What's a Tensor?

Dan Fleisch briefly explains some vector and tensor concepts from A Student's Guide to Vectors and Tensors

[@gilzam9183]

I love how he says at the end thank you very much for your time, when in reality he went out of his way to try and express tensor in laymen terms for us.

[@blancaroca8786]

It seems a very traditional reply that many people say without thinking much. It probably would be better to say "Hope you enjoyed or learned something." The thanks for your time thing I often find a bit misplaced but maybe I am missing something? Maybe better used by a headmaster at a parent teacher meeting when asking parents to volunteer for a sponsored run or similar and advising them about school rules and such.

[@NuthanPrasanna]

True knowledge brings humility!

[@Erik20766]

blanca roca with ads in the video the creator does get something from the viewers. Although not much from one individual it could potentially accumulate a considerable about while the phrase also only needs to be uttered once

[@pasijutaulietuviuesas9174]

@@Erik20766 You don't need to get monetary value to be thankful. Back when KZbin didn't offer any ad revenue, people still enjoyed to have audiences. It gives a feeling of accomplishment, a feeling of being wanted and respect. Back in the old KZbin days, people still thanked the audience for watching even though they'd get nothing tangible out of it. It's the satisfaction that's enough. Especially in the educational area. When you teach someone something, even with nothing in return, you still feel a satisfaction, especially if the person you teach expresses a sense of curiosity.

[@patarnababan6440]

@@pasijutaulietuviuesas9174 yes! Iagree with you, it's nice to have audience

[@antoinefdu]

"Thanks very much for your time" Dude. Thank us!? Thank YOU for your time!

[@DynamoDeniro]

Right On!

[@siquod]

Must have taken quite a while to arrange all 81 basis vectors in just the right way. And I noticed that he took care to make the big vector 5 basis vectors long, so he could use that (3, 4, 5) is a Pythagorean triple.

[@calculator4482]

Right! How dare he thank us

[@JonathanPoynter]

Outstanding. There are teachers, Teachers and TEACHERS; and TEACHERS like this are all too rare. It’s a special treat to take a masterclass from someone so thoroughly across the material, that they have the ability to simplify the complex. Thank you.

[@NotDogda]

As a theoretical physicist it's one of the best simple explanations for a tensor I've ever seen. Thank YOU for your time. :-)

[@supa3ek]

If you were a real theoretical physicist you wouldnt be looking at this on youtube......you try hard wanna be !!!!!! So sick of idiots starting a comment with.....As a *.* (engineer, rocket scientist..........)......blah blah blah

[@NotDogda]

@@supa3ek Ok. That´s _one_ way you to give a comment to a real compliment. Well done! 🤡 So. Calm down. The tryin-hard-idiot was now the last time here on your channel.

[@sieni221]

As a math student I hate this video

[@hubertbonnisseur-de-la-bat4858]

@@sieni221 Yeah I don't get why so many enthousiastic people commenting. Like what is the difference between a tensor and a matrix ?

[@jonathanjackson7047]

@@sieni221 As a physicist, I also hate this video. It perpetuates the notion that vectors are arrows. Directed line segments are certainly one application of vectors, but not the only use. It's the usual misconception of equating a mathematical object with one of it's representation and considering them one and the same,

[@RalphDratman]

This video is a good example of the educational riches accumulating on youtube. In fact I hate to use the word educational, because it might evoke stiff, uncommunicative academia. But the learning culture on youtube is just the opposite of that. Here we find people passionate to pass their understanding along, filling in a thousand little corners of knowledge. I think we are in the early beginnings of a historical period in which the internet rewrites most of the rules and experiences of human society. I find it thrilling to be living in such an amazing moment.

[@MichaelHarrisIreland]

+Ralph Dratman Agree with you 100% Not that I understand it but now I know about it.

[@mickelodiansurname9578]

+Ralph Dratman This is where I learned to use Adobe Illustrator rather than paying a graphic designer... its also where I learned to play the guitar properly, its where I learned a million different things that I simply would not have had the finance or ability to learn in years past... when I was a kid the library was great...but you didn't get a musician turn up on a moments notice... or a chef turn up to show you how to tenderize and cook the perfect steak... I agree... education in the next decade will explode out of the box even for kids from 'what you must learn' to 'what would you like to know' ....and both support one another..... Kudos to +Dan Fleisch for taking the time....

