To be fair to physicists, the standard physicists' definition isn't "a tensor is a thing that transforms like a tensor", it's "a tensor is a mathematical object that transforms in the following way <....explanation of the specific characteristics that mean that a tensor transforms in a way that's independent of the chosen coordinate system...>".

When people say "a tensor is a thing that transforms like a tensor" they're using a convenient shorthand for the bit that I put in angle brackets above.

My favourite explanation is that "Tensors are the facts of the universe" which comes from Lillian Lieber, and is a reference to the idea that the reality of the tensor (eg the stress in a steel beam or something) is independent of the coordinate system chosen by the observer. The transformation characteristic means that no matter how you choose your coordinates, the bases of the tensor will transform such that it "means" the same thing in your new coordinates as it did in the old ones, which is pretty nifty.

https://www.youtube.com/watch?v=f5liqUk0ZTw&pp=ygURdGVuc29yc...

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lisper 2 days ago | parent | next [-]

> a convenient shorthand for the bit that I put in angle brackets above.

Yes, but the "convenient shorthand" only makes sense if you already know what a tensor is. That renders the "definition" useless as an explanation or as pedagogy. It's only useful as a social signal to let others know that you understand what a tensor is (or at least you think you do).

> My favourite explanation is that "Tensors are the facts of the universe"

That's not much better. "The earth revolves around the sun" is a fact of the universe, but that doesn't help me understand what a tensor is.

What matters about tensors are the properties that distinguish them from other mathematical objects, and in particular, what distinguishes them from closely related mathematical objects like vectors and arrays. Finding a cogent description of that on the internet is nearly impossible.

> the reality of the tensor ... is independent of the coordinate system chosen by the observer

Now you're getting closer, but this still misses the mark. What is "the reality of a tensor"? Tensors are mathematical objects. They don't have "reality" any more than numbers do.

> no matter how you choose your coordinates, the bases of the tensor will transform such that it "means" the same thing in your new coordinates as it did in the old ones

That is closer still. But I would go with something more like: tensors are a way to represent vectors so that the representation of a given vector is the same no matter what basis (or coordinate system) you choose for your vector space.

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seanhunter 2 days ago | parent [-]

> But I would go with something more like: tensors are a way to represent vectors so that the representation of a given vector is the same no matter what basis (or coordinate system) you choose for your vector space.

That's just incorrect though for a couple of reasons. Firstly, a vector in the sense in which it is used in physics is a rank 1 tensor so it has this transformation behaviour just like other higher order tensors. Secondly the representation is the thing that changes, but the meaning of that representation in the old basis and the new basis is the same. For example, if I take the displacement from me to the top of the Eiffel tower, I can represent that in xyz Cartesian coordinates or in spherical or cylindrical coordinates, or I can measure it relative to an origin that starts with me or at sea level at 0 latlong. The representation will be very different in each case, but the actual displacement from me to the top of the Eiffel tower doesn't change. What has happened is the basis vectors transform in exactly such a way as to make that happen. It's a rank 1 tensor in 3 dimensions because there is a magnitude and one direction (one set of 3 basis vectors) in whatever case.

Now if I want an example of a rank 2 tensor think about a stress tensor. I have a steel beam which is clamped at both ends and a weight is on top of it. This is a tensor field. For every point in the beam there are different forces acting in each direction. So you could imagine the beam as made up of a grid of little rubik's cubes. On each face of each cube you have different net forces. (eg at the middle of the beam the forces are mainly downwards due to gravity, at the ends of the beam the fact that the middle of the beam is bowing downards will lead to the "faces" that point to the middle of the beam to be being pulled towards the middle (transverse to the beam and slightly downwards) whereas the opposite face is pulled in the opposite direction because the ends of the beam are clamped. So I need two sets of basis vectors. One set indicates the "face" experiencing the force, one set indicates the direction of the force. Now just like the vector/rank one tensor case I can represent those in whatever coordinate system I want, and my representation will be different in each case, but will mean the same sets of forces in the same directions and applied to the same directions because both sets of basis vectors will transform to make that true. I would call that a rank 2 tensor field because I would express it as a function from a set of spatial coordinates to a thing which has a magnitude and 2 directions (that's what I think of as the tensor). However I understand physicists and civil engineers and stuff just call the whole thing the stress tensor (not the stress tensor field). I could be wrong.

So what I mean when I talk about the reality of the tensor I mean whatever it is the tensor is expressing in the physical universe (eg the displacement from me to the tower or the stress in the beam). From a mathematical point of view I agree of course, mathematical objects themselves are purely arbitrary and abstract. But if you have a bridge and you want to make sure it doesn't buckle and fall down, the stress tensor in the bridge is a real and important fact of the universe that you need to have a decent understanding of.

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lisper 2 days ago | parent [-]

> That's just incorrect though

Quite possible. But that's in no small measure because I have yet to find an actual cogent definition of "tensor" that distinguishes a tensor from an array. (I have a similar problem with monads.)

> So what I mean when I talk about the reality of the tensor I mean whatever it is the tensor is expressing in the physical universe

OK, but then "the reality of a tensor" not depending on the coordinate system has nothing to do with tensors, and becomes a vacuous observation. It is simply a fact that actual physical quantities don't depend on how you write them down, and hence don't change when you write them down in different ways.

	▲	seanhunter a day ago \| parent [-]
		No it’s very important for physics to have a mathematical object that doesn’t change so that you can represent these characteristics of the universe that don’t change. For every observer in every reference frame even though they will use different basis vectors and different components, the combination of basis vectors and components will be the same. That’s extremely powerful. Try the video I linked a few posts above for what I think is a really excellent explanation of what a tensor is (using practical household objects to illustrate everything practically). I think you’ll get it.

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denotational 2 days ago | parent | prev [-]

Right, but if you fill in the shorthand there’s no reason to think it’s circular; it’s just a normal definition at that point, albeit one without much motivation.

	▲	lisper 2 days ago \| parent [-]
		But it's not possible to fill in the shorthand unless you already know what it stands for. Hence: the shorthand is not useful for communicating information, only for social signaling.