# What is a tensor? **URL:** \ **Category:** Factual Questions\ **Created:** [September 26, 2004, 2:24am UTC](https://boards.straightdope.com/t/what-is-a-tensor/266112 "2004-09-26T02:24:11Z")\ **Posts on this page:** 20\ **Page:** 3
**Author:** ![Complex\_Conjugate](https://avatars.discourse-cdn.com/v4/letter/c/3bc359/32.png) [@Complex\_Conjugate](https://boards.straightdope.com/u/Complex_Conjugate)\ **Post date:** [September 27, 2004, 8:09pm UTC](https://boards.straightdope.com/t/what-is-a-tensor/266112/41 "2004-09-27T20:09:05Z")
OOps presse ‘subm,it’ instaed of preview 9the above post was in answr to tim’s question by the way) and the ‘d’ in the equation is actually a ‘partial’ symbol. The matrix transformation by mapping a vector onto itself if anything behaves like a tensor of type (1,1). ---
**Author:** ![Bryan\_Ekers](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/bryan_ekers/32/183_2.png) [@Bryan\_Ekers](https://boards.straightdope.com/u/Bryan_Ekers)\ **Post date:** [September 27, 2004, 8:12pm UTC](https://boards.straightdope.com/t/what-is-a-tensor/266112/42 "2004-09-27T20:12:11Z")
Of course, now I’ve got that damn song from _The Demolished Man_ in my head. ---
**Author:** ![Complex\_Conjugate](https://avatars.discourse-cdn.com/v4/letter/c/3bc359/32.png) [@Complex\_Conjugate](https://boards.straightdope.com/u/Complex_Conjugate)\ **Post date:** [September 27, 2004, 8:13pm UTC](https://boards.straightdope.com/t/what-is-a-tensor/266112/43 "2004-09-27T20:13:10Z")
That should be \*a transformation matrix. ---
**Author:** ![Complex\_Conjugate](https://avatars.discourse-cdn.com/v4/letter/c/3bc359/32.png) [@Complex\_Conjugate](https://boards.straightdope.com/u/Complex_Conjugate)\ **Post date:** [September 27, 2004, 8:24pm UTC](https://boards.straightdope.com/t/what-is-a-tensor/266112/44 "2004-09-27T20:24:54Z")
So anyway getting back to the orignal question, it does seem there are other equivalent ways of defining tensors which I was unware of (i.e. **Mathocists** defintion from catergory theory). ---
**Author:** ![Complex\_Conjugate](https://avatars.discourse-cdn.com/v4/letter/c/3bc359/32.png) [@Complex\_Conjugate](https://boards.straightdope.com/u/Complex_Conjugate)\ **Post date:** [September 27, 2004, 8:37pm UTC](https://boards.straightdope.com/t/what-is-a-tensor/266112/45 "2004-09-27T20:37:25Z")
> [@matt](#): > > I can’t do complete abstraction at all. I hit things with hammers. It would take me a few days and a lot of reading and thinking just to get to grips with the idea of vector space, or a field of numbers for that matter! I don’t have the concepts in place to follow what you’re saying. > > This is good for me. It reminds me to have patience with people who are bewildered by the terminology I use on a daily basis. The concepts of vector spaces and fuields in themsleves are very simple concepts, to understand the axioms of a field takes no advanced math at all ditto for a vector space. Yes they are completely abstract concepts, but the most fmalir example of a field is the real numbers (wrt additon and muplication) and the most famlair example of a vector space is the set of radius vectors in a plane (which forms a vector space over the real numbers). If you’ve got enough math to understand tensors in certain settings then you have more than enough math to understand the axioms of a field and the axioms of a vector space. ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [September 27, 2004, 10:13pm UTC](https://boards.straightdope.com/t/what-is-a-tensor/266112/46 "2004-09-27T22:13:12Z")
This is all well and good but you’re assuming a few things without mentioning them. First of all, that your vector spaces are finite-dimensional. Most interesting vector spaces are infinite dimensional. A subtler point (not applicable when you specialize to **R** ) is that you’re tacitly assuming char F = 0. Bad things happen when the character of F divides the dimension of the representation. ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [September 27, 2004, 10:15pm UTC](https://boards.straightdope.com/t/what-is-a-tensor/266112/47 "2004-09-27T22:15:03Z")
