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!
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.
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!! :)
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
Thank you for presenting in dark mode, easy on the eyes. Looking for dark mode presentations of electrodynamics with dipole radiation, and accelerating charge.
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.
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.
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.
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
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...
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!
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.
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
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!
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!
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.
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.
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...🙏🙏🙏
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! 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 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
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?
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!
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.
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!!
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.
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.
This was such an exciting video to watch, I can't wait for more! Me: "Okay, there's less than a minute left to the video. Surely there are no more revelations. There's no way it can deepen my foundational understanding of tensors with a slight change to the demonstration..." udiprod: "Let's now drop the requirement for a symmetric matrix. These are now general second-order tensors." Me: :O
Wow, awesome explanation with great presentation, the best introduction to tensors I've ever seen. One small question/nitpick though. When showing the scaling of rows based on vector components around 5:50, it is implied that the vector is a unit vector, right? It doesn't seem to make sense otherwise, so maybe it's obvious enough, but I still feel like it could be mentioned explicitly, just in case.
this video is how i know my university meche department has no clue how to teach. in 12 minutes this video explains away most of the gaps from stress analysis because instead of using tensor math we just goofed off with random simplified equations
Amazing. It has taken me. At least I can start working more on it. It is just adding another 3D-axis on an existing 3D axis. Thanks for such a hardwork
Finally, a proper explanation in plain English. Been trying to wrap my head around this for over a year now.
why?
@@timurgabdyshev1139Let's just say that people don't joke about the definition of a tensor being "a tensor" for no reason.
@@timurgabdyshev1139most of the time, it was defined without any motivation
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.
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!
@@not2tiredbruhh
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 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.
Most intuitive and simple explanation of a tensor you can come across on UA-cam!
as an engineering student those 12 minutes here have more blessing than a 4 month course of my engineering program in university. Just Perfect
This is the world's easiest explanation of tensors. I wanted to see it when I was a college student over 30 years ago.
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.
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.
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.
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......
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
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.
?
Of all the videos I have watched on tensors, this is the first time I've *actually* understood them. Outstanding work!
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!
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.
Freaking amazing bro. Much appreciated. Can't thank you enough for taking the time out to make this for us!
RIGHT?!!!
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.
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
it's crazy how it's simple to understand tensors after this video, Nice work
OMG!! I do believe I now know how important it is to calculate the Eigenvalues and Eigenvectors of a tensor field. Thanks.
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!
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!
came for tensor, staying for stunning visuals for material mechanics! great job!
Great video. It elevates my understanding of tensors to the level of intuition. Love your graphical presentation.
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.
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
Awesome visuals and explanations. This is the video I’ve been waiting for all my life. Thanks for producing this.
you know what? this is the first video ever (among zillions) that made me grasp what a tensor is about. 👍👌
Very cool video. Even after taking two classes involving tensors (years ago), I never fully got them. This was very easy to understand naturally.
Well done. Fantastic visualization. I was watching a UA-cam tutorial by an MIT prof that was slow and boring. In comparison this is brilliant. Thanks!
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
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.
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.
FINALLY I understand Tensors, after 10 years of gratuation and all day watching ton of videos on YT haha, thanks!!!
Wish I saw visualizations like these before I had to tackle the subjects. Very well done, thank you!
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 the greatest video that was ever made.
This is incredible. I wish I had your UA-cam page when I was in college.
This is the best explanation I have come across. Please release the second part.
holy cow. I realized I didn't understand tensor in my Physics degree. Brilliant work! 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 beyond cool. Please continue with the series!
Best video on the stress tensor I’ve seen so far. Thanks!
This deserves a standing ovation, a masterfully excellent video!!!
My mind is completely blown by your explanation!!! amazing work! you are a real gem to humanity! 💕✌✌👍👍
that's one of the greatest videos on UA-cam for mechanical engineers!
Thank you
Beautiful physical interpretation of tensors, keep doing this great job!
Very well explained . Making clear what the geniuses have always understood .
Superb. This is the kind of videos that expand my mind. 👏👏👏👏👏
This was perfect. The physical intuition of a stress tensor really helped make everything click!
Excellent. Best intro to the concept of tensors I’ve seen. Very helpful
Best explanation of the tensors I found After years, Thank you so much!
Hi. As a complete curious amateur, this was perfectly clear. Well done.
That is how you know a teacher knows his stuff.
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.
Incredible visualization. Thank you for making this video.
Eagerly waiting for part 2 buddy....
always struggled with understanding tensors and this is the first one that's making it click, thanks so much!
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.
Such a clean, concise explanation and visualization. Well done!
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
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?
This is such a remarkable introduction. Bravo 👏👏👏
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!
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.
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!!
This is the tensor visualization I have been waiting for. Thanks!
Very simple and powerful explanation. Keep making more videos.
Beautiful video, especially in showing the construction of a tensor space as the product of vector spaces
Hi, Thank you for making this great effort. Unselfish and straightforward video. ❤
This is very interesting, especially seeing electromagnetism as a tensor.
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.
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.
WOW ! after all these years I finally understand what a tensor is ! 😲 great video. Thanks from France 🇫🇷
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.
Great explanation and demonstration. Visualization is key to understanding.
I'm excited for part 2! You make great anumations! Keep it up!
Nice visualization, it makes the details a lot more memorable.
This is totally magnificent ... thank you. I'll make sure my students follow this video and the channel
A very intersting and comprehensive explanation!
Excelent video, makes visual everything that needs to be visualized. Congrats
This was such an exciting video to watch, I can't wait for more!
Me: "Okay, there's less than a minute left to the video. Surely there are no more revelations. There's no way it can deepen my foundational understanding of tensors with a slight change to the demonstration..."
udiprod: "Let's now drop the requirement for a symmetric matrix. These are now general second-order tensors."
Me: :O
Wow, awesome explanation with great presentation, the best introduction to tensors I've ever seen. One small question/nitpick though. When showing the scaling of rows based on vector components around 5:50, it is implied that the vector is a unit vector, right? It doesn't seem to make sense otherwise, so maybe it's obvious enough, but I still feel like it could be mentioned explicitly, just in case.
This video is most credible great fantastic representation of tensor ❤❤❤
Didática excelente! Vou assistir todos os vídeos do canal!
this video is how i know my university meche department has no clue how to teach. in 12 minutes this video explains away most of the gaps from stress analysis because instead of using tensor math we just goofed off with random simplified equations
Really good vídeo, one of the best!
Very didactic.
Thank You !
No way no way no way , can i cry just bcuz i understand it ,daamnnn you did exellent explanation man ❤❤❤❤❤❤❤
Thank you so much for the great effort in this video, all the best, and keep it up
I don't know how to appreciate your effort. Thanks a lot. You truly saved me.:))
Amazing. It has taken me. At least I can start working more on it. It is just adding another 3D-axis on an existing 3D axis. Thanks for such a hardwork
i rlly like this style of video, keep it up!
Awesome! Every time I'll teach linear algebra I'll make sure everyone see this video!
Most excellent explanations and visualization!!
Finally, a proper explanation of Tensors
Good job buddy, keep it up. Looking forward to part 2!
This video series should be included in linear algebra courses