[@RalphDratman]

Mick Wright It may be a difficult transition, but our educational system cannot go on with business as usual.

[@mickelodiansurname9578]

Ralph Dratman To be fair the existing methods have been in place a long time..So it won't be any surprise to find legislators the last to wise up.... But there was a time n the past when all children were thought what they needed to know by their community... it got them by... but then obviously picking fruit and hunting antelope is not a skill we really need these days... The problem I experience online, not sure if you'll agree is that you learn more by doing than by listening and nodding your head..... and the web just doesn't have that facility... It is great for learning if you are an adult... because adults being mature are willing to put the work in, even if the work is a little harder and initiative is called for... What I also found missing having done a few courses online is the feature of human interaction... when I was in university I also had friends, a lifestyle, it was more of a life experience, and it helped me to get through the work. I'm not sure how that would translate to distance education.

[@RalphDratman]

Mick Wright Yes, I agree that human interaction is necessary, particularly in the early stages of learning.

[@postalci95]

Brilliant and simple explanation. It reminds me the quote: "If you can't explain it simply, you don't understand it well enough."

[@narendrnayak3629]

Feyman isn't it

[@ramka5861]

@@narendrnayak3629 Albert Einstein ne kahaa thaa yee baath.

[@rome8726]

Well, here's also a quote from the same guy: " We can explain things as simply as possible, but not simpler. "

[@Imran52Feb]

For example Ayzx represents the forces in z and x direction on the area formed by x-z plane whose area vector is in y direction and the vector sum of Ax and Az will be the shear force on the face x-z . Am I right or wrong ? SOMEBODY PLZ REPLY

[@AbhiChenghayi]

@@Imran52Feb Force is not a rank 3 tensor. He gave that example for Axx ( 2 Indices )

[@AdhamMGhaly]

I am a Space Operations Engineer. I have been using vectors for most of my professional life. And I have to say that this is by far the best and most dedicated explanations I have ever seen in my life, either online or offline. I bow to you sir.

[@Anutosh13]

Nilesat

[@samhudson1169]

Looks like you haven't posted for a while, so I hope you're still doing OK. Just letting you know that people are STILL getting huge value out of this, 11 years after you made it. Thank you so much

[@lurantheluranthe6406]

As of 2022 Dr. Fleisch was serving as the resident Grand Canyon Astronomer, so unless something dramatic happened in the last year, I don't think there's much to be concerned about

[@ashleybaker852]

@@lurantheluranthe6406 I just met him this week and he’s still doing awesome! ❤

[@markgreen2170]

@@lurantheluranthe6406 "...something dramatic happened" ...oh my god, you don't think he was abducted by aliens do you?

[@toddpnewton]

No Mr. Fleisch, thank you for YOUR time. Without exaggeration, this is one of the best educational videos I have seen on any subject: evenly paced, concise, clear and direct.

[@rickevans33050]

I agree with Todd Newton; no exaggeration.

[@null_corporation]

i agree with rick evans; no exaggeration

[@ANunes06]

I found myself really appreciating the consistency of the angles for each of the basis vectors on-camera. He established his coordinate system and, by jove, he was committed to it. I wish I'd been able to watch this when I was learning Structural Analysis. For me that was just a book-keeping exercise that eventually spat out "you'll need 8 of them there quarter inch rebars in the concrete here." and "This floor has a safety factor of 2.4" This reminder that all of these values I was keeping track of are related to "real" tensors on the surfaces of the structure sure would've been handy.

[@thebroadjeff17]

I agree with Filipe; no exaggeration.

[@chrisns9712]

I broadly agree with Jeff; no exaggeration.

[@vigneshr2438]

Mildly teared up when he says "Thank you very much for your time"

[@FilipCodes]

Lmao same

[@marlopainter8246]

It shows how grateful he is to teach and his respect for others.

[@temporarychannel4339]

@@marlopainter8246 he meant he was sad it was over

[@tushargumber5655]

Today I studied tensors for the first time in my msc maths class and didn't understand anything but when I searched about it on yt this masterpiece video appeared and now I'd say I know what is tensors. Thanks a lot sir for this practical demonstration of tensors. We need teachers like you in our indian education system.

[@revenevan11]

Teachers like him are sadly few in every education system that I've seen. Here in the USA where I'm from there aren't anywhere near enough of them either. But with the internet, even just a few can make a big difference!