> [@matt](#): > > I’m glad it’s excellent, but to be honest I don’t understand the terminology you use! V is a vector space. GL(V) is the general linear group on V: the collection of all invertible linear transformations sending V to itself. Changing variables in a finite-dimensional space like you’re considering is effected by the action of an element of this group. ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [September 27, 2004, 10:19pm UTC](https://boards.straightdope.com/t/what-is-a-tensor/266112/48 "2004-09-27T22:19:25Z")
> [@Complex Conjugate](#): > > Are yo famliar with the Einstein summation convetion? > > The componets of a tensor transform: > > A’[sup]mn…[/sup][sub]rs…[/sub] = dx’[sup]m[/sup]/dx[sup]i[/sup] dx’[sup]n[/sup]/dx[sup]j[/sup]… dx[sup]k[/sup]/dx’[sup]r[/sup] dx[sup]l[/sup]/dx’[sup]s[/sup]…A[sup]ij[/sup][sub]kl[/sub] This is for changing coordinates systems by x’[sup]m[/sup]=f[sup]m/sup in a manifold. The map T(f) sending T[sub]p[/sub] to itself is the Jacobian (written in the bases determined by the coordinate systems) T(f)[sup]m[/sup][sub]i[/sub] = dx’[sup]m[/sup]/dx[sup]i[/sup] The notion of a transformation acting on a tensor product is more general. ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [September 27, 2004, 10:30pm UTC](https://boards.straightdope.com/t/what-is-a-tensor/266112/49 "2004-09-27T22:30:35Z")
> [@Complex Conjugate](#): > > So anyway getting back to the orignal question, it does seem there are other equivalent ways of defining tensors which I was unware of (i.e. **Mathocists** defintion from catergory theory). Well, to join the two, consider this: Let R be a ring, M be a right R-module, and N be a left R-module. A balanced product is a bilinear map f from M×N to an abelian group A such that f(mr,n) = f(m,rn) for all m in M, n in N, and r in R. A morphism of balanced products f[sub]1[/sub] and f[sub]2[/sub] is a linear map g from A[sub]1[/sub] to A[sub]2[/sub] such that g(f[sub]1/sub) = f[sub]2/sub. The tensor product over R – M?[sub]R[/sub]N – is a universal such map (well, technically couniversal, but it doesn’t matter). That is, a balanced product t:M×N -\> M?[sub]R[/sub]N such that for any other balanced product f:M×N -\> A there exists a unique linear map g:M?[sub]R[/sub]N -\> A such that f is the composition of t and g. Since this map is unique, any other universal balanced product must be isomorphic to M?N, so this is effectively uniquely defined. My definition of a tensoring category modules over a small linear category is a generalization of this: a ring is a linear category with a single object. ---
**Author:** ![NewHeart\_Male](https://avatars.discourse-cdn.com/v4/letter/n/bc8723/32.png) [@NewHeart\_Male](https://boards.straightdope.com/u/NewHeart_Male)\ **Post date:** [September 28, 2004, 12:22am UTC](https://boards.straightdope.com/t/what-is-a-tensor/266112/50 "2004-09-28T00:22:00Z")
**Mathochist** , I hope I’m not being incredibly rude, but may I ask how old you are? (for e.g. in your 30’s or 40’s or 50’s if you don’t wanna give a direct answer). Your thorough mathematical knowledge is astounding. How is it you can come to know so much on such a broad subject area? ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [September 28, 2004, 12:32am UTC](https://boards.straightdope.com/t/what-is-a-tensor/266112/51 "2004-09-28T00:32:10Z")
> [@NewHeart(Male)](#): > > Your thorough mathematical knowledge is astounding. How is it you can come to know so much on such a broad subject area? No offense at all taken. I’m 25. A little more than halfway to 26 if that makes you feel better. 😃 As for how I came to this, I’m soon enough to be Dr. Mathochist and this is pretty near dead on my areas of interest. I directly study representations of small linear categories (as I said before in this thread, I just extended the notion of tensoring over a ring last week). I’m sort of a hobbyist in mathematical physics, so the GR is something I’ve been through quite a while back. In fact, my major application of category representation is to topological and geometric problems (I once thought I was a geometer before I came out as a category theorist ;)) so I had to become well-versed in differential geometry in general. It’s amazing what you pick up when you devote your life to studying something. ---
