Visualization of tensors

Visualization of tensors - part 1

Рет қаралды 565,763

Жыл бұрын

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.

Пікірлер: 526

@HitAndMissLab Жыл бұрын

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!

@Kumurajiva Жыл бұрын

This is why pure math is simpler than physics, or applied math.

@gulamm1 2 ай бұрын

Pure math may be simpler but how it is applied to understand physical phenomena is more important.

@Phi1618033 3 ай бұрын

Finally, a proper explanation in plain English. Been trying to wrap my head around this for over a year now.

@timurgabdyshev1139 3 ай бұрын

why?

@TunaBear64 9 күн бұрын

@@timurgabdyshev1139Let's just say that people don't joke about the definition of a tensor being "a tensor" for no reason.

@jamierbb Жыл бұрын

Time to get smart on a Thursday morning.

@Neerzzz Жыл бұрын

Here its 21:32

@bryanb2653 Жыл бұрын

Big learn

@corrompido7680 Жыл бұрын

why of course the first coment I see in a completely random maths-physics video is one with homestuck pfp sure why not

@Aecor Жыл бұрын

Friday baby

@justanotherguy469 Жыл бұрын

2:10 a.m. Fri. morning. Lovin it!

@user-yi7hl8tt2w 3 ай бұрын

This is the world's easiest explanation of tensors. I wanted to see it when I was a college student over 30 years ago.

@scene2much Жыл бұрын

I've been trying to get a PERFECT understanding of Tensors for decades.... this was Wonderful!

@sethhhhhhhhhhhhhhh Жыл бұрын

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!! :)

@bencrossley647 Жыл бұрын

Worth sharing with your lecturer. They often appreciate a good resource!

@juniorcyans2988 Жыл бұрын

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😂

@bobbwc7011 Жыл бұрын

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

@-danR 11 ай бұрын

This video is the natural second step after Dan (no relation) Fleisch's video introduction to tensors.

@timothyjohnson1511 5 ай бұрын

Thank you for presenting in dark mode, easy on the eyes. Looking for dark mode presentations of electrodynamics with dipole radiation, and accelerating charge.

@davidgoldgruber8541 Жыл бұрын

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.

@not2tired 3 ай бұрын

Studying stress tensors at unicycle!

@yousefamar288 Жыл бұрын

as an engineering student those 12 minutes here have more blessing than a 4 month course of my engineering program in university. Just Perfect

@mikip3242 Жыл бұрын

Finally!!! Someone in KZfaq decides to make the concepto of Tensor easier and meaningful to mortals.

@justanotherguy469 Жыл бұрын

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.

@anjanavabiswas8835 6 ай бұрын

@@justanotherguy469 Fair enough.

@dialectphilosophy Жыл бұрын

Most intuitive and simple explanation of a tensor you can come across on KZfaq!

@visualchallenge2413 Жыл бұрын

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.

@yevgeniygorbachev5152 Жыл бұрын

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.

@cykkm Жыл бұрын

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...

@justanotherguy469 Жыл бұрын

He is amazing and very detailed.

@Oxygenationatom 2 ай бұрын

@pianojay5146 Жыл бұрын

came for tensor, staying for stunning visuals for material mechanics! great job!

@AlexBesogonov Жыл бұрын

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.

@sumairahmad9464 Жыл бұрын

Freaking amazing bro. Much appreciated. Can't thank you enough for taking the time out to make this for us!

@justanotherguy469 Жыл бұрын

RIGHT?!!!

@t3db0t97 5 ай бұрын

Of all the videos I have watched on tensors, this is the first time I've *actually* understood them. Outstanding work!

@gabitheancient7664 Жыл бұрын

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

@rantorakoto8785 Жыл бұрын

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.

@mathscottage5785 11 ай бұрын

Exactly 💯💯💯💯...just amazing...no words...Bravo Bravo......

@laurenwrubleski7204 Жыл бұрын

I've been looking for a good explanation of tensors for years! I'm so excited for the rest of this series!!

