# 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:** 2
**Author:** ![David\_Simmons](https://avatars.discourse-cdn.com/v4/letter/d/9de053/32.png) [@David\_Simmons](https://boards.straightdope.com/u/David_Simmons)\ **Post date:** [September 27, 2004, 2:28am UTC](https://boards.straightdope.com/t/what-is-a-tensor/266112/21 "2004-09-27T02:28:11Z")
> [@rjk](#): > > I have nothing against math, or mathematicians either, and I knew it was a math thread from the start. It’s just that I came in here hoping for a nice simple layman’s definition I could understand, maybe with a trivial example or two. But then I just barely got through elementary calculus. I should know better. I’m not a mathematician so this will be sketchy at best. Tensors are tied up in the idea of the measurement of distances between two points in a generalized coordinate system. In a rectangular coordinate system the distance from the origin, s, of the point p(x,y) is s[sup]2[/sup] = x[sup]2[/sup] + y[sup]2[/sup]. This is the familiar Pythagorean formula. From Analytic Geomety the distance between the points p[sub]1/sub and p[sub]2/sub is: s[sup]2[/sup] = [x[sub]2[/sub] - x[sub]1[/sub])[sup]2[/sup] + (y[sub]2[/sub] - y[sub]1[/sub])[sup]2[/sup] For ease of writing them out we’ll call (x[sub]2[/sub] - x[sub]1[/sub]) dx and (y[sub]2[/sub] - y[sub]1[/sub]) dy. Now a rectangular coordinate system isn’t the only one around. For instance there is the curvilinear coordinate system of latitude and longitude that is used to locate points on the surface of the earth. So we can talk about a generalized coordinate system of a set of curves of any shape that intersect another set of curves. The two sets of curves need not be regularly spaced and they can intersect each other at any old angle. If we number the lines in the sets and call one set u[sub]1[/sub] and the other u[sub]2[/sub] (so that we don’t confuse them with x and y of the rectangular system) then the distance between any two points in this set will be: s[sup]2[/sup] = du[sub]1[/sub][sup]2[/sup] + du[sub]2[/sub][sup]2[/sup], by analogy to the rectangular case. Once we assign a metric to the coordinate numbers we can compute a distance in that metric. This is for a coordinate system in a plane. Carl Freidrich Gauss developed a metric analogous to the one dimentional case for distances between two points on a surface. I am unable to follow his method very well and the following is from Edna E. Kramer, Professor Emeritus of the Polytechnic Institute of Brooklyn in her book _The Nature and Growth of Modern Mathematics_. The Gauss formula for distances between points on a surface is: s[sup]2[/sup] = g[sub]11[/sub]du[sub]1[/sub][sup]2[/sup] + 2g[sub]12[/sub]du[sub]1[/sub]du[sub]2[/sub] + g[sub]22[/sub]du[sub]22[/sub][sup]2[/sup]. u[sub]1[/sub] and u[sub]2[/sub] are the two coordinates on the surface, for example latitude and longitude, and the g’s are constants. Since the du’s are just furniture we can assume they are present and the metric is determined by the value of the g’s which can be put into a matrix. (g[sub]11[/sub]) (g[sub]12[/sub]) (g[sub]12[/sub]) (g[sub]22[/sub]) When this matrix is accompanied by a law of transformation that allows transforming g into g’ in all permissible meshes or frames of reference it is called a tensor. I’ll leave it to the mathematicians to explain what “all permissible frames of reference” is all about. ---
**Author:** ![tim314](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/tim314/32/3468_2.png) [@tim314](https://boards.straightdope.com/u/tim314)\ **Post date:** [September 27, 2004, 3:13am UTC](https://boards.straightdope.com/t/what-is-a-tensor/266112/22 "2004-09-27T03:13:48Z")
> [@David Simmons](#): > > When this matrix is accompanied by a law of transformation that allows transforming g into g’ in all permissible meshes or frames of reference it is called a tensor. > > I’ll leave it to the mathematicians to explain what “all permissible frames of reference” is all about. Well IANAM, but I know you can’t use a Lorentz transformation to transform a time-like interval into a space-like interval, for example. So while you and I might disagree on an object’s speed if we’re in different reference frames, you won’t have a situation where something’s moving slower than light from my perspective but faster than light from your perspective. I think this is related to the fact that no reference frame can be moving faster than light speed, so /[Gamma] = 1 / sqrt(1-v[sup]2[/sup]/c[sup]2[/sup]) (a factor that appears in Lorentz transforms involving boosts) is always positive. So I guess you could say an inertial reference frame that’s moving faster than light speed relative to another inertial frame is not “permissible.” To revise my comment above, I guess it’s more correct to say that a tensor maps one kind of mathematical object to another, and the scalar, vector, matrix, what-have-you is a representation of this map. But, again, IANAM. ---