[@julesm3334]

Without a computer graphic or computer for that matter anywhere, this was probably the clearest illumination of vectors and tensors I've ever watched. Outstanding. Genius in it's simplicity.

[@yokcos]

Nononono Dan, thanks for *your* time.

[@VeteranVandal]

Yeah.

[@DavidAndrewsPEC]

Agreed wholeheartedly!

[@Ogremanji123]

Yup. Can't thumbs up this comment enough. Thank you for taking the time to upload this.

[@SisterPegasus]

EXACTLY!

[@zenbauhaus1345]

EXACTLY!!!!

[@richardrichards9180]

This guy is proof that if you want to teach, it's not enough for YOU to know your stuff-you have to have the ability to communicate it to those who DON'T know.

[@edoedo8686]

Exactly... Being a student is also an art...

[@trouty7947]

Yup, you need to be able to known how people that *don't know* think, so you can explain in a way that answers their questions. Some people really struggle with visualisations and work much better with the numbers, where as others need to actually *see* what's happening to understand.

[@richardrichards9180]

@@trouty7947 Absolutely-which is why so many teachers are crap &where they go wrong-failing to realise that THEY need to understand WHY a student does not "get it"& going over&over in the same way won't help-it's stage by stage,simple steps,as here.

[@hebermoreno7963]

Couldn't agree more!

[@DemoniteBL]

z\ou ma

[@tearsintherain6311]

This is maybe the best video or teacher interaction I have had in my life , I can’t believe it, it’s so well crafted and I understood it perfectly thanks to his way of explaining and showing it in the real world. I envy the students of this man.

[@rolfvanderbijl1972]

I think only someone who loves his field of interest and is actually compasionate and understanding about it, can explain so calmly and clearly. In my experience, that's the best way to learn. Thank you for this explanation Mr. Fleish!

[@uva1312]

Damn the twelve minutes flew by in an instant. Incredible explanation.

[@CLASSICALFAN100]

Wow, you didn't say "awesome'. I thought it was *REQUIRED*, nowadays...lol

[@PrimataFalante]

Such a masterpiece in science communication! Saw it the first time years ago, when I first started to learn Tensor Calculus, but I still learn something every time I watch it again.

[@MsBobsnob]

que surpresa não inesperada tu por aqui hahaha. Estou no 3 ano do bacharelado em física e esse video é ótimo tanto pelo visual quanto pela coesão do senhor em explicar.

[@ericlindell317]

you can say that again

[@bieldozap]

Santo Deus! Um Primata Falante!

[@sergiolucas38]

isnt it so? :D

[@tomahzo]

During my 5 years at university going through my undergrad studies I might have had _one_ teacher like this. Going through higher level education is like an endurance test. If you succeed it's _despite_ the teachers rather than _thanks_ to them. If every teacher was like this guy (or if the teachers at least tried) then it would have been a joy through and through. Sad state of affairs, really, but it's very nice to see someone teach who knows how to do it properly :).

[@gregfavret3404]

Really well said!

[@darnthatdreamx]

I have long thought that tenure at universities should be obtainable by publishing _or_ by teaching. We'd get many more excellent teachers and a lot fewer (forced and lame) published papers.

[@jared_bowden]

Your comment about school being an endurance test really resonates with me: My undergrad program's with a university that has good hiring rates out of their technical programs; however, and it has become painfully obvious to me and other students that the main reason for this isn't that the teachers are great as much as it is the work is difficult, and since your Lectures/Professors/TAs/Textbooks are often little help you have to learn everything on your own. This creates a sort of filter where the only people who are able to graduate are the ones that happen to be grittiest and the best at self-teaching, and _of course_ these students are going to be good workers. So, as far as the businesses are concerned, the schools pumping out good students, so as for as the _schools_ are concerned, they're doing it right, even if most the learning is coming from free KZbin videos such as this rather than their insanely expensive 'services.' I'm sure my school isn't uniquely bad in this regard (in fact, it might be better than many) and that this is more of a general problem with technical education around the world. I don't want to just complain without giving any solutions, but it's a really complex issue. @darnthatdreamx 's comment on changing tenure reqs would probably help a lot. In general, making schools more incentivized on teaching (since that's what people are actually paying for) is what's needed.