@IAmNotARobotPinkySwear 6 ай бұрын

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.

@adamb7088 Жыл бұрын

OMG!! I do believe I now know how important it is to calculate the Eigenvalues and Eigenvectors of a tensor field. Thanks.

@christopherneufelt8971 Жыл бұрын

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.

@adrian_sp6def Жыл бұрын

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!

@eqwerewrqwerqre Жыл бұрын

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

@dank3k Жыл бұрын

This is beyond cool. Please continue with the series!

@joejoejoe3 Жыл бұрын

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!

@billvvoods Жыл бұрын

Awesome visuals and explanations. This is the video I’ve been waiting for all my life. Thanks for producing this.

@justanotherguy469 Жыл бұрын

Great video. It elevates my understanding of tensors to the level of intuition. Love your graphical presentation.

@kimchi_taco Жыл бұрын

holy cow. I realized I didn't understand tensor in my Physics degree. Brilliant work! Thanks!

@evancarpenter Жыл бұрын

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!

@robocop30301 Жыл бұрын

This is the greatest video that was ever made.

@us-Bahn 24 күн бұрын

There’s room for much more graphic instruction in physics & math than what is conventionally used.

@tedsheridan8725 Жыл бұрын

Very cool video. Even after taking two classes involving tensors (years ago), I never fully got them. This was very easy to understand naturally.

@Rodrigoviverosa 3 ай бұрын

FINALLY I understand Tensors, after 10 years of gratuation and all day watching ton of videos on YT haha, thanks!!!

@timothywelk3244 5 ай бұрын

This is incredible. I wish I had your KZfaq page when I was in college.

@tim40gabby25 Жыл бұрын

Hi. As a complete curious amateur, this was perfectly clear. Well done.

@justanotherguy469 Жыл бұрын

That is how you know a teacher knows his stuff.

@fitness60plus52 9 ай бұрын

you know what? this is the first video ever (among zillions) that made me grasp what a tensor is about. 👍👌

@gparizoto Жыл бұрын

Beautiful physical interpretation of tensors, keep doing this great job!

@IoldxmanI Жыл бұрын

This is the best explanation I have come across. Please release the second part.

@JoeyFaller 21 күн бұрын

Beautiful video, especially in showing the construction of a tensor space as the product of vector spaces

@LucyHealthy97 Жыл бұрын

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

@thomaschiesa8202 Жыл бұрын

This deserves a standing ovation, a masterfully excellent video!!!

@rudypieplenbosch6752 Жыл бұрын

This is very interesting, especially seeing electromagnetism as a tensor.

@avocado1998 9 ай бұрын

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.

@vanderkarl3927 Жыл бұрын

I don't know why I wasn't subscribed before considering I watched and enjoyed all the halting problem and Bell Inequalities related videos!

@DevRajyaguru-lx8pi Жыл бұрын

Best explanation of the tensors I found After years, Thank you so much!

@enricolucarelli816 Жыл бұрын

Superb. This is the kind of videos that expand my mind. 👏👏👏👏👏

@marcfruchtman9473 Жыл бұрын

Incredible visualization. Thank you for making this video.

@Ammar2b Жыл бұрын

that's one of the greatest videos on KZfaq for mechanical engineers! Thank you

@wanderrodrigues499 Жыл бұрын

Didática excelente! Vou assistir todos os vídeos do canal!

@alexandreboulch5455 3 ай бұрын

Best video on the stress tensor I’ve seen so far. Thanks!

@davidjones5319 Жыл бұрын

Excellent. Best intro to the concept of tensors I’ve seen. Very helpful

@apoorvmishra6992 11 ай бұрын

Eagerly waiting for part 2 buddy....

@StefanosZormpas 7 ай бұрын

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

@khajiit92 Жыл бұрын

always struggled with understanding tensors and this is the first one that's making it click, thanks so much!

@michaelblankenau6598 9 ай бұрын

Very well explained . Making clear what the geniuses have always understood .