**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, 3:30am UTC](https://boards.straightdope.com/t/what-is-a-tensor/266112/23 "2004-09-27T03:30:52Z")
> [@David Simmons](#): > > Tensors are tied up in the idea of the measurement of distances between two points in a generalized coordinate system. The standard application of tensors to differential geometry (and thus GR) is what you’re talking about. Specifically, these are tensors over the tangent vector space at a given point of a manifold (T[sub]p[/sub]). Changing coordinate systems acts in the tangent space as an invertible linear transformation: an element of GL(T[sub]p[/sub]) _like I said before_. > [@tim314](#): > > I know you can’t use a Lorentz transformation to transform a time-like interval into a space-like interval, for example That’s because in the context of SR the relevant group action is SO(3,1), not GL(4). > [@Hyperelastic](#): > > A tensor is a rule that associates every vector to another vector. This is an element of V?V[sup]\*[/sup], not a tensor in general. I hate to be so authoritatian here, but everyone who’s said that it has little or nothing to do with transformation properties is flat-out wrong. Tensor products are intimately connected with group (or generally: algebra; or more generally: category) actions. If the dual vector space doesn’t transform according to the contragrediant representation of the appropriate group it’s not a contravariant tensor. If the new vector space constructed from the two previous ones doesn’t transform via the action guaranteed by the coalgebra structure on **C** [G], it’s not the tensor product. ---
**Author:** ![Standup\_Karmic](https://avatars.discourse-cdn.com/v4/letter/s/ac91a4/32.png) [@Standup\_Karmic](https://boards.straightdope.com/u/Standup_Karmic)\ **Post date:** [September 27, 2004, 3:34am UTC](https://boards.straightdope.com/t/what-is-a-tensor/266112/24 "2004-09-27T03:34:52Z")
> [@tim314](#): > > I couldn’t tell if **Standup Karmic** was kidding or not . . . but it seems no more reasonable to expect a “Caution: Math Thread” label than, say, a “Caution: Art History Thread” label. My apologies to **tim314** et al. who may have taken offence. No, mathematicians are not really jerks. Yes, my commentary was meant to be humorous. Note to self: When making jokes to mathematicians, figure out how to present the punchline in the form of matrices or as imaginary numbers. ---
**Author:** ![astro](https://avatars.discourse-cdn.com/v4/letter/a/9dc877/32.png) [@astro](https://boards.straightdope.com/u/astro)\ **Post date:** [September 27, 2004, 3:52am UTC](https://boards.straightdope.com/t/what-is-a-tensor/266112/25 "2004-09-27T03:52:49Z")
[Time for math jokes!](http://www.workjoke.com/projoke22.htm) > [@](#): > > The Flood is over and the ark has landed. Noah lets all the animals out and says, “Go forth and multiply.” > > A few months later, Noah decides to take a stroll and see how the animals are doing. Everywhere he looks he finds baby animals. Everyone is doing fine except for one pair of little snakes. “What’s the problem?” says Noah. > “Cut down some trees and let us live there”, say the snakes. > > Noah follows their advice. Several more weeks pass. Noah checks on the snakes again. Lots of little snakes, everybody is happy. Noah asks, “Want to tell me how the trees helped?” > > “Certainly”, say the snakes. “We’re adders, so we need logs to multiply.” > [@](#): > > A mathematician is in Africa trying to capture a lion. When he spots one he proceeds to build a fence around himself and says, “I define this to be outside!” > [@](#): > > One day a mathematician decides that he is sick of math. So, he walks down to the fire department and announces that he wants to become a fireman. > The fire chief says, “Well, you look like a good guy. I’d be glad to hire you, but first I have to give you a little test.” > > The firechief takes the mathematcian to the alley behind the fire department which contains a dumpster, a spigot, and a hose. The chief then says, “OK, you’re walking in the alley and you see the dumpster here is on fire. What do you do?” > The mathematician replies, “Well, I hook up the hose to the spigot, turn the water on, and put out the fire.” > > The chief says, “That’s great… perfect. Now I have to ask you just one more question. What do you do if you’re walking down the alley and you see the dumpster is not on fire?” > The mathematician puzzles over the question for awhile and he finally says, “I light the dumpster on fire.” > The chief yells, “What? That’s horrible! Why would you light the dumpster on fire?” > The mathematician replies, “Well, that way I reduce the problem to one I’ve already solved.” ---
