringen, Fung ...). The stress and strain tensors are the most frequently used words related to "tensor" . If one has a full understanding of tensor, he yet was puzzled what is stress/strain tensor. Then tensor is useless.

7 References. I can list some roughly by my memory, but they may not be referenced by you and they are mostly only for tensor. Maybe some more references on the prequisite knowledge for understanding the tensor geometry will be better.

Schaum (2 books, one by Spiegel, the other by Kay),
Dover (3 books, one translated by Silverman, one by lovelock, one by Bishop and Goldburg),
Sokolnikoff (1964 tensor analysis),
Malvern (1969, very good continuum mechanics book which explained the tensor for its purpose),
Fung (1965, foundation of solid mechanics, again a good book on continuum mechanics like Malvern's book, which explained the tenosr in a very concise manner)
Eringen (edited in continuum physics, and it claims to be a treatise)
Tensor geometry (Christopher T.J. Dodson and Timothy Poston)
Frankel, Geometry of physics (as Chris Hillman suggested)
A brief on tensor analysis (UTM for undergraduate)
Mathemetical physics (tensors are explained in many such texts)

8 Finally, you should put a link in each of your own texts on different definitions. Not everyone is so lucky to see this thread with the three links together.


UU

 

[pmb_phy]

My overall impression of your sites is okay, but there are a few things I'd like to point out:
1) You give 2 different definitions of tensors. Wouldn't it be nice to show that they actually amount to the same thing? I feel that's quite important.

Thanks I forgot to do that. Although I'm not sure where to place that. Any ideas?

2) Concerning the "analytic" approach: I find that rather unsatisfying and unconcinving.

The web page is not intended to convince you of something per se. Its supposed to teach. But I am confused about your use of the term "unconcinving." What was it that you were unconvinced about?

The little jump you execute when you talk about the infinitesimal arc length (around Eq.14) and its approximations to suddenly having a completely well-behaved nice formula makes me personally feel like I'm missing the point.

Take a guess as to what you think the reader was supposed to learn from that.

Frankly speaking, I don't think that in the context of the analytic description you would naturally introduce the concept of tensors. It seems almost completely useless.

The use of the analytic view is widely used and therefore I wrote a page explaining it. If the person who reads that page is then no longer totally confused when he reads material which contains that type of math then the page did its job.

3) Concerning the geometric approach: At the very beginning you present the definition of a manifold, although that isn't used later on.

Thanks. I was going to use it but forgot. I'll go back and do what I had intended to do.

The vectors do not have to be 4-vectors.

Thanks. I'm so used to referring to 4-vectors I forgot the web page was about tensors in general. Thanks.

I have one more question: Are you familiar with the more abstract stuff like vector spaces?

I should. I have a BA in math.

Thanks

Pete

 

[pmb_phy]

pmb_phy,

First great thanks to the linked websites by your efforts. I hope that that your websites will eventually help everyone understand tensor completely.

Thanks. Each web page on my website is still a work in progress.

I have some pieces of advice (which may improve in your websites) as follows

1 What is prequisite knowledge in order to learn tensor. I think for different levels, the prequisite will differ, e.g. one can learn the tensor by the coordinate transformation with calculus and basic matrix/linear algebra...

Calculus and linear algebra.

2 As Cliowa pointed out, it is important to show what is difference(or relation) of the definitions of tensors. Are they describing the same thing?

Yes. Otherwise I would never have called them different views. The name of each page is "Tensors". They each go on to say whether the specific page is discussing it from the analytic view or from the geometric view. Was that unclear to you from the title of the pages?

3 The different definitions of tensors may include e.g. under coordinate transformations (you call it analytical), vector spaces (multilinear algebra object, you call it geometric),in the expressions of component (sometimes matrix) and component-free (such as dyadic) description

I don't follow what you're saying. Please elaborate.

4 As to that under coordinate transformations used by many physics and engineering applications, the word "admissible transformation" seems often omitted often.Therefore, such a definition is "conditional"? Or the word "transformation" also indicated "admissible" ?

Yep. You are 100% correct. I was going to mention that but I forgot to. Thanks.

7 References. I can list some roughly by my memory, but they may not be referenced by you and they are mostly only for tensor. Maybe some more references on the prequisite knowledge for understanding the tensor geometry will be better.

I'll look into it.

