1085:(4) Flesh out the sections on notation. Half the battle is just understanding the notation. For example, the main tensor page has a section on the various notations. I assume that the "Abstract Index Notation" is the "main" notation used now days, because it is listed first. But when you go to that stub, it doesn't actually show you the notation at all. It just says that Penrose invented it and it is "better." Similarly, if you follow the next link to the Einstein notation, you learn about Einstein's summation convention, but nothing about the basic tensor notation. Finally, after a lot of jumping around, I found the "intermediate" page, which gives the best explanation of the basic notation (although I am still scratching my head after following the links to the covariant and contravariant links).
1079:(2) I would suggest that you gently introduce other highly technical supporting subject matter by limiting the number of cross links early in the discussion. Of course, when the subject itself is complex (like tensors), there is obviously going to be certain prerequisite areas of knowledge. Perhaps it would be best to explicitly state that the reader should be familiar with x, y, and z (with supporting links) in order to understand the following discussion. I would keep this list as short as possible.
782:, and poorly formatted, but the modern treatment is mixed up in it. No!!! This is completely inaccessible! Who did this!? It must be undone. I'm sure there may be some good pieces in this new part, but that doesn't justify all the other crap. Whoever it was, please revert, or someone else will have to go thru the trouble of doing so. And be a little more considerate and less self-righteous next time. Pay more attention to presention, rather than brute information.
568:. If the reader cannot clearly identify their equavalency, which is explicitly stated, clearly marked, and geometrically neccessary, then it is clear that they have not geometrically comprehended the material, and thus the manner of presentation is failing. It is of little practicality for one to recognize the equivalence of two things that they cannot understand. However, once they understand them, the equavalence is obvious and trivial.
1082:(3) I know that this is a somewhat nebulous concept, but I would try to cultivate a "beginners mind" when sharing your knowlege. In other words, keep in mind that a lot of us have no idea of what you are talking about once you get beyond basic algebra (especially in reference to mathematical formalisms). For example, the "dual vector space U*" is introduced with no explanation and no cross-link. I still don't know what this is.
1076:(1) I would urge you to present some very simple and "intuitive" example of tensors (or whatever concept you are trying to explain) early in the non-technical introduction. For example, the page on eigenvectors presents a marvelous and very intuitive example (the "face") early in the presentation. If a tensor is "like" a matrix, show me a tensor, and the corresponding matrix. How do I get from the tensor to the matrix, etc.
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inner product operation (which would involve a metric tensor in
Riemmanian geometry). Covariant, on the other hand, would be where both tensors are superscripted or both are subscripted, resulting in something like a diffeomorphism. Or an outer product, depending on whether indices are shared between (or among) the tensors.
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Really?? I suppose a tensor of rank 0 would not be the product of rank 1 tensors, but can you name any other examples? Saying it's not always the product is like saying that a natural number is not always the sum of adding 1 with itself multiple times. And that directly contradicts the definition
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In any event, I have a basic background in math and science (a couple of years of calculus, diff. eq., classical physics, etc.) but nothing very advanced. For example, while I have heard the terms, I never studied manifolds, eigenvalues, tensors, groups, etc. So, that is where I am. Would I not be
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So that you can understand where I am coming from, I will give you a little background on myself. Many years ago, I got a BS in engineering, but I don't work professionaly as an engineer. Since I was a kid, I have been fascinated by science (primarily physics) and have read many popular accounts of
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The political agenda reference was not pointed at you. Since the classical method is being deprecated, there are some who think that it should be quit altogether. It is very difficult, however, to learn the modern approach without a developed geometric intuition, which the classical approach does a
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I've always understood the prefixes contra- and co- to refer to a relationship between two things. I've understood contravariant to be a relationship between two tensors (not an absolute property of one), such that one of the tensors is a superscripted and the other is subscripted, resulting in an
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I don't understand where your reference to a political agenda comes from. The modern and the classical approach are really talking about the same thing, but from different angles. They need to be explained in the same article so that the reader sees how and why they are about the same thing. By the
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In general, the value of a tensor field at an event in spacetime is an element of a vector space which is the tensor product of multiple copies of the tangent space (contravariant vectors) and multiple copies of the cotangent space (covariant vectors). As such, it is a smooth (C∞) mapping from the
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I think the point is not that an n-rank tensor is neccesarily a tensor product of rank 1 tensors, but rather that one can construct an n-rank tensor in this fashion. It should be intuitively clear that, since a tensor product increases the number of neccessary dimensions to specify the resultant
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You obviously are in agreement with me that both methods must be presented. We are, then, only in disagreement on a more subtle point: whether the two treatments( which are, ofcourse, about the same thing) should be presented simultaneously or in parrallel. I would image that an encyclopedia in
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Before I get into all of that, let me say how much I appreciate that "you people" (the smart guys who are writing all of this stuff) have taken the time to share their knowledge in this forum. This is truly a fantastic resource, and demonstrates how the
Internet can be used for a great and
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This page (the
Classical Treatment of Tensors) is impossible to understand from the perspective of someone (me) with a B.S. in Math but who does not already know what a tensor is. Also, why does the discussion button for the "classical treatment of tensors" redirect to a page titled
1088:(5) I do think that the whole tensor section has gotten a little too difuse and disconnected (with the whole "classical," "intermediate" and "modern" approaches). Perhaps this is necessary, but it makes things rather confusing when you are first approaching the subject like I am.