**Author:** ![David\_Simmons](https://avatars.discourse-cdn.com/v4/letter/d/9de053/32.png) [@David\_Simmons](https://boards.straightdope.com/u/David_Simmons)\ **Post date:** [September 27, 2004, 5:38am UTC](https://boards.straightdope.com/t/what-is-a-tensor/266112/26 "2004-09-27T05:38:25Z")
> [@Mathochist](#): > > The standard application of tensors to differential geometry (and thus GR) is what you’re talking about. Specifically, these are tensors over the tangent vector space at a given point of a manifold (T[sub]p[/sub]). Changing coordinate systems acts in the tangent space as an invertible linear transformation: an element of GL(T[sub]p[/sub]) _like I said before_. Yeah I know, but few of us understand the lingo. For example “tangent vecor space” and “manifold” are specialized mathematical usages that don’t have a lot of meaning to most of us. Application to General Relativity is one use of tensors. What are some others? I think applications that give a sense of what tensors _do_ are what **rjk** and those of us without advanced mathematical training could use. For example, having read Dr. Kramer’s book I have a general feel for what tensors do in General Relativity by analogy with ordinary Analytic Geometry. Likewise the transformation of coordinates from, say, rectangular to polar gives me a small sense of the Lorenz transformation but I certainly can’t use it with any confidence. ---
**Author:** ![matt](https://avatars.discourse-cdn.com/v4/letter/m/a6a055/32.png) [@matt](https://boards.straightdope.com/u/matt)\ **Post date:** [September 27, 2004, 10:21am UTC](https://boards.straightdope.com/t/what-is-a-tensor/266112/27 "2004-09-27T10:21:50Z")
> [@rjk](#): > > > [@Mathochist](#): > > > > In physics, generally a “tensor” is a multicomponent quantity where the different components transform among each other subject to certain relations with a change of basis vectors. Modern physicists are starting to finally settle into seeing such “tensors” as elements of “tensor product spaces”, which are the tensor product of some number of representation spaces of some group with the diagonal action of the group. > > I was OK up to “transform”… > > …It’s just that I came in here hoping for a nice simple layman’s definition I could understand, maybe with a trivial example or two. I’m not a mathmetician or even a physicist, but I’ll have a shot at getting past “transform”, hopefully as far as “basis vectors”. Physicists and non-physicists alike occasionally measure things, and express the measurement using numbers. Examples are temperature, altitude, weight, wind strength, whatever. A lot of these things can be expressed using only one number, e.g. temperature and altitude. They are _single component_ quantities or _scalar_ quantities. Wind strength is slightly different - it has both a magnitude (the wind speed) and a direction. Generally we say stuff like “wind speed 10mph in a north east direction”. Quantities with magnitude and direction are called _vector_ quantities. To specify vector quantities, we need a co-ordinate system which includes directions. We have such a system in common use - the compass directions, N, S, E and W, plus UP and DOWN. These are our reference directions, and we can express any general direction in terms of them. (6 feet DOWN and 3 inches NORTH for every mile WEST is an example of a “general direction”). The reference directions N, S, E, W, UP and DOWN are the set of **basis vectors** we use to express direction. In fact, since we can use negative numbers, we can reduce them to N, E and UP, and express S, W and DOWN as “-N”, “-E” and “-UP”. So my previous " general direction" becomes “-6 feet UP and 3 inches NORTH for every minus-mile EAST”. To express wind strength properly in terms of our basis vectors, we need to use a minimum of three numbers - “wind speed 10mph in a north-east direction” becomes “wind speeds 7.07mph N, 7.07mph E, 0mph UP”. This is the case for all vector quantities in our three dimensional universe - you can