[@tomahzo]

@@jared_bowden Just as long as they get dedicated teachers into higher education somehow that would solve most of the issue (actually getting there is not trivial of course, but at least the end goal is clear). Also, you don't necessarily need to be the one with the most grit to succeed (even though you need some of that _too_) : Forming study groups is how I pulled through :). Everything's easier if you can work things through together with others.

[@gregfavret3404]

@@jared_bowden Excellent points. Really hit the nail on the head. Thank you for writing this. Videos like this and comments like the ones on this video are the real substantive stuff of learning.

[@stevensenger]

Wow... tenured math prof here. I've always struggled with tensor explanations as either too abstract or too concrete. You've struck a wonderful balance here! Thank you for your hard work!

[@juanfranciscojesushernande3396]

👌👏👏👏👏👏👏👏👏👏👏👏👏👏 this is a great methodology, it is a mix of old school and new media, but what really makes this awesome is the professor's attitude and charisma. Keep on doing an excellent job. 👍

[@SisterPegasus]

I'd love to be one of his students!

[@meghajoshi6442]

I totally agree

[@lindhe]

Juan Francisco Jesús Hernández López I agree 100%

[@brendanlarson3432]

This man is so wholesome. We must protect him at all costs.

[@ytg6663]

Im ready lets go

[@ad2181]

Who are we protecting him from? The ignorant dark forces of chaos and evil like "urban youths", ANTIFA, BLM (Burn Loot Murder). Mathematics has been declared racist. Its the tool of the white man's suppression. And must be erased. Darks times are coming.

[@ad2181]

@Kabir Goyal I'm not joking, BLM wants to deconstruct western ideology.

[@calvinsylveste8474]

@@ad2181 So? Why does this deconstruction of western ideology bother you so much?

[@zahraakhalife9150]

@@ad2181 What you've said is so stupid and its really annoying in a stupid way. I can't argue because what you've said is totally nonsense and silly.... (Why are you commenting on a Mathematics video if you hate Mathematics)

[@Ysehporp]

This is incredible. I just finished a course on data visualization from one of the most accomplished professors in data visualization. Some of his works have many thousands of citations. Despite that, his explanation of tensors was nearly incoherent so I went through the whole class without understanding them. Yet your 12 minute video did what his whole course couldn't. Thank you so much!

[@ahmedovsky7166]

I am a master's student in physics, and this is the first time I understand vectors and tensor with such clarity Thank you for this effort and your time I opened your channel and saw that the last video was uploaded 3 years ago I hope you are well.

[@iron_blood7]

At the end, he said "Thank you very much for your time" but actually we have to thank You for your time!

[@nicorobin7666]

You can feel him putting a good heart into his explanation, ❤

[@Methadras]

How about a great mind behind this explanation. Distilling the complex into the simple.

[@NateROCKS112]

@@Methadras why not both?

[@bhaskartripathi]

In understood this 11 years later after watching it many times when I really paid attention and felt the visualization internally. The video deserves an Oscar!

[@markmiguel7508]

Sir, you are exactly what's missing from education. It is so seldom that one gets such excellent explanations. I really value how you take the time to build up on these concepts and leave away the maths for a moment to explain exactly what a tensor is. Far too often we have lecturers diving in headlong into the mathematics of it all without explaining what it all actually means. Thank you for your willingness to share your knowledge with the world. To you, sir, I take off my hat!

[@ZoeyTangCRE]

I’m in love. KZbin actually had quality content that was worth subscribing for, 10 years ago

[@prajwaldeepkamble6617]

Can anyone take a minute to appreciate how beautifull he made us visualise vectors without animations like 3b1b You really can't tell which is better both are great teacher. Sir Feynman must be smiling from above.

[@fortuna19]

Agreed. I was honestly smiling like a child when the idea of tensors clicked. I’ve studied so much physics, mathematics, and astrophysics while mindlessly computing numbers without a solid, intuitive understanding of what I was even doing (biggest example would be tensors in EM and GR)

[@stibiumowl8644]

Especialy using non-digital techniques make it much more easy to understand, I have at least the feeling. May be because you can allways recap his "pencil-puzzle-animations" if you're about to forget what the vid said, but with AI-Animations, you would fail to reconstruct them in your brain and will forget.