Schaum (2 books, one by Spiegel, the other by Kay),
Dover (3 books, one translated by Silverman, one by lovelock, one by Bishop and Goldburg),
Sokolnikoff (1964 tensor analysis),
Malvern (1969, very good continuum mechanics book which explained the tensor for its purpose),
Fung (1965, foundation of solid mechanics, again a good book on continuum mechanics like Malvern's book, which explained the tenosr in a very concise manner)
Eringen (edited in continuum physics, and it claims to be a treatise)
Tensor geometry (Christopher T.J. Dodson and Timothy Poston)
a brief on tensor analysis (UTM for undergraduate)
Mathemetical physics (tensors are explained in many such texts)

Sorry but I have a rule that I do not reference a text I've never read or never heard of.

8 Finally, you should put a link in each of your own texts on different definitions. Not everyone is so lucky to see this thread with the three links together.

What three links are you talking about?

Thank you. You've been quite helpful.

Best regards

Pete

 

[cliowa]

The web page is not intended to convince you of something per se. Its supposed to teach. But I am confused about your use of the term "unconcinving." What was it that you were unconvinced about?

The analytic thing is just not rigorous math. If you knew where you were going, you could have done it yourself, as a student, but if you didn't, there's no chance you would have ever come up with the ds stuff. My point simply is that one introduces here objects which don't have any rigorous meaning in that context; what's a infinitesimal arc length supposed to be?
I'm well aware of the fact that this is not your fault, as it's done often. To me it seems that anybody doing that should put a disclaimer somewhere saying that it's all just motivation for studying things rigorously. Thereby you can avoid that the reader gets the feeling he's not understanding the explanation properly, because there simply are some parts which he cannot really understand, you know what I mean? Otherwise it seems to me you're fooling people a little.

Take a guess as to what you think the reader was supposed to learn from that.

I have absolutely no idea. Should it show them, that it's all not really ok? If so, I personally would state that explicitly.

The use of the analytic view is widely used and therefore I wrote a page explaining it. If the person who reads that page is then no longer totally confused when he reads material which contains that type of math then the page did its job.

As I wrote, if you add a disclaimer it's all perfect.

I should. I have a BA in math.

That is great. Then I would definitely introduce tensors in a more general fashion, taking just some arbitrary vector space. I feel that makes it more clear what part is tensors and what part is due to a certain underlying space. Take your definitions of vectors as an example: The reader may be inclined to think that all this vector stuff is important for the concept of tensors, because you build up the page in that spirit ("First we need to understand vectors, 1-forms, then we can understand tensors").
Best regards...Cliowa

 

[uiulic]

pmb_phy ,

Thanks for the clarification of the points.

I need to elaborate the following things after your first reply.

1. Regarding the geometrical approach to the definition of "tensor", I only feel that your are dealing with the "geometry of the tensor components" because you can show to visualize e.g. covarient and contravarient components (can be visualised if it is a "vector") given the basis vectors. However, does tensor itself depend upon any reference basis vectors (or any background coordinate systems) ? If tensor itself depends on such references, then why physicists design tensor analysis in physical sense? Therefore, my conclusion is that the geometrical approach to the tensor defined by physicists (coordination transformation, as well as used in differential geometry) is impossible in this sense. I would like to say Physicists "feel" that there is such an quantities since their experiments and intuition tell them that time a scalar, force a vector and stress a tensor... Therefore, they "create" things like tensors and we are forced to use the rules the physicists specify. As to convariance and contravarience, that is designed for convenience, so that in dealing with non-Rectangular-Cartesian coordinatons the treatment can make them "comfortable". While such treatment seems more physical than mathematical, because you cannot prove MATHEMATICALLY to us why a tensorial quantity is one coordinate system is like this and in another coordinate system is like that and their components in different coordinate systems transform like that, EXCEPT that you put some extra conditions to the quantities you physicists described. Mathematicians. however, do not seem like this idea :extensively taking it as a formal mathematical definitions in a general sense? While in differential geometry, the story may be different in mathematicians' eyes. And, tensor defined in vector space has mathematical rigor. If you don't agree, then you need to show me force is vector (stress is a tensor) according to your definitions and let others see what there may be the "definition'''s problem.

2 What's more, the different approaches to the definitions have been argued upon by cliowa. You need to "deal with" cliowa first. If the tensor defined by yours mean the same things exactly, then...