reduce their expression to three numbers, but no fewer. You would normally write such a vector as (7.07, 7.07, 0). Now, on to transformations of basis vectors! N, E and UP are all very well if you’re looking at the weather map or going sailing, but they’re not very convenient if you’re trying to land a plane on a north-east runway. You’d prefer have the wind strength expressed as TAILWIND, CROSSWIND (left to right) and DOWNDRAFT. Remember “different components transform(ing) among each other subject to certain relations with a change of basis vectors”? In this case it is simply the three numbers of the vector changing as you switch from using N, E, UP to TAIL, CROSS and DDRAFT. Wind speed 10mph in a north east direction may be (7.07, 7.07, 0) using N, E and UP but it becomes (10, 0, 0) using TAIL, CROSS and DDRAFT for a NE runway as your basis vectors. Now, it is obvious that the transformation of the three numbers of the Wind Strength vector from one set of basis vectors to the other is a matter of geometry. In fact, if you remember how to do matrix algebra, there is a 3 x 3 matrix that you can multiply the Wind Strength vector by to transform one to the other. That matrix is the _transformation tensor_ and it relates the Wind Strength expressed using two different sets of basis vectors. Revisiting measurements and quantities for a moment: scalars are single component quantities,and we give them a single number, which is technically a 1 x 1 matrix. We don’t normally bother to think about it as a matrix, just as we don’t normally bother to write 11.6 as 11.60000000… with the zeros going on forever, but in this context it is a matrix. Vectors are three-component quantities, and we give them three numbers, making a 3 x 1 matrix. But there are other quantities that have even more components. Stress measured at a point in a material has NINE components - three plain stress components and six shear stress components, and it is described with nine numbers in a 3 x 3 matrix. Quantities with more than three components are often referred to as “tensor quantities”, kind of a step up from scalar and vector quantities. This is a misnomer really - scalars and vectors are tensor quantities as well. So - a tensor is a quantity with components. (Even a scalar, with only one component.) The components can be written as a matrix. Each component expresses a magnitude in terms of basis vectors. (A scalar technically has ONE basis vector - the “number line”.) If you change the basis vectors, then all the values of the components of the tensor change in some well-defined way e.g. as Euclidian geometry dictates. The change from one set of basis vectors to another can be described by a matrix, a “transformation tensor”. I’m not too delighted by the use of the word “tensor” both for the type of quantity and the matrix used for transformations, but it seems quite common. Mathematicians of course have no truck with this wishy-washy, stuck-in-the-real-world type of thinking. They are quite happy to imagine tensors in spaces with more than three dimensions, tensor transformations using non-Euclidian geometry, and all kinds of general cases of which real-world applications are only a tiny subset. I’m glad they’re around to think about that stuff, so I don’t have to! ---
**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, 12:19pm UTC](https://boards.straightdope.com/t/what-is-a-tensor/266112/28 "2004-09-27T12:19:30Z")
> [@astro](#): > > A mathematician is in Africa trying to capture a lion. When he spots one he proceeds to build a fence around himself and says, “I define this to be outside!” Very astutely proving the _existence_ of a lion on the other side of the fence first. ---
**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, 12:21pm UTC](https://boards.straightdope.com/t/what-is-a-tensor/266112/29 "2004-09-27T12:21:36Z")
> [@David Simmons](#): > > Yeah I know, but few of us understand the lingo. For example “tangent vecor space” and “manifold” are specialized mathematical usages that don’t have a lot of meaning to most of us. Note that the OP specifically mentioned having taken differential geometry. Besides the question is what a tensor _is_, not what it _does_. ---
