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Intro to tensor material  If you have time please... 
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#1
Jul2407, 06:08 PM

P: 2,954

Hi folks
I am reworking my web site 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_wor...a_analytic.htm http://www.geocities.com/physics_wor..._geometric.htm All constructive criticism is welcome. Thank you. 


#2
Jul2507, 03:42 AM

P: 191

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 wellbehaved 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 4vectors. 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. 


#3
Jul2507, 06:10 AM

P: 96

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 componentfree (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 mechancis (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 


#4
Jul2607, 01:20 AM

P: 2,954

Intro to tensor material  If you have time please...
Thanks Pete 


#5
Jul2607, 01:34 AM

P: 2,954

Thank you. You've been quite helpful. Best regards Pete 


#6
Jul2607, 03:39 AM

P: 191

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. Best regards...Cliowa 


#7
Jul2607, 08:16 AM

P: 96

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 nonRectangularCartesian 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 


#8
Jul2607, 08:52 PM

P: 2,954

Note: The web pages on my web site 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.
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 1forms 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, 1forms, 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 1form. 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 1form. Best regards Pete 


#9
Jul2607, 10:12 PM

P: 2,954

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 


#10
Jul2707, 02:53 AM

P: 191

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:
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. 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. 


#11
Jul2707, 03:07 AM

P: 96

"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 web site 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. 


#12
Jul2707, 03:19 AM

P: 96

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. 


#13
Jul2707, 05:26 AM

P: 2,954

Best regards Pete 


#14
Jul2707, 08:03 AM

P: 191

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


#15
Jul2707, 08:08 PM

P: 2,954

Next  I've overhauled another web page. The link is http://www.geocities.com/physics_wor...llel_transport Pete 


#16
Jul2707, 08:24 PM

P: 2,954

I'll PM my address you you now. Thanks cliowa Best regards Pete 


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