3 Dyadic (a linear combination of dyad, used in Malvern's book). I would rather not explain it to make things worse at the moment.

4 The contemporary mathematical tensor seems to be created by Weyl (or Cartan in some aspects). The previous physicists seem to adopt that defined by Ricci and in differential geometry.

5 Forget about the three links I mentioned. I later found that your website is very good, and I can actually read your photo (like a professor) as well as many other interesting topics if I go back from the address you gave.

GG

 

[pmb_phy]

Note: The web pages on my website are not written to be of the same vigour as one might see in a text. It is a reference page for me (or whoever) to point to if/when someone asks Huh? What's a tensor?? It has no other purpose besides that other than for me to practice writing a good physcs web page. As I said before, these web pages are living breathing pages and will evolve in time when a reader is unable to understand what is written on a page or if I believe I can increase the rigour and still asume the average reader is the same. These pages are constructed to the best of my ability given I have no idea of the readers background. I prefer not to place prerequisites before the document. I am hoping that people will tell me where things are unclear. If/when that time comes I'll try to find a better way to rephrase things.

The analytic thing is just not rigorous math.

Rigorous enough for who? Its a main problem writing these kinds of pages because I don't know the background of the reader. It'd seem to me best to try to get to the largest crowd I can while not introducing higher level math than is required. When I learned about tensors from texts who taught them the analytic way I really didn't have a problem understanding it. So far I see no major problems.

If you knew where you were going, you could have done it yourself, as a student, but if you didn't, there's no chance you would have ever come up with the ds stuff.

Since I disagree with your assertion that I don't where I'm going then I will not be unable to respond to that comment.

My point simply is that one introduces here objects which don't have any rigorous meaning in that context; what's a infinitesimal arc length supposed to be?

Consider a curve on a manifold on which there is a metric defined. In infinisesimal arc length is a infinitesimally small section (arc) of a curve. This section is referred to as an infinitesimal arc. The length of this section, as defined by the metric, is referred to as the infinitesimal arc-length. I remain confused as to your comments on this. It seems to me that you knew the answer. Are you trying to say that I didn't explain that? If so then its not a problem in my opinion because I assume the reader can visualize that an infinitesimal arc length approximates a line and the length of that line is definied in terms of the Euclidean metric. I suppose it might have been better to add that to the page. Therefore I will put it on my list of changes. Thank you. But next time please be more clear and let's not play games. If you really don't know what an infinitesimal arc length is then I doubt that you're versed enough to give constructive criticism on the subject.

I'm well aware of the fact that this is not your fault, as it's done often.

I also see nothing wrong with what's there and I understand that you do. Therefore on this point we'll have to agree to disagree.

Thereby you can avoid that the reader gets the feeling he's not understanding the explanation properly, because there simply are some parts which he cannot really understand, you know what I mean?

Not really, no. especially when you're speaking about a typical reader who might come across this page. I don't see what you mean. Please take note of what I stated in the web page that right before the definition of the metric tensor I stated The ideas above motivate the following definition...

That is great. Then I would definitely introduce tensors in a more general fashion, taking just some arbitrary vector space.

What reason do I have to assume that an arbitrary reader is familiar with linear algebra? I do believe that the web page needs work in this area in that the argibtary reader may not understand why any vector can be expanded as a linear sum of basis vectors. I suppose that I can state than when I first write the expression. I believe that will clear things up.

By the way. You referred to stress as a tensor. To really know that to be true you'd have had to show that the components of the stress tensor transform according to the definition of an tensor, i.e. as their components transform under a change in basis. The nice thing about the geometric view is that all one needs to show is that the supposed tensor can map 1-forms and vectors to the set of real numbers. The relationship I was lax in describing. So thanks for noting that (it was you right?)

I feel that makes it more clear what part is tensors and what part is due to a certain underlying space. Take your definitions of vectors as an example: The reader may be inclined to think that all this vector stuff is important for the concept of tensors, because you build up the page in that spirit ("First we need to understand vectors, 1-forms, then we can understand tensors").
Best regards...Cliowa[/QUOTE]Tensors are defined by the way their components transform and that comes from the transformation equations for the vector and 1-form. I believe its a bit different in the geometric view in that the transformation equations are derived from the transformations of the basis vectors and basis 1-form.

