Visualization of tensors - part 1
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- čas přidán 21. 12. 2022
- This video visualizes tensors. It shows some introduction to tensor theory and demonstrates it with the Cauchy stress tensor.
Future parts of this series will show more theory and more examples.
It talks about the term 'tensor' as used in physics and math. In the field of AI the term 'tensor' was borrowed with a simplified meaning. In AI it simply means a multi-dimensional array. So for example the array "double a[4][6]" (4 by 6 matrix of doubles) is called a second-order tensor, but there's no special relationship to vector spaces, as shown in the video.
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Time to get smart on a Thursday morning.
Here its 21:32
Big learn
why of course
the first coment I see in a completely random maths-physics video is one with homestuck pfp
sure why not
Friday baby
2:10 a.m. Fri. morning. Lovin it!
This explanation with red and blue spaces is so good.
It takes usually a whole year of studying to reach this level of intuition, and this video does it all in 10 minutes.
Thanks for making the effort!
This is why pure math is simpler than physics, or applied math.
Pure math may be simpler but how it is applied to understand physical phenomena is more important.
Finally, a proper explanation in plain English. Been trying to wrap my head around this for over a year now.
why?
This is a wonderful introduction!! This past semester, I took linear algebra and differential equations. Tensors were hiding in the background, as boogeymen that our teacher warned us we would come across later.
Your visualizations were both beautiful and clear. Excited for the rest of the series!! :)
Worth sharing with your lecturer. They often appreciate a good resource!
Oh I’m gonna take differential equations this coming semester and linear algebra next semester. This is the first video I watched in 2023! I was wondering where I would encounter tensor, then you leaked a hint😂
Tensors are like John Wick ...it's not just the boogeyman, it's the thing you sent to kill the fucking boogeyman.
Also, this is a very "accessible" introduction via physics and engineering. I've had a mathematician as professor when I took tensor calculus as a university-level engineering scientist.
An absolute nightmare because the first half of the semester was pure mathematics, tensorproduct, tensor bundles, blargh, and only after a Guantanamo-level of torture - when you had gone blind from all the superscript and subscript indexes - the "practical" stuff started to show itself on the horizon.
That class "Tensorrechnung and Tensoranalysis für Ingenieure" / "tensor algebra and tensor calculus for engineers" was my happiest 3.0 ("satisfactory", C) ever and I was the third best in class with that result LOL (such results used to be normality at German universities in the old pre-Bologna system).
Sometimes mathematicians are simply evil people. :D
This video is the natural second step after Dan (no relation) Fleisch's video introduction to tensors.
Thank you for presenting in dark mode, easy on the eyes. Looking for dark mode presentations of electrodynamics with dipole radiation, and accelerating charge.
I've been trying to get a PERFECT understanding of Tensors for decades.... this was Wonderful!
Best and most compact explanation/visualization of the Cauchy stress tensor I have seen so far. Wish this video had existed a few years ago when I studied the topic at uni.
Studying stress tensors at unicycle!
as an engineering student those 12 minutes here have more blessing than a 4 month course of my engineering program in university. Just Perfect
Finally!!! Someone in CZcams decides to make the concepto of Tensor easier and meaningful to mortals.
You know, I was just thinking the same thing, but then it dawned on me that most mortals do not even know what a scalar is.
@@justanotherguy469 Fair enough.
This is the world's easiest explanation of tensors. I wanted to see it when I was a college student over 30 years ago.
Most intuitive and simple explanation of a tensor you can come across on CZcams!
It is amazing what you can learn in 12 minutes ! I have no use for this information but I watched this video in order to get rid of the mystification about this subject that bothered me for more than 30 years. Thank you for this great effort of explanation.
I'm working through the Eigenchris series right now, and I'm really excited to see other math/physics youtubers take a crack at it.
Eigenchris' lectures are amazing! He disposes with a few common conventions, both in notation (simply AB instead of common A⊗B); and, especially important, coordinate conventions (he doesn't normalize the unit in the direction of θ by 1/r), which makes their action the same as that of partial derivative. They _are_ PD's! The first time I saw that, I thought, wowzers, why do the most textbooks manage to kill this correspondence, so natural? First by tearing the 1/r out of the object, then sticky-taping it to it, because from now on you're bound to carry them together. And they don't mean anything anymore...
He is amazing and very detailed.
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This is the best introduction I've seen! People typically try to introduce tensors from the other direction, from the abstract side, and only at the end moving to matrix representation.
OMG!! I do believe I now know how important it is to calculate the Eigenvalues and Eigenvectors of a tensor field. Thanks.
you have no idea how good I feel seeing the springs moving with that specific sound, pls never abandon this sound, I need it together with springs, springs are so much cooler with this sound added
came for tensor, staying for stunning visuals for material mechanics! great job!