**Author:** ![NewHeart\_Male](https://avatars.discourse-cdn.com/v4/letter/n/bc8723/32.png) [@NewHeart\_Male](https://boards.straightdope.com/u/NewHeart_Male)\ **Post date:** [September 28, 2004, 12:43am UTC](https://boards.straightdope.com/t/what-is-a-tensor/266112/52 "2004-09-28T00:43:30Z")
> [@Mathochist](#): > > No offense at all taken. I’m 25. A little more than halfway to 26 if that makes you feel better. 😃 > > As for how I came to this, I’m soon enough to be Dr. Mathochist and this is pretty near dead on my areas of interest. I directly study representations of small linear categories (as I said before in this thread, I just extended the notion of tensoring over a ring last week). I’m sort of a hobbyist in mathematical physics, so the GR is something I’ve been through quite a while back. In fact, my major application of category representation is to topological and geometric problems (I once thought I was a geometer before I came out as a category theorist ;)) so I had to become well-versed in differential geometry in general. > > It’s amazing what you pick up when you devote your life to studying something. Wow!!! That’s amazing! I **never** would have expected you to be sooooooooooo young. Granted I’m still a couple of years younger than you but jeeez, I hope I’m half as well-versed in my field(s) as you are in yours by the time I’m your age 🙂 ---
**Author:** ![tracer](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/tracer/32/20578_2.png) [@tracer](https://boards.straightdope.com/u/tracer)\ **Post date:** [September 28, 2004, 1:48am UTC](https://boards.straightdope.com/t/what-is-a-tensor/266112/53 "2004-09-28T01:48:38Z")
Ahem. Tensor is the name of [the lamp on my desk](http://www.tensorlamp.com/scripts/ListSearch.asp?IC=5). ---
**Author:** ![Hyperelastic](https://avatars.discourse-cdn.com/v4/letter/h/a3d4f5/32.png) [@Hyperelastic](https://boards.straightdope.com/u/Hyperelastic)\ **Post date:** [September 28, 2004, 4:50am UTC](https://boards.straightdope.com/t/what-is-a-tensor/266112/54 "2004-09-28T04:50:30Z")
**Mathochist** , while your precision is commendable, you surely realize that your posts are indistiguishable from nonsense for anyone without doctoral-level training in mathematics, which is about 99.99% of the readership. If you can manage to bring it down a few notches, you’ll connect with a lot more people. ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [September 28, 2004, 6:38am UTC](https://boards.straightdope.com/t/what-is-a-tensor/266112/55 "2004-09-28T06:38:58Z")
> [@Hyperelastic](#): > > **Mathochist** , while your precision is commendable, you surely realize that your posts are indistiguishable from nonsense for anyone without doctoral-level training in mathematics, which is about 99.99% of the readership. If you can manage to bring it down a few notches, you’ll connect with a lot more people. My first post was directly in response to the directive in the OP; to wit, “i.e. post the definition of a tensor as you understand it”. Since then I’ve generally aimed my responses at the posts I’ve been responding to. Someone posting claiming a knowledge of GR gets a differential geometry answer. Someone talking about vector spaces gets a vector space answer. If someone says they don’t understand a part of my response, they get a clarification of the points they’ve questioned. On the other hand, you may have a valid argument. After all, dumbing it down is what got this country to the pinnacle of mathematical… nevermind. ---
