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0 pointstagrun4y ago0 comments

Which mathematician disagrees with what exactly?

Tensors are introduced by physicists to ensure various physical quantities (which involve coordinates and their derivatives) do not depend on the arbitrarily chosen coordinate system. This is ensured through the transformation properties of tensors.

The name tensor itself comes from the theory of elasticity, Cauchy stress tensor, which BTW is uniform in many practical cases, and obeys the following tensor transformation rule:

https://en.wikipedia.org/wiki/Cauchy_stress_tensor#Transform...

like any other (contravariant) tensor must.

Matrices are not examples of tensors. Matrices can be used for representation of tensors, in which case tensor product becomes Kronecker product, but matrices in general don't have to represent tensors. You can put anything, including your favorite colors or a list of random numbers, in a matrix, and it won't be a tensor in general, not unless it must transform like a tensor under coordinate system changes.

Similarly, TensorFlow "tensor" is just a multidimensional data array, with no transformation rules enforced on it, and therefore is not a tensor.

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clairity4y ago

i did a whole phd-level course (a long time ago) in deformable materials, which was entirely based on tensors, and i still don’t know how to differentiate one from vectors/matrices. even the idea that tensors must obey coordinate transforms doesn’t really do it, since the practical applications of vectors/matrices do so as well.

it’s like some people invented a new word and won’t tell you what it actually means in sufficient detail to differentiate it from all the other words you know. so you keep using it with others in the hopes that contextual information will finally make it clear. one day.

abdullahkhalids4y ago

I just taught a module on Tensors in one of my physics courses. The mistake lots of people make is not show examples of matrices that are not tensors. But this is really difficult to do in physics courses because all physics matrix-like-objects must be tensors. Any theory that has non-tensor like objects in it will necessarily fail as soon as you change your coordinate system.

Thankfully, there is a great historical example of this. The electric field vector \vec{E} = (E_x, E_y, E_z), is not a tensor. It doesn't obey the tensor-transformation law. Similarly, the magnetic field vector is not a tensor. These are matrices, but not tensors.

As you know the Electromagnetic tensor [1] is the tensor that correctly transforms under coordinate transformation, and hence allows different observers to agree with each other.

[1] https://en.wikipedia.org/wiki/Electromagnetic_tensor

clairity4y ago

thanks for the examples. i vaguely recognize the term 'electromagnetic tensor', but i have to admit i didn't know that it was special in that way. i can barely spell 'tensor' at this point.

ps - one thing that always annoyed me was the limitation of linearity in so many of these models (which i totally understand why, but still). all the interesting real-world stuff happens non-linearly...

abdullahkhalids4y ago

Indeed it is annoying. But the models are linear not because physicists mistakenly believe that the world is linear, but because in most cases linear models are the only ones that one can solve to get qualitative predictions out. Non-linear models can be constructed as well, and then numerically solved on computers to get exact answers. But a physicist is one who understand the essential qualitative features of the world, rather than one who can compute understanding-free numerical answers.

In the words of Dirac, "I consider that I understand an equation when I can predict the properties of its solutions, without actually solving it." This usually only works if your equations are linear.

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vcxy4y ago

Interesting, you use the word matrix differently than me. The way I use it, a 2d array isn't necessarily a matrix. It's only a matrix if it represents a linear map between vector spaces with respect to chosen bases. Then again, I'm not much of a programmer, but I've taught linear algebra a few times. My head is just in a different place I guess.

tagrunOP4y ago

Any 2D array of data is a valid matrix.

But not any multidimensional data is a valid tensor.

vcxy4y ago

I'm not sure I buy this after you just used an example of a 2d array of you're favorite colors just a few minutes ago. Maybe I'm missing something. What kind of linear transformation does that represent, and between what vector spaces?

xyzzyz4y ago

(the exchange between you two is a perfect example of the disconnect between how mathematicians see linear algebra, and how physicists do)

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tagrunOP4y ago

My favorite colors are 1, 2, 3 and 4 in some 8-bit grayscale color space. Here's a matrix

  [ 1 2 ]
  [ 3 4 ]

This represents some linear transformation, and is a matrix. Are you trying to argue otherwise?

FYI, matrices transform between vectors within a vector space.

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clairity4y ago

abdullahkhalids4y ago

As you know the Electromagnetic tensor [1] is the tensor that correctly transforms under coordinate transformation, and hence allows different observers to agree with each other.

[1] https://en.wikipedia.org/wiki/Electromagnetic_tensor

clairity4y ago

thanks for the examples. i vaguely recognize the term 'electromagnetic tensor', but i have to admit i didn't know that it was special in that way. i can barely spell 'tensor' at this point.

abdullahkhalids4y ago

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vcxy4y ago

tagrunOP4y ago

Any 2D array of data is a valid matrix.

But not any multidimensional data is a valid tensor.

vcxy4y ago

xyzzyz4y ago

(the exchange between you two is a perfect example of the disconnect between how mathematicians see linear algebra, and how physicists do)

1 more reply

tagrunOP4y ago

My favorite colors are 1, 2, 3 and 4 in some 8-bit grayscale color space. Here's a matrix

  [ 1 2 ]
  [ 3 4 ]

This represents some linear transformation, and is a matrix. Are you trying to argue otherwise?

FYI, matrices transform between vectors within a vector space.

2 more replies

j / k navigate · click thread line to collapse