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Why tensors? A beginner's perspective

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Re: Why tensors? A beginner's perspective

#101

Post by abdullahkhalids »

clairity wrote: post #93
tagrun wrote: post #74

Earlier quoted context omitted.

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…

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 su…

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

Re: Why tensors? A beginner's perspective

#102

Post by mettamage »

725686 wrote: post #21

A wonderful little video to understand what tensors are, by Daniel Fleish: https://www.youtube.com/watch?v=f5liqUk0ZTw Very simple and basic. Edit: incorrectly wrote vectors instead of tensors.

Wow, that's such a good video. Thanks! Haha, mind blown really. And other than graph theory, I never took a college level math course (I artfully skipped almost all math during my CS degree), I'm doing pre-calculus at the moment, because I want to get better at it.

Re: Why tensors? A beginner's perspective

#103

Post by clairity »

abdullahkhalids wrote: post #101
clairity wrote: post #93

Earlier quoted context omitted.

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 su…

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…

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...

Re: Why tensors? A beginner's perspective

#104

Post by abdullahkhalids »

clairity wrote: post #103
abdullahkhalids wrote: post #101

Earlier quoted context omitted.

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…

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...

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.

Re: Why tensors? A beginner's perspective

#105

Post by clairity »

abdullahkhalids wrote: post #104
clairity wrote: post #103

Earlier quoted context omitted.

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...

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 w…

yah, as an engineer[0], i totally get the solvability angle, and even the physicist's core desire to be able to test (and predict via) the math rather than the physical manifestations (which may be impossible to test directly), but i'm eager to see us advance deeper into the non-linear, since that's where it gets really interesting. like, how do proteins really work? or multi-body energy fields? we're in the infancy of really understanding all this stuff. the future is stochastic and non-linear. in a thousand years, people might look back with amusement on how ignorant we were with our puny little linear models and deterministic computers. =)

[0]: but at this point, not really. even in grad school, i only did linear modeling, and relatively rudimentary ones, at that.

Re: Why tensors? A beginner's perspective

#106

Post by hinkley »

groby_b wrote: post #90
hinkley wrote: post #37

Earlier quoted context omitted.

Randall Monroe has a comic about how most people need enough math to be able to handle a birthday dinner where the guests split the bill for the birthday boy/girl evenly and pay for their meals and tip separately. That’s a pretty good bar and I wonder if we could just cut to that chase earlier. But I also believe that people need enough math to see when they’re being cheated, and I feel like you could just tell middl…

Except that humanity desperately needs a better understanding of probabilities and non-linear relationships. We don't use more than division because we haven't succeeded teaching more, not because nobody needs it.

"Thinking Fast and Slow" makes a quite extended argument that our brains have a faulty intuition about probabilities, and I nodded along thinking of all the bad decisions I've watched my teams make over the years (either noticed retrospectively, or presaged by myself or some other old hat).

If you have a way to fix this, you would be set for life, going around playing a sort of corduroy jacketed Robin Hood, keeping the rich from stealing from the poor.

Re: Why tensors? A beginner's perspective

#107

Post by bmitc »

billfruit wrote: post #96
bmitc wrote: post #56

Earlier quoted context omitted.

I’m not sure I understand the question enough to answer. Do you mean something like differential geometry? There, the theory is built upon vectors and covectors (i.e., differential forms) that are associated with tangent spaces and cotangent spaces, respectively. But that is modern differential geometry and not classical geometry.

I was asking for a classical geometry book which is using a treatment using vectors. Usually classical geometry is treated without resorting to vectors.

I'm still not really sure what you mean.

The most "modern" treatment of classical geometry that I know of is Geometry by Brannan, Esplen, Gray. It might be worth a look.

Re: Why tensors? A beginner's perspective

#108

Post by billfruit »

bmitc wrote: post #107
billfruit wrote: post #96

Earlier quoted context omitted.

I was asking for a classical geometry book which is using a treatment using vectors. Usually classical geometry is treated without resorting to vectors.

I'm still not really sure what you mean. The most "modern" treatment of classical geometry that I know of is Geometry by Brannan, Esplen, Gray. It might be worth a look.

That is still not using vectors as much. My question is if classical geometric problems can be solved using vectors, using vector concepts for example using cross products for area expressions?

Re: Why tensors? A beginner's perspective

#109

Post by groby_b »

hinkley wrote: post #106
groby_b wrote: post #90

Earlier quoted context omitted.

Except that humanity desperately needs a better understanding of probabilities and non-linear relationships. We don't use more than division because we haven't succeeded teaching more, not because nobody needs it.

"Thinking Fast and Slow" makes a quite extended argument that our brains have a faulty intuition about probabilities, and I nodded along thinking of all the bad decisions I've watched my teams make over the years (either noticed retrospectively, or presaged by myself or some other old hat). If you have a way to fix this, you would be set for life, going around playing a sort of corduroy jacketed Robin Hood, keeping t…

I have a corduroy jacket... so, halfway there? ;)

Realistically, it is (somewhat) fixable in small contexts. I've worked (and continue to work) on teams that are somewhat decent at risks and probabilities, but it's definitely an exceptional experience.

I don't know how to widely teach that. But I'm not yet ready to give up and say "it can't be taught", because those folks on my team are the counterexamples.

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