**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, 12:32pm UTC](https://boards.straightdope.com/t/what-is-a-tensor/266112/30 "2004-09-27T12:32:21Z")
> [@matt](#): > > So - a tensor is a quantity with components. (Even a scalar, with only one component.) The components can be written as a matrix. Each component expresses a magnitude in terms of basis vectors. (A scalar technically has ONE basis vector - the “number line”.) If you change the basis vectors, then all the values of the components of the tensor change in some well-defined way Excellent, when considering the GL(V) action on V. The only thing I’d add is the emphasis that the tensor measures something that is there in the absence of a coordinate system, while the components are artifacts of the choice of coordinates. Changing the basis changes the components, but since there’s something “really there” independant of the coordinate system the way the coordinates change is fixed by the nature of the object. > [@](#): > > The change from one set of basis vectors to another can be described by a matrix, a “transformation tensor”. I’m not too delighted by the use of the word “tensor” both for the type of quantity and the matrix used for transformations, but it seems quite common. Maybe in physics or engineering usage. To a mathematician it’s the matrix representation of the element of GL(V) in the basis chosen, nothing more. ---
**Author:** ![David\_Simmons](https://avatars.discourse-cdn.com/v4/letter/d/9de053/32.png) [@David\_Simmons](https://boards.straightdope.com/u/David_Simmons)\ **Post date:** [September 27, 2004, 1:06pm UTC](https://boards.straightdope.com/t/what-is-a-tensor/266112/31 "2004-09-27T13:06:11Z")
> [@Mathochist](#): > > Note that the OP specifically mentioned having taken differential geometry. Besides the question is what a tensor _is_, not what it _does_. Of course. My attempt at an illustration of the use of tensors was in response to **rjk**. I think **Complex Conjugate** has had enough math that I can be of no help whatsoever to him. ---
**Author:** ![gtbiehle](https://avatars.discourse-cdn.com/v4/letter/g/5f8ce5/32.png) [@gtbiehle](https://boards.straightdope.com/u/gtbiehle)\ **Post date:** [September 27, 2004, 1:52pm UTC](https://boards.straightdope.com/t/what-is-a-tensor/266112/32 "2004-09-27T13:52:51Z")
IAAP(physicist) We would not call the transformation matrix a transformation tensor for the simple reason that it is not one. Its elements transform under change of basis but do not transform as a tensor’s would. Otherwise, Matt, that was a good explanation. ---
**Author:** ![Ethilrist](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/ethilrist/32/4604_2.png) [@Ethilrist](https://boards.straightdope.com/u/Ethilrist)\ **Post date:** [September 27, 2004, 2:17pm UTC](https://boards.straightdope.com/t/what-is-a-tensor/266112/33 "2004-09-27T14:17:54Z")
> [@Standup Karmic](#): > > Geez. Now my brain is full. > > What the hell ever happened to, “Pan-fried Semen”, “Is there any reason that semen would be bad for my fish?” and other non-brain-filling questions? > > Seriously. You people know way too much and you’re going to hurt yourselves. You just watch. Tensor’s a wizard in Greyhawk. Creator of the massively over-munchkinned spell, _Tensor’s Floating Disk_ and _Tensor’s Transformation,_ among others. There. Did that drive out some of that pesky knowledge? ---
**Author:** ![rjk](https://avatars.discourse-cdn.com/v4/letter/r/ed655f/32.png) [@rjk](https://boards.straightdope.com/u/rjk)\ **Post date:** [September 27, 2004, 4:18pm UTC](https://boards.straightdope.com/t/what-is-a-tensor/266112/34 "2004-09-27T16:18:34Z")
First, thanks to all for the explanations. I think I’m starting to understand! > [@Mathochist](#): > > Excellent, when considering the GL(V) action on V. The only thing I’d add is the emphasis that the tensor measures something that is there in the absence of a coordinate system, while the components are artifacts of the choice of coordinates. Changing the basis changes the components, but since there’s something “really there” independant of the coordinate system the way the coordinates change is fixed by the nature of the object. My first reaction to this (as for many math posts) was “Say whaaa?” but a little thought leads me to yet another question. (I assume we’re in abstract math here, or at least back to GR.) Do you mean that the tensor defines the transformation rules in a general way, no matter what coordinate system we might choose later? My brain hurts. I must be a masochist! :rolleyes: (I do like the name, BTW.) ---