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This article is confusing. First statement: tensors generalise vectors and matrices. Fine: a vector is a 1-dimensional array of numbers, a matrix is a 2-dimensional array of numbers. The only obvious generalisation is to an n-dimensional array of numbers, n integral, possibly :
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The reason it was split from 'tensor', is that it is a different treatment. Merging the two would be like merging 'Marxism' with 'Plato's
Republic'. It would be very confusing, and also would make for an excessively long article. If this article is unclear, then the
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There is a small, recurrent contradiction in this article. If a (p,q) tensor has contravariant rank p and covariant rank q, then the subscript and superscript indices have been mixed up. It should be in the superscript and in the subscript, right?
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I image it's to be meta-syntactical. I suppose they can be just taken out because people probably get that they can put anything there. A better question: why does the classical treatment of tensors's talk page redirect to this one?
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Surely if the tensor is the tensor product of lots of rank-1 tensors, each with d.o.f. = dimension, it's the latter? On the other hand, if we think about the representing array, it's the former, but am I missing something here? --
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Also, keep in mind that this is the starting page for the classical treatment, not the final page. You write as if they already knew everything about tensors, in which case, why the hell would they be reading this?
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Failed to parse (syntax error): {\displaystyle \mathcal{U}^{p,q} = \overbrace{\mathcal{U}\otimes\ldots\otimes\mathcal{U}}^{p\mbox{ times}} \otimes \overbrace{\mathcal{U}^*\otimes\ldots\otimes\mathcal{U}^*}^{q\mbox{
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Instead of trying to push a political agenda, maybe we should attempt to clearly present the information, with a focus on developing concepts( as opposed to constructing a rigourous self-referential mathematical
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a transformation, for example, of coordinates. I'm not sure that representation is identity; however, perhaps this is just quibbling. Provided the representation is unique, that may do for practical purposes.
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So, in relativity say, dimension = 4 (space-time dimensions). A curvature tensor has four indices. What does that mean? It means we have 4x4x4x4 components to account for. The exponential is right, therefore.
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physical theories (relativity, QED, etc). In fact, I am currently reading Roger
Penrose's book (Road to Reality). Trying to understand some of the concepts in that book is what brought me to this page.
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There was plenty of noisy discussion years ago. This and the other pages formed would now be called 'POV forks', a somewhat deprecated notion. We might revisit the whole issue. (Please sign with ~~~~).
1810:{\displaystyle {\hat {T}}_{\,j_{1}\ldots j_{q}}^{i_{1}\ldots i_{p}}=A^{i_{1}}{}_{k_{1}}\cdots A^{i_{p}}{}_{k_{p}}B^{l_{1}}{}_{j_{1}}\cdots B^{l_{q}}{}_{j_{q}}T_{l_{1}\ldots l_{q}}^{k_{1}\ldots k_{p}}?}
1522:{\displaystyle {\hat {T}}_{\,j_{1}\ldots j_{q}}^{i_{1}\ldots i_{p}}=A^{i_{1}}{}_{k_{1}}\cdots A^{i_{q}}{}_{k_{q}}B^{l_{1}}{}_{j_{1}}\cdots B^{l_{p}}{}_{j_{p}}T_{l_{1}\ldots l_{q}}^{k_{1}\ldots k_{p}}}
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tensor, the relation does not, in general, work the other way around. If this is not intuitive to the general reader, however, then perhaps a clarification should be appended.
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It can't be true to say that a rank n tensor is 'simply' a tensor product of rank 1 tensors. In some or other sense it may be a linear combination of that kind of product.
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I notice the indices are enclosed in brackets. In the classical notation square bracked means skew symmetrize and round brackets symmetrize. Why are they here?
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on the right is undefined. And why must a transformation necessarily take the form of a partial derivative? Similar problems afflict the second definition.
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thing it needs is to be entangled in a mass of ideas related to another procedure altogether, and especially one which is currently far more unclear.
