Visualization of tensors - part 1
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- Опубліковано 21 гру 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.
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?
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!
I've been trying to get a PERFECT understanding of Tensors for decades.... this was Wonderful!
This is the world's easiest explanation of tensors. I wanted to see it when I was a college student over 30 years ago.
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.
as an engineering student those 12 minutes here have more blessing than a 4 month course of my engineering program in university. Just Perfect
Most intuitive and simple explanation of a tensor you can come across on UA-cam!
Finally!!! Someone in UA-cam 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.
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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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......
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.
Thanks! I feel my IQ has increased by 50% in just 10 minutes
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
Of all the videos I have watched on tensors, this is the first time I've *actually* understood them. Outstanding work!
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.
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.
came for tensor, staying for stunning visuals for material mechanics! great job!
OMG!! I do believe I now know how important it is to calculate the Eigenvalues and Eigenvectors of a tensor field. Thanks.
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!
I've been looking for a good explanation of tensors for years! I'm so excited for the rest of this series!!
Very well explained . Making clear what the geniuses have always understood .
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
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
Freaking amazing bro. Much appreciated. Can't thank you enough for taking the time out to make this for us!
RIGHT?!!!
you know what? this is the first video ever (among zillions) that made me grasp what a tensor is about. 👍👌
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!
Eagerly waiting for part 2 buddy....
Awesome visuals and explanations. This is the video I’ve been waiting for all my life. Thanks for producing this.
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!
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.
This is incredible. I wish I had your UA-cam page when I was in college.
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
FINALLY I understand Tensors, after 10 years of gratuation and all day watching ton of videos on YT haha, thanks!!!
holy cow. I realized I didn't understand tensor in my Physics degree. Brilliant work! Thanks!
Superb. This is the kind of videos that expand my mind. 👏👏👏👏👏
Hi. As a complete curious amateur, this was perfectly clear. Well done.
That is how you know a teacher knows his stuff.
This is very interesting, especially seeing electromagnetism as a tensor.
My mind is completely blown by your explanation!!! amazing work! you are a real gem to humanity! 💕✌✌👍👍
This is beyond cool. Please continue with the series!
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 the greatest video that was ever made.
Very cool video. Even after taking two classes involving tensors (years ago), I never fully got them. This was very easy to understand naturally.
Didática excelente! Vou assistir todos os vídeos do canal!
Beautiful physical interpretation of tensors, keep doing this great job!
This is the best explanation I have come across. Please release the second part.
Excellent. Best intro to the concept of tensors I’ve seen. Very helpful
This deserves a standing ovation, a masterfully excellent video!!!
Best video on the stress tensor I’ve seen so far. Thanks!
Really high-quality explanation. Thanks!
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.
Incredible visualization. Thank you for making this video.
Best explanation of the tensors I found After years, Thank you so much!
that's one of the greatest videos on UA-cam for mechanical engineers!
Thank you
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.
Missed it by as many as 50 years!
Such a clean, concise explanation and visualization. Well done!
No way no way no way , can i cry just bcuz i understand it ,daamnnn you did exellent explanation man ❤❤❤❤❤❤❤
This is such a remarkable introduction. Bravo 👏👏👏
This is the tensor visualization I have been waiting for. Thanks!
always struggled with understanding tensors and this is the first one that's making it click, thanks so much!
This is the best video about tensors
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...🙏🙏🙏
I can't wait for part 2!
Nice visualization, it makes the details a lot more memorable.
This video is most credible great fantastic representation of tensor ❤❤❤
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.
I'm excited for part 2! You make great anumations! Keep it up!
This is totally magnificent ... thank you. I'll make sure my students follow this video and the channel
Good job buddy, keep it up. Looking forward to part 2!
Great explanation and demonstration. Visualization is key to understanding.
Excelent video, makes visual everything that needs to be visualized. Congrats
Thank you sir. Please more on Tensors. 🙏🙏🙏
awesome vid broski.
This video series should be included in linear algebra courses
Most excellent explanations and visualization!!
A very intersting and comprehensive explanation!
Excellent visualisation and clear 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.
FANTASTIC INTRO!!! THANK YOU VERY MUCH!!!
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.
Wow...exceptional. So well done. Kudos.
Appreciate your work and this video so deeply. Thank you!
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.
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.
Can't wait for part 2!
Beautiful clarity. Masterpiece.
Ooh man, exciting to find such a good video, but also scary to realize it is 8 months old already without part 2. 😭
Thanks! Part 2 should be ready in a few weeks.
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
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!
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!!
one of the best videos on tensor...thank you...
Nice animations! Looking forward to more!
Wow! This is probably the best demo and explanation that I've seen on YT. I wish that tensors and its application in maxwell's equations could have been covered in undergrad EE rather than basing our knowledge on using the derived formulas. Should I have went for physics instead, or do all undergrad programs ignore tensors?
Great explanation, fantastic work.
Very well done Udi! Thank you!
Really good vídeo, one of the best!
Very didactic.
Thank You !