Timeline for answer to Most harmful heuristic? by Darsh Ranjan
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| Nov 6, 2020 at 22:38 | comment | added | Yongyi Chen | A bit late, but the definition of tensors that I got the most mileage of wrapping my head around (and connecting to the other more rigorous definitions) was that a tensor (for two vector spaces) simply a linear combination of the symbols $e_i\otimes f_j$ where $\{e_i\},\{f_j\}$ are a basis of the two vector spaces. | |
| Oct 27, 2018 at 1:09 | comment | added | ziggurism | The definition of tensors as linear maps, although common, is not the "right" definition. It requires you to define (1,0) rank tensors as linear maps from the dual space, and relies on the isomorphism between a vector space and its double dual. But that isomorphism fails unless your vector space is finite dimensional. A tensor should not be a linear map, It should just be an element of the appropriate tensor product. | |
| Sep 20, 2017 at 4:56 | comment | added | Tim Carson | The pedagogical difficulty with tensors is that there are many more possible operations with them than with matrices and vectors. If I'm comfortable with composition of linear maps and smooth vector fields, then it's not a big jump to tensors on a manifold with the right definition. But then my teacher has to actually compute something, and the quickest way to write down tensor contractions is to use these multi-index beasts, and we have to go through an explanation of these to get the calculation, and by this point I've forgotten the point and just see indices. | |
| Jan 19, 2017 at 0:15 | comment | added | user21349 | [...] It also doesn't give a definition of a scalar that maps nicely onto how physicists think about scalars. E.g., a physicist does not think of a time coordinate as a scalar, but a time coordinate is certainly a "smooth function on M." A scalar has to be invariant under a change of coordinates. | |
| Jan 19, 2017 at 0:14 | comment | added | user21349 | An advantage of the "physicist's definition" is that it treats all tensors, of any rank, on an equal footing. With the definition favored by Darsh Ranjan and Vectornaut, we have to first build up the notions of scalar, vector, and covector, then define all the other ranks, and then go back and do some more equivalence classing to deal with the fact that the definition is not consistent, because a vector is not literally the same as the dual of its own dual.[...] | |
| Nov 3, 2016 at 14:09 | comment | added | David Fernandez-Breton | I completely agree with @DarshRanjan: in fact, one of the main reasons why I never really learnt general relativity, is because I never understood that definition (how can numbers vary? Shouldn't it be, in any case, a multidimensional array of functions? Any assumptions on smoothness of those functions? I never found satisfactory answers to those questions). I can finally say that I'm starting to get the meaning of a tensor, thanks to the definition that you wrote in this answer, which I just read. | |
| Jun 26, 2015 at 17:57 | comment | added | Darsh Ranjan | @Vamsi - from personal experience: maybe not wrong intuitions, but rather absence of intuition. | |
| May 29, 2015 at 0:24 | comment | added | Vamsi | I agree with Scott Aaronson. In fact, the physicist way of defining tensors as things that change correctly under coordinates gives a nice way to define tensor fields on manifolds (Simply a smooth collection of multi-index beasts on different open sets such that on the intersection they are related by an appropriate transformation (the transition functions of the tensor bundle)). I am not sure if this "heuristic" actually gives rise to wrong intuitions. | |
| Feb 15, 2015 at 2:54 | history | edited | Darsh Ranjan | CC BY-SA 3.0 |
Clarification
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| Apr 18, 2013 at 5:26 | comment | added | Scott Aaronson | The trouble I have is that none of the alternative definitions on offer seem accessible to someone first learning about tensors! Related to that (in my mind), they don't make clear how one would actually represent a tensor on a computer (e.g., how many degrees of freedom are there, and what do we do with them?). So, is there a way to explain what tensors are that satisfies those constraints but also leads to fewer wrong intuitions? | |
| Oct 3, 2010 at 3:39 | comment | added | Cosmonut | I second, third and fourth that, Darsh. That particular definition of tensor set back my understanding of differential geometry by at least a year. | |
| Aug 18, 2010 at 21:54 | comment | added | Thierry Zell | For that matter, teaching linear algebra and doing echelon forms and so on does not strike me as very enlightening. When I saw a matrix for the first time, it was already in the context of linear maps and for given bases of source and target spaces. | |
| Jun 16, 2010 at 15:08 | comment | added | Michael Hardy | OK, a proposed intuitive point of view: say you have an ordered pair of vectors. If you multiply one of them by a nonzero scalar and the other by the reciprocal of that scalar, you've got a different ordered pair of vectors. But in both cases you have the same tensor. All the other algebraic requirements that are supposed to be satisfied are just there to make sure the algebra works out neatly the way it should. But this one is where the basic intuition is. | |
| May 22, 2010 at 22:01 | comment | added | Vectornaut | I second this wholeheartedly!!!!!! @Victor Protsak: For me, "the real definition of a tensor" is something like the following. "Let M be a smooth manifold; let FM be the space of smooth functions on M, let VM be the space of smooth vector fields on M, and let VM be the space of smooth covector fields on M. A (k,l) tensor is a multilinear map from (VM)^k x (VM)^l to FM." There might be more abstract and versatile definitions, but this one seems to work pretty well in the context of general relativity, which is where the definition Darsh Ranjan quoted tends to show up (in my experience). | |
| May 22, 2010 at 15:09 | comment | added | Victor Protsak | What's "the real definition of a tensor"? Element of a tensor product? A section of a tensor bundle? | |
| Oct 31, 2009 at 19:59 | history | answered | Darsh Ranjan | CC BY-SA 2.5 |