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This article is part of a series of closely related articles for which I would like to clarify the interrelations. Please contribute your ideas at
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It is not true that tensor are always transformation (i.e. classical mass: tensor of rank(0/0); how this can be considered as a transformation?)
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This equation does not work at the moment, so I will paraphrase it for now. Moving it here for future reference when \overbrace is fixed:
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base space of a vector bundle to the total space which when projected back onto the base space has returned to its starting point."
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I suspect that I am the target audience for this (and similar) articles. So, I thought I would share my perspective with you.
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It could be the sum of such tensors, couldn't it (as opposed to being simply an outer product of vectors and 1-forms)?
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I agree with the two sentiments expressed above, this article is very unclear. I think it is best to merge it with
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I think what he means to say is that you look nice today. (Forgive him, he doesn't know the dialect very well.)
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Define the contravariant/covariant component by showing how they transform under a change of coodinate system.
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This article text has mostly been replaced by material adapted from the PlanetMath GFDL article on tensors.
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Introduce the two species: contravariant/covariant, introduce notation and ranks (~= number of indices).
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Then explain that tensors allow to express physical laws in a form that apply to any coordinate systems.
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It seems to me there is a mistake here, a general tensor is not always the product of rank 1 tensors.
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a classical approach presented in the tensor section, but it was 'phased out'; replaced.
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I know that these must be difficult concepts to convey, and I appreciate your efforts.
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An earlier version of this article was adapted from the GFDL article on tensors at
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good job of establishing. For this, and other reasons, and esp. that this is an
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ACK! What happened!?! This is unreadable now! The page used to be very
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Now, on to my comments (which apply to all of the entries on tensors):
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Can we put things in english or not at all please? Thank you.
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is something that does not change under a set of transformations
873:- is there anything the wikipedia community can do about this?
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Sorry to be annoying, but this article have several problems:
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Knowledge TeX transformations:
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http://encyclopedia.thefreedictionary.com/Tensor-classical
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required for the specification of a particular tensor"
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include a typo in the indices? Shouldn't it rather be
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Knowledge talk:WikiProject
Mathematics/related articles
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398:
397:
367:
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317:A tensor is an
257:Special cases:
196:A tensor is an
194:the sentence: "
188:
121:
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12:
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5:
1897:
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1169:General tensor
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1035:discussion at
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685:\ve should be
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630:\bv should be
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604:\cU should be
580:
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308:
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289:
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206:the entry for
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1100:
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1080:
1077:
1074:
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1067:
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1059:
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1034:
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1024:
1023:
1022:
1020:
999:
996:
993:
986:
962:
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942:
937:
935:
926:
922:
921:
920:
914:
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489:
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453:
443:
417:
407:
381:
371:
345:
340:
329:
326:
322:
320:
313:
307:
302:
298:
297:Metric tensor
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123:Find sources:
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110:Verifiability
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52:Learn to edit
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19:This is the
1821:Jamesmelody
876:—Preceding
861:Page copied
575:2003.05.02
468:, the term
148:free images
31:not a forum
1232:Kevin Baas
1186:Kevin Baas
1160:Kevin Baas
852:Kevin Baas
828:Kevin Baas
791:Kevin Baas
784:Kevin Baas
594:PlanetMath
573:Kevin Baas
558:used to be
534:Kevin Baas
1019:The Anome
934:The Anome
770:Rewritten
764:The Anome
753:times}}.}
545:AxelBoldt
517:AxelBoldt
325:represent
319:invariant
280:one-forms
214:invariant
208:invariant
198:invariant
88:if needed
71:Be polite
27:redirect.
21:talk page
1876:Theowoll
1149:Billlion
1129:Halberdo
1107:contribs
1095:unsigned
890:contribs
878:unsigned
586:Credit:
56:get help
29:This is
780:bloated
269:vectors
262:scalars
154:WP refs
142:scholar
1175:Ylebru
1048:MarSch
978:, not
541:tensor
532:soup).
513:tensor
222:" !@?
210:say: "
126:Google
1834:Also
925:times
776:clean
592:from
169:JSTOR
130:books
84:Seek
1880:talk
1854:and
1825:talk
1199:talk
1133:talk
1103:talk
1046:. --
941:must
886:talk
526:last
396:and
244:and
162:FENS
136:news
73:and
919:or
826:--
311:-->
212:An
176:TWL
1882:)
1827:)
1790:…
1765:…
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1550:^
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1480:…
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1360:⋯
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1284:…
1265:^
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1135:)
1105:•
888:•
721:^
718:ε
693:ε
668:^
665:α
571:--
483:¯
447:¯
411:¯
375:¯
299:,
295:,
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264:,
172:·
166:·
158:·
151:·
145:·
139:·
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128:(
58:.
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