Did you have enough? Are you happy? Finished? I am ready to have tears in my eyes, this is probably the best description of a complicated subject that I ever seen. Really beautifully made with clarity and insight. Bravo, bravo.
Eagerly waiting for part 2 buddy....
From the bottom of my heart, thank you 😭❤️ you simply did an amazing work. I've been unsuccessfully trying to understand this for 3 years, I eventually dropped out and just moved forward using tensors without having any idea of what it really is, but now, thanks to you, after 11 minutes, It's finally clear to me, so thank you very much.
Exactly 💯💯💯💯...just amazing...no words...Bravo Bravo......
Of all the videos I have watched on tensors, this is the first time I've *actually* understood them. Outstanding work!
I've been looking for a good explanation of tensors for years! I'm so excited for the rest of this series!!
This is like discovering the holy grail of explanation videos.
Saved, and I'm gonna download an offline copy just in case.
Thank you so much, subscribed.
The shortest and best vid aboud tensors! I really love short and compact videos like this. I do not have enough time on my life to watch >10min videos. In this video there is only knowledge without unnecessery staff around. BIG THANKS!
Pleaaasseee pleaaasssee post more of this series. There's such a lack of intelligible introductions to tensors on the internet. I've seen eigenchris's videos too and I think this is the best possible addition to his work as it approaches from an entirely different angle (visualization) and is extremely valuable to me. I would be eager to see why EM tensors are antisymmetric or really any continuation of this series, this was a great video. I've subscribed in the hopes of more
At the end of my Calc III course a few weeks back I randomly stumped upon the idea of a tensor (namely trying to figure out if ∇𝐅 was meaningful,) so this couldn't have come out a better time. Definitely going to keep up with the series!
Holy smokes, this is the explanation that’s making it click for me
After years of hearing how it’s a map and invariant and all the other things, seeing the relationship between the two spaces and exploring the relationship between the components helps a lot, incredible video!
Very well explained . Making clear what the geniuses have always understood .
Best video I've ever watched. So clearly explained and the 3D visualisation is incredibly helpful when you're trying to learn this stuff. It's so hard to learn this just on paper in 2D without animations.
Awesome visuals and explanations. This is the video I’ve been waiting for all my life. Thanks for producing this.
Didática excelente! Vou assistir todos os vídeos do canal!
Great video. It elevates my understanding of tensors to the level of intuition. Love your graphical presentation.
You have a beautiful brain! I feel so excited to watch your video. Please post visualization of tensors - part 2. I'm waiting for it
Great video for engineers. If you move beyond vector a arrows, and say, replace “Z” with cosine theta, then vectors are the things that can be rotated amongst themselves, and require a 360 degree rotation to remain unchanged. There are 3 of them.
Rank 2 tensors are such things that require 180 deg. There are five of them.
Rank 3 require 120 degrees, there are 7 of them.
Keep going, and you have the spherical harmonics, which are a great way to visualize tensors, since you can draw them.
Btw: the reason we even care about vectors and tensors is exactly these property under rotation, and ps: there are things that require 720 degrees of rotation to be unchanged. Hint: there are 2 of them.
Freaking amazing bro. Much appreciated. Can't thank you enough for taking the time out to make this for us!
RIGHT?!!!
Very cool video. Even after taking two classes involving tensors (years ago), I never fully got them. This was very easy to understand naturally.
holy cow. I realized I didn't understand tensor in my Physics degree. Brilliant work! Thanks!
you know what? this is the first video ever (among zillions) that made me grasp what a tensor is about. 👍👌
This is the best explanation I have come across. Please release the second part.
This is beyond cool. Please continue with the series!
This is incredible. I wish I had your CZcams page when I was in college.
Best video on the stress tensor I’ve seen so far. Thanks!
Hi. As a complete curious amateur, this was perfectly clear. Well done.
That is how you know a teacher knows his stuff.
Incredible visualization. Thank you for making this video.
Beautiful physical interpretation of tensors, keep doing this great job!
Excellent. Best intro to the concept of tensors I’ve seen. Very helpful
Superb. This is the kind of videos that expand my mind. 👏👏👏👏👏
Best explanation of the tensors I found After years, Thank you so much!
My mind is completely blown by your explanation!!! amazing work! you are a real gem to humanity! 💕✌✌👍👍
This deserves a standing ovation, a masterfully excellent video!!!
that's one of the greatest videos on CZcams for mechanical engineers!
Thank you
This is the greatest video that was ever made.
FINALLY I understand Tensors, after 10 years of gratuation and all day watching ton of videos on YT haha, thanks!!!
always struggled with understanding tensors and this is the first one that's making it click, thanks so much!