Best regards

Pete

 

[pmb_phy]

pmb_phy ,

Thanks for the clarification of the points.

Thanks for your help.

I need to elaborate the following things after your first reply.

1. Regarding the geometrical approach to the definition of "tensor", I only feel that your are dealing with the "geometry of the tensor components" ...

The section called "Tensors" states the definition of tensors in the most accurate way that I know of. There is no mention of components there. The only place where I get into components is the section entitled Tensor Components

...because you can show to visualize e.g. covarient and contravarient components (can be visualised if it is a "vector") given the basis vectors.

I created a web page which includes a visual representation of vectors. Thanks for pointing this out. It seems to have disappeared from the web site. I'll correct that mistake. I'll create a new web page to get into this. However the visual representation of 1-forms is something I never quite liked. Perhaps I'll include it so that I will understand why I don't like it.

However, does tensor itself depend upon any reference basis vectors (or any background coordinate systems) ?

Not in principle. Otherwise it wouldn't be called a geometric object. It may us components when it comes down to actual calculations though. Hence my reasons for making that section of Tensor Components.

If tensor itself depends on such references, ...

If you mean coordinate systems then I've explained above that the definition of a tensor doesn't depend on that. Therefore I've ignored the comments which assumes dependence on coordinate systems.

Therefore, my conclusion is that the geometrical approach to the tensor defined by physicists (coordination transformation, as well as used in differential geometry) is impossible in this sense.

Have you ever been exposed to the geometric view of tensors? IT can be found in texts such as Gravitation, by Misner, Thorne and Wheeler. Its worth reading. If you don't have the text I would gladly scan the parts of the definition of "tensor" from the geometric point of view. I'll place it on my website if I can find space. So when you tell me that you're interested I will do exactly that.

I would like to say Physicists "feel" that there is such an quantities since their experiments and intuition tell them that time a scalar, force a vector and stress a tensor...

Is that what I feel?

Therefore, they "create" things like tensors and we are forced to use the rules the physicists specify.

I believe it was mathematicians who defined the concept first but I'm not sure.

As to convariance and contravarience, that is designed for convenience, so that in dealing with non-Rectangular-Cartesian coordinatons the treatment can make them "comfortable".

They do it because the physics/geometry makes more sense that way.

While such treatment seems more physical than mathematical, because you cannot prove MATHEMATICALLY to us why a tensorial quantity is one coordinate system is like this and in another coordinate system is like that and their components in different coordinate systems transform like that, ..

It need not be proved since it is part of the definition or simply derived from it.

Dyadic..

I learned the term a very long time ago but don't recall ever having to use it, at least under that name.

I later found that your website is very good, and I can actually read your photo (like a professor) as well as many other interesting topics if I go back from the address you gave.

That's very kind of you to say. Thanks.

Best wishes

Pete

ps - I won't be able to get around to the changes I mentioned that I should make since at the moment I'm working on redoing the web page on the Christoffel symbols

 

[cliowa]

I'm sorry to hear that you feel I'm playing games, which clearly was not what I was intending to do. As you observed, there are some issues on which we'll disagree. Let's leave it that way. There is however one thing on which I would like to elaborate:

Consider a curve on a manifold on which there is a metric defined. In infinisesimal arc length is a infinitesimally small section (arc) of a curve. This section is referred to as an infinitesimal arc. The length of this section, as defined by the metric, is referred to as the infinitesimal arc-length.

1st problem: If you take a section of a curve on a manifold, i.e. some part of the curve lying in the manifold, you can't possibly measure its length using the metric. The metric operates on tangent vectors.
2nd problem: Wikipedia states that "Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them.". So, are you saying the length of your infinitesimal arc will be some finite number? If that's the case the two of us have a very different understanding of infinitesimal.

If you really don't know what an infinitesimal arc length is then I doubt that you're versed enough to give constructive criticism on the subject.

I really have no rigorous understanding of that concept. But why shouldn't I be able to give constructive criticism?
Best regards...Cliowa

P.S.: One more thing: If you're talking about tensors on manifolds, wouldn't you also have to require that they be linear over the functions on the manifold? I think so.

 

[uiulic]

"Gravitation, by Misner, Thorne and Wheeler. Its worth reading.If you don't have the text I would gladly scan the parts of the definition of "tensor" from the geometric point of view. I'll place it on my website if I can find space. So when you tell me that you're interested I will do exactly that."