@grindstm Жыл бұрын

This is the tensor visualization I have been waiting for. Thanks!

@megri3441 Жыл бұрын

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.

@alfahad7433 11 ай бұрын

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...🙏🙏🙏

@kalpitkatpara2720 10 ай бұрын

My mind is completely blown by your explanation!!! amazing work! you are a real gem to humanity! 💕✌✌👍👍

@KalebPeters99 Жыл бұрын

This is such a remarkable introduction. Bravo 👏👏👏

@Midazc 8 ай бұрын

Such a clean, concise explanation and visualization. Well done!

@DrDeuteron 7 ай бұрын

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.

@floppy8568 Жыл бұрын

I'm excited for part 2! You make great anumations! Keep it up!

@fbkintanar Жыл бұрын

Nice visualization, it makes the details a lot more memorable.

@patrick1532 Жыл бұрын

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.

@bassamxp Жыл бұрын

This is totally magnificent ... thank you. I'll make sure my students follow this video and the channel

@Edu888777 Жыл бұрын

Really good vídeo, one of the best! Very didactic. Thank You !

@GanerRL Жыл бұрын

i rlly like this style of video, keep it up!

@petersimon985 Ай бұрын

Hi, Thank you for making this great effort. Unselfish and straightforward video. ❤

@ProkofievAMD Жыл бұрын

WOW ! after all these years I finally understand what a tensor is ! 😲 great video. Thanks from France 🇫🇷

@BernardoMartins_ 2 ай бұрын

Beautiful clarity. Masterpiece.

@tf2excession Жыл бұрын

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

@danielobambelo1411 Жыл бұрын

Yes, finally new video, havent even seen it but i know that it will be great, also those who say first are stupid.

@LuisGarcia1992_ Жыл бұрын

Most excellent explanations and visualization!!

@rockstonic52 Жыл бұрын

cant right now honey, new udiprod about math i dont need to know

@najahsaeed2006 8 ай бұрын

Thank you so much for the great effort in this video, all the best, and keep it up

@traderalex655 Жыл бұрын

Great explanation and demonstration. Visualization is key to understanding.

@NoNTr1v1aL Жыл бұрын

Absolutely amazing video!

@gomdangdang Жыл бұрын

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!!

@jeremiahvanhook1114 Жыл бұрын

Visualization is such a key component of understanding all mathematical concepts. Part 2 PaRt 2 PART 2!!!!

@joserodrigomorenolopez6651 Жыл бұрын

Excelent video, makes visual everything that needs to be visualized. Congrats

@EMLtheViewer Жыл бұрын

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!

@chej9 Жыл бұрын

Wait what. That electromagnetic fields part is THE best cliffhanger ive seen in a while.

@maximpavlov5379 Жыл бұрын

A very intersting and comprehensive explanation!

@tinkeringengr Жыл бұрын

Nice animations! Looking forward to more!

@carlosfelipe766 Жыл бұрын

Good job buddy, keep it up. Looking forward to part 2!

@Lleruelu Жыл бұрын

FANTASTIC INTRO!!! THANK YOU VERY MUCH!!!

@Kart-sl2qq Жыл бұрын

This video was just sensational, thank you

@jaybingham3711 Жыл бұрын

Wow...exceptional. So well done. Kudos.

@Songvbm Жыл бұрын

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.

@ale8088 Жыл бұрын

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.

@rohan.fernando Жыл бұрын

Excellent visualisation and clear explanation.

@leonstenutz6003 Жыл бұрын

Appreciate your work and this video so deeply. Thank you!

@MCSteve_ Жыл бұрын

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.

@gorgolyt 7 ай бұрын

Great explanation, fantastic work.

@SwarthmoreDB 3 ай бұрын

Very very informational! Subscribed, and thank you for the lesson! 👍🏻😊

@richardcasey4439 11 ай бұрын

This video series should be included in linear algebra courses

@colafish2152 Жыл бұрын

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.