4:24 Ugh, dude, I had to figure this out for myself when I took a mechanical design course in school. Prof just jumped straight into doing problems without explaining anything about what she was doing. Glad my understanding of it has been validated though.
awesome! awesome! awesome! For the first time I understood the topic a bit. Please continue the series as I believe that there are a huge number of students feeling like me...🙏🙏🙏
This is such a remarkable introduction. Bravo 👏👏👏
Such a clean, concise explanation and visualization. Well done!
the Chinese say "a picture, a thousand words" and I say "a video of yours, five years of university". My congratulations, keep up the excellent work
This is the tensor visualization I have been waiting for. Thanks!
Thank you for this excellent demonstration. I'm able to imagine how machine learning models learn from the data. It was looking like a magical mystrious for me.
Nice visualization, it makes the details a lot more memorable.
Really high-quality explanation. Thanks!
I don't know why I wasn't subscribed before considering I watched and enjoyed all the halting problem and Bell Inequalities related videos!
This is very interesting, especially seeing electromagnetism as a tensor.
Great explanation and demonstration. Visualization is key to understanding.
FANTASTIC INTRO!!! THANK YOU VERY MUCH!!!
Beautiful clarity. Masterpiece.
A very intersting and comprehensive explanation!
Wow! Thanks a lot for this valuable content Sir!
"2 vector spaces" related together by a correspondence between "label", the weighted raw combination of the matrix to obtain the vector that acts
on a particular inclined slice (matrix product: namely a linear transformation) and
the example of the sphere, where every direction in the first space have a realted direction in the second space,
make finally reasonable understand why, in abstract math, tensors are defined directly like linear map between dual spaces.
Whith only a table of 9 number (6 due to simmetry of this pshysic problem) it is possible to manage any slice no matter its orientation.
Obvious: only if we assume hypothesis of linearity is true (and locally it is).
If we think about this, it results amazing how mr Cauchy, a great Franch engeneer, in middle of 1800, realized all those ideas
without linear algebra, rather founding linear algebra itself and matrix calculus.
This is totally magnificent ... thank you. I'll make sure my students follow this video and the channel
Most excellent explanations and visualization!!
Definitely a valuable video tutorial for me. My request to the uploader is to provide the visualization of "tensors applied in the domain of Statistics & ML", if possible. Thank you.
As an engineering major who has seen this math in many structures classes, but it's always been presented as simply a matrix representation of a system of linear equations. I've never realized that I was using tensors before.
Missed it by as many as 50 years!
I'm excited for part 2! You make great anumations! Keep it up!
Excellent visualisation and clear explanation.
Visualization is such a key component of understanding all mathematical concepts. Part 2 PaRt 2 PART 2!!!!
Excelent video, makes visual everything that needs to be visualized. Congrats
This is an incredible video, I see so many people have struggles with tensors & change of basis. If you can bring in some intuition for the universal property as well, this will be the single canonical series to explain them
Great explanation, fantastic work.
You are amazing. I heard that tensor is the object which does not depend on change of coordinate. Now it is somehow clear to me, but I am really looking forward to the next video. Thank you!!
Very well done Udi! Thank you!
Can't wait for part 2!
Thank you so much for the great effort in this video, all the best, and keep it up
I imagine you can find the the new coordinate system with no apparent matrix sheering by figuring out the eigenvectors.
Also with that anti-symmetric matrix, I imagine it has the property of no divergence yet has curl, which satisfies the property of electromagnetic fields (according to Maxwell's equations). I just don't understand what makes this connection of symmetry and zero divergence.
Amazing video, I think I finally understand the idea of tensors.
I cannot stress enough how much this video helped me. (pun unintended, but welcome)
Just reading about tensors was not enough, and I always wondered what exactly made them different from matrices. Thanks for this explanation!
one of the best videos on tensor...thank you...
Appreciate your work and this video so deeply. Thank you!
Good job buddy, keep it up. Looking forward to part 2!
I am waiting the part 2 since this video uploaded. Please continue this incredible video
Thanks! Part 2 should be ready in a few weeks.
Wow..amazing animation..and explanation.
Nice animations! Looking forward to more!
Absolutely amazing video!
I can't wait for part 2!
Amazing video, it give a vivid visual explanation to some very abstract mathematics-physics concept. Actually, I spend lot's of time to study these concept by myself, but I still can't fully understand these concept, what I learn from this video it worth then these book i read.
Wow...exceptional. So well done. Kudos.
No way no way no way , can i cry just bcuz i understand it ,daamnnn you did exellent explanation man ❤❤❤❤❤❤❤
Really good vídeo, one of the best!
Very didactic.
Thank You !
This is the best video about tensors
Really good introduction!!!