Thanks for the reference. It is enough for me to remember it down for my future reference. You needn't do it specially for me, but thank you for your kindness anyway.

 

[uiulic]

Pete, I don't see cliowa is playing games. His criticism may be a great help.You, as a physicist, should convince mathematicians of your good definition as much as possible.

cliowa, But you seem to have criticize too seriously in some aspect. Pete's intention of the websites is very good.

I hope there is no further misunderstanding.

 

[pmb_phy]

1st problem: If you take a section of a curve on a manifold, i.e. some part of the curve lying in the manifold, you can't possibly measure its length using the metric. The metric operates on tangent vectors.

This is what I was referring to before. This is one of the main staples of differential geometry and tensors, especially the metric. You obviously knew what the arc length of a curve was. You just thought it didn't have a meaning in that you believe that "The metric operates on tangent vectors" prohibits. The infinitesimal arc length, ds, is defined by ds^2 = g(dx, dx). So while the displacement vector may lie on the tangent plane one can still add then arc lengths. By the way, when someone is discussing a curve, the curve is always in the manifold.

2nd problem: Wikipedia states that "Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them.". So, are you saying the length of your infinitesimal arc will be some finite number? If that's the case the two of us have a very different understanding of infinitesimal.

I never go by Wikipedia as a valid reference. But surely you're aware of what the integral of ds means? The infinite sum of infinitesimals is finite number. That's calculus 101! If you need more then I recommend Tensors, Differential Forms, and Variational Principles by LoveLock and Rund. Go to section labeled Riemannian Geometry and se page 241.

I really have no rigorous understanding of that concept. But why shouldn't I be able to give constructive criticism?

Nobody said you couldn't.

P.S.: One more thing: If you're talking about tensors on manifolds, wouldn't you also have to require that they be linear over the functions on the manifold? I think so.

Linearity applies to all vectors and 1-forms which are defined at the same point where the tensor is defined (A tensor being defined over the manifold is obviously a tensor field) and any give tensor at any given point on the manifold is always a multilinear function of its arguements . Is that what you meant.

Best regards

Pete

 

[cliowa]

It now seems pretty clear to me where our opinions diverge. You accept "inifitesimals" and their manipulation as a mathematically sound concept and I don't. I'll try to explain my point of view in the following, so you can choose to ignore it or comment on it.

As far as I understand, all this dx stuff does not have a rigorous basis as long as you don't talk about the dual space to...etc.

The infinitesimal arc length, ds, is defined by ds^2 = g(dx, dx).

To me, the ds^2 is purely notation for the sum of those g(dx,dx), not an infinitesimal squared (whatever that might be...).

The infinite sum of infinitesimals is finite number.

To me this is meaningless.

Linearity applies to all vectors and 1-forms which are defined at the same point where the tensor is defined (A tensor being defined over the manifold is obviously a tensor field) and any give tensor at any given point on the manifold is always a multilinear function of its arguements . Is that what you meant.

No, actually that's not what I meant. I wanted to express that a tensor on a manifold is a multilinear mapping, which is multilinear over the functions on the manifold, i.e. functions can be factored out of the arguments. Not everything that is multilinear over the reals is a tensor on a manifold.

Thanks for your reference. In case you want to look up my statements, I can only recommend the fantastic book "Semi-Riemannian Geometry with Applications to Relativity" by Barrett O'Neill. It's got an extremely well written chapter on tensors on manifolds.

Best regards...Cliowa

 

[pmb_phy]

It now seems pretty clear to me where our opinions diverge.

I agree. So let us agree to disagree and end this thread. Thanks for your input. It was helpful annd I will make some changes to it due to comments made in this thread. In fact I plan on making a new page to introduce the definition of manifolds then change the other pages so bhe more accurate by using that terminology. Perhaps it will help the beginning reader. At least that is the goal.

Next - I've overhauled another web page. The link is
http://www.geocities.com/physics_world/ma/parallel_transport

Pete

 

[pmb_phy]

In case you want to look up my statements, I can only recommend the fantastic book "Semi-Riemannian Geometry with Applications to Relativity" by Barrett O'Neill.

I'd love to read about what you were saying from that text. Can you send it to me someway? E.g. can you xerox it and send it to my home or scan it and e-mail the scan to me?

I'll PM my address you you now.

Thanks cliowa

Best regards

Pete