**Author:** ![matt](https://avatars.discourse-cdn.com/v4/letter/m/a6a055/32.png) [@matt](https://boards.straightdope.com/u/matt)\ **Post date:** [September 27, 2004, 6:24pm UTC](https://boards.straightdope.com/t/what-is-a-tensor/266112/35 "2004-09-27T18:24:46Z")
> [@Mathochist](#): > > Excellent, when considering the GL(V) action on V. I’m glad it’s excellent, but to be honest I don’t understand the terminology you use! I was introduced to tensors as a concept purely because I’m a materials scientist and we had to use them to describe stress. That’s a very specific “real world” application - any generalisation or mathematical depth was avoided like the plague. > [@Mathochist](#): > > The only thing I’d add is the emphasis that the tensor measures something that is there in the absence of a coordinate system, while the components are artifacts of the choice of coordinates. Changing the basis changes the components, but since there’s something “really there” independant of the coordinate system the way the coordinates change is fixed by the nature of the object. I wish I’d said that. I probably will, somewhere else. > [@gtbiehle](#): > > We would not call the transformation matrix a transformation tensor for the simple reason that it is not one I wish I could blame my engineering textbooks and my lecturers, but the fault may well lie with my own capricious memory… ---
**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, 6:40pm UTC](https://boards.straightdope.com/t/what-is-a-tensor/266112/36 "2004-09-27T18:40:41Z")
Before I do anything else I’ll put a basic explanation of what a tensor is (or at least how I like to define a tensor) for anyone whose interested (e.g. **matt** ): A [vector space](http://mathworld.wolfram.com/VectorSpace.html), simply put is just a set of objects called vectors (and a field of numbers like for example the real numbers) that satisfies certain axioms (the concept of a ‘vector’ in this instance is a complete abstraction, so don’t think they have to be vectors corresponding to physical quantities, but an example of a vector space is the set of all radius vectors in real space). A one-form (aka co-vector, covariant vector and several other names) is a (linear) function that acts on a vector and produces a number. The set of all one-forms for a given vector space also satisfies the axioms of a vector space, so we call vector space of vectors and the vector space of one-forms ‘dual vector spaces’. For each vector there is a corresponding one-form, the two making up what is called a pair of dual vectors. Given a vector **A** and a one-form **B~( )** (I’ve put the empty brackets to signify the fact that it is a fucntion of a vector) we can obtain some number **B~** ( **A** ), this number is called the dot product of the vector **A** and the vector **B** ( **B** **B~** ( ) form a pair of dual vectors). A one-form is a tensor of type (0,1). A tensor of type (r,s) is a (multilinear) function that take r vectors and s one-forms and produces a number. To give an example, the metric tensor **g** ( , ) (a tensor of type (0,2) is a very important tensor. Given two vectors **A** , **B** the metric tensor will produce a real number **g** ( **A** , **B** ) this number is called the dot product of the vector **A** and the vector **B**. Comparing this to the one-form: **g** ( **A** , **B** ) = **B~** ( **A** **g** ( **B** , ) = **B~** ( ) Infact vectors themsleves are tensors of type (1,0) threfore they can take a one-form a produce a real number: [a]A\*\*( **B~** ) = **B~** ( **A** ). What this physically means in n-diemnsional space is that a tensor of type (r,s) has n[sup]r + s[/sup] componets so for example a tensor of rank 2 (r + s =2) in some (recatasngular) basis in three dimensional space has 9 componets: xx, xy, xz, yx, yy, yz, zx, zy (normal matrix notation does not work for tensors of rank 3 and higher). Under a transformation the components of a tensor will tranform in a way that depends on their type. ---
**Author:** ![tim314](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/tim314/32/3468_2.png) [@tim314](https://boards.straightdope.com/u/tim314)\ **Post date:** [September 27, 2004, 7:29pm UTC](https://boards.straightdope.com/t/what-is-a-tensor/266112/37 "2004-09-27T19:29:56Z")
> [@gtbiehle](#): > > IAAP(physicist) We would not call the transformation matrix a transformation tensor for the simple reason that it is not one. Its elements transform under change of basis but do not transform as a tensor’s would. I would like to see some elaboration on this point, if someone is willing. What are the possible ways a tensor can transform under change of basis, and how does the transformation matrix transform? I’ve seen this stuff before, but I need a refresher. ---