**Author:** ![Hyperelastic](https://avatars.discourse-cdn.com/v4/letter/h/a3d4f5/32.png) [@Hyperelastic](https://boards.straightdope.com/u/Hyperelastic)\ **Post date:** [September 28, 2004, 8:31pm UTC](https://boards.straightdope.com/t/what-is-a-tensor/266112/56 "2004-09-28T20:31:33Z")
> [@Mathochist](#): > > On the other hand, you may have a valid argument. After all, dumbing it down is what got this country to the pinnacle of mathematical… nevermind. If, by “dumbing it down”, you mean explaining things inaccurately such that later your audience has to “unlearn” things, that is not what I’m suggesting. I’ve found that the overwhelming majority of people learn mathematics best by generalizing from special cases. When you get to a certain level, it becomes more efficient to define the object of interest rigorously and then deduce its properties, but that level is pretty high up the scale. By high up, I mean that most Ph.D.'s in the sciences never reach that level unless they are in a particularly mathematical discipline, like theoretical physics. On the other hand, it’s harder to impress the impressionable when you do it my way. ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [September 28, 2004, 9:38pm UTC](https://boards.straightdope.com/t/what-is-a-tensor/266112/57 "2004-09-28T21:38:32Z")
Bourbaki would strenuously disagree with this position. ---
**Author:** ![Complex\_Conjugate](https://avatars.discourse-cdn.com/v4/letter/c/3bc359/32.png) [@Complex\_Conjugate](https://boards.straightdope.com/u/Complex_Conjugate)\ **Post date:** [September 28, 2004, 10:24pm UTC](https://boards.straightdope.com/t/what-is-a-tensor/266112/58 "2004-09-28T22:24:58Z")
Well, I’d rather be in possesion of the defintion that just some examples, after all we are talking about abstract mathematical objects and they only are what they are defined to be. Of course examples never hurt though. I admit I can’t understand _all_ of what **Mathocists** is saying, but I can understand _some_ of what he is saying and I don’t have a phd in math. But **Hyperelastic** your explanation of a tensor is pretty unelightening, by just restrictng your explantion to some of the properties of tensors of type (1,1), it will not give anybody any idea of what a tensor is as a class of object. It would be like trying to define the complex numbers, by just talking about some of the proeprties of the proper subset of half-integers. ---
**Author:** ![rjk](https://avatars.discourse-cdn.com/v4/letter/r/ed655f/32.png) [@rjk](https://boards.straightdope.com/u/rjk)\ **Post date:** [September 28, 2004, 11:42pm UTC](https://boards.straightdope.com/t/what-is-a-tensor/266112/59 "2004-09-28T23:42:28Z")
> [@Hyperelastic](#): > > **Mathochist** , while your precision is commendable, you surely realize that your posts are indistiguishable from nonsense for anyone without doctoral-level training in mathematics, which is about 99.99% of the readership. If you can manage to bring it down a few notches, you’ll connect with a lot more people. I’m pretty much in agreement with this, and with his later post. We non-mathematicians don’t need “dumbing down” to the point that we need to unlearn things later, but we _do_ need some sort of superficial explanation (with hand-waving as needed) that gives us an idea of what the object under discussion is, what it does, and how people can use it. Mathematical notation is very terse and abstract even if we have definitions for the terms used, so when I see a post like **Mathichist** ’s first reply (post #3), my eyes glaze over. This whole discussion reminds me of a mathematician joke: A math prof is up in front of the class, filling board after board with a long complicated proof, explaining as he goes. Halfway down the fourth or fifth board he says “So then it’s obvious that …” He stops, stares at the board for a moment, and rushes out of the room. Fifteen minutes later he comes back in, stands in front of the board, and says “Yes! It _is_ obvious!” and goes on with the lecture. I often feel like a student in that class. ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [September 28, 2004, 11:53pm UTC](https://boards.straightdope.com/t/what-is-a-tensor/266112/60 "2004-09-28T23:53:09Z")
> [@rjk](#): > > so when I see a post like **Mathichist** ’s first reply (post #3), my eyes glaze over. I quote again from the OP, “i.e. post the definition of a tensor as you understand it”. [Previous page](https://boards.straightdope.com/t/what-is-a-tensor/266112.md?page=2) [Next page](https://boards.straightdope.com/t/what-is-a-tensor/266112.md?page=4)