**Author:** ![matt](https://avatars.discourse-cdn.com/v4/letter/m/a6a055/32.png) [@matt](https://boards.straightdope.com/u/matt)\ **Post date:** [September 27, 2004, 7:39pm UTC](https://boards.straightdope.com/t/what-is-a-tensor/266112/38 "2004-09-27T19:39:25Z")
> [@rjk](#): > > My first reaction to this (as for many math posts) was “Say whaaa?” but a little thought leads me to yet another question. (I assume we’re in abstract math here, or at least back to GR.) Do you mean that the tensor defines the transformation rules in a general way, no matter what coordinate system we might choose later? If my understanding is correct, a tensor quantity is one that can be described by a matrix, and the elements of that matrix depend on the co-ordinate system being used. The matrix itself is referred to as a tensor. Unitless quantities, such as the number of M & Ms in a jar, or the ratio of a circle’s circumference to its diameter, are not tensor quantities - there is no co-ordinate system, although they are **real things** and measurable. Scalar quantities e.g. altitude, with units such as feet, are tensor quantities, but their matrices contain only one component. Of course, you don’t tend to think of a single value as a one-element matrix, nor should you have to. The “co-ordinate system” of a scalar is the unit of measurent, and you can transform from one unit to another with a conversion factor, e.g. from feet to metres. The conversion factor is the single component of the transformation matrix between the two units. This is the same as switching to a basis vector with a different magnitude. In my previous example with wind strength I neglected the possibility of changing the _magnitude_ of the basis vectors, dealing only with their orientation. Vector quantities in three-dimensional space have to have three components, which will depend on the co-ordinate system chosen. I used the compass directions because this is a co-ordinate system that everyone is familiar with, and the example of a skewed runway to show why you might want to transform to some other co-ordinate system. The point \*\* Mathochist\*\* was making is that the steadily blowing wind is a **real thing** - the tensor (the matrix of three values) used to describe it depends on the co-ordinate system chosen. The co-ordinate system includes both the directions and _magnitudes_ of the basis vectors. in my previous example I made the implicit assumption that the basis vectors were a “mile per hour” long. Transformation rules from one co-ordinate system to another depend on the _relationship between the co-ordinate systems_, and have nothing to do with the **real thing** being measured. For example, if you’re using N, E and UP as your basis vectors but want to convert to basis vectors a “kilometre per hour” long, the relationship between the two co-ordinate systems is simply a multiplication of the basis vector magnitudes by 1.6. The transformation is described by a 3x3 matrix with 1.6 down the top-left to bottom-right diagonal and the rest filled with zeros - a diagonal matrix. If you wanted to convert to basis vectors of km/h for N and E but feet per second for the UP direction, you can. There is no reason why your basis vectors need to be of the same magnitude. They don’t even have to be at right angles to each other - you can still describe arbitary vectors with them. The co-ordinate system is up to you, the components of the resulting tensor describing the **real thing** depends on what you choose. ---
**Author:** ![matt](https://avatars.discourse-cdn.com/v4/letter/m/a6a055/32.png) [@matt](https://boards.straightdope.com/u/matt)\ **Post date:** [September 27, 2004, 7:53pm UTC](https://boards.straightdope.com/t/what-is-a-tensor/266112/39 "2004-09-27T19:53:17Z")
> [@Complex Conjugate](#): > > A [vector space](http://mathworld.wolfram.com/VectorSpace.html), simply put is just a set of objects called vectors (and a field of numbers like for example the real numbers) that satisfies certain axioms (the concept of a ‘vector’ in this instance is a complete abstraction, so don’t think they have to be vectors corresponding to physical quantities, but an example of a vector space is the set of all radius vectors in real space). 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. ---
**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, 7:55pm UTC](https://boards.straightdope.com/t/what-is-a-tensor/266112/40 "2004-09-27T19:55:13Z")
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] [Previous page](https://boards.straightdope.com/t/what-is-a-tensor/266112.md?page=1) [Next page](https://boards.straightdope.com/t/what-is-a-tensor/266112.md?page=3)