[@mastershooter64]

@@stibiumowl8644 but it's super easy to visualize both animations and explanations using physical objects

[@falconbmstutorials6496]

You deserve a show in TV with an entire crew of assistance and a audience! Thank you sir!!!

[@biturboism]

Pure education with passion for knowledge and love for teaching. No matter how many years of experience one has, it’s always a pleasure to see new ways of presenting these basics.

[@OmNamaShivay79]

Extremely well explained...I wish we had more teachers like him.

[@sasoptimis]

I have many many textbooks, and I have never seen anyone explain vectors and tensors as well as him.

[@aammssaamm]

Cause books written for those who is capable of abstract thinking. Most likely you are not.

[@Quantum-Bullet]

Anne Smith oof

[@seankelly8906]

Yep. I've done a lot of math but not physics so much, and always heard physics students talk about tensors as if they were mystical or powerful. (The invariant properties are, yes) I was surprised that I never had the occasion to work with or really know what they are in any of my math courses. But now that I see this, I realize they actually did match any intuitive on what they should be all along.

[@seankelly8906]

@@aammssaamm not at all. Tensors are only common in specific contexts

[@aammssaamm]

@@seankelly8906 It's the way you look at them for the same reason those people cannot understand books. Math is abstraction.

[@philosopher_sage_07]

Thank you so much for taking the time to do this beautiful explanation. These are times that make me glad that I chose the field I did. Tensors are indeed facts of the universe.

[@Nsimalen]

After watching several popular videos about tensors without understanding them, I finally did it with this one. Just FANTASTIC!!! 👏👏👏

[@AShiga]

15 years after college and finally understood it right off a random yt suggestion. I came here with no goal and learned something really complex! Thank YOU Mr. Fleisch!

[@johnames6430]

This is why we need to re-do the entire education system. This is the way we should be learning. By up-voting videos on different concepts. Eventually you "crowd-source" what is the best way to learn.

[@AIQuotient]

This is probably the best explanation of a 'Tensor' out there! Great illustration of the 'Feynman Technique'. Thanks, Dan for all the effort!

[@jhenafiberafin]

I've just been introduced to tensors in my university mechanics class and didn't understand anything and have been struggling to find a good explanation of what it is and now I finally did, thank you very much!!!

[@solomondawit8866]

Your in-depth knowledge of the subject made this complex topic so simple that even grandparents could understand it. What a masterpiece!

[@19ghost73]

624 scalars disliked this video! ;)

[@ErikB605]

The other 41 are probably some arrogant quaternions.

[@wjblake]

Scalars have no bases? Ha!

[@losthor1zon]

@@wjblake - Well, they seem to be an aimless bunch. They clearly have no direction.

[@ofeyofey]

Currently 708 people don't like this out of some 42,000, so about 1.5% dislike it which means 98.5% like it. That's good going for Prof. Fleisch.

[@4c00h]

Err.. I think you mean "Scholars". **woosh**

[@MazdakNation]

This is excellent. Only wished had found it 30 years ago.

[@guitarttimman]

over 30 years ago, I got an A in a calculus based physics class that utilized this information. Why do you wish? If you're still alive, it's not too late.

[@abhyaskanaujia3862]

kzbin.info/www/bejne/r5iZmmqkpdWed9k I'd recommend

[@vasukhandelwal2233]

Same here too

[@aammssaamm]

At that time it used to be a niche math mostly theoretical due to low power of computers. Internet with big data made a use of it.

[@cjk9988]

One of the best videos on Tensors, no complicated animation and none of the vague jargon. Thank you for making this video!

[@eltonago2200]

As a student researching astrophysics whose struggling with tensors, this was EXACTLY what i needed. thank you for the time and effort put into this video

[@kleinemonnik]

I really wish I had this video when I studied astrophysics two decades ago and was learning about tensors and struggling with them too. It would have helped so much!

[@mrgaudy1954]

Educators like this are worth their weight in gold, particularly when dealing with the often abstract and seemingly intangible concepts of mathematics. Great demonstration.

[@MartinLaBelleIV]

You gotta love a guy that teaches you something and then thanks you for your time.

[@Guesthunter]

And I will never forget what I have learned in this video. That's how well it was executed and how smooth and on point the explanation was.

[@ShrimplyPibblesJr]

You sir, have an amazing gift at illustrating in a way where anyone can understand the maths of basic physics. I was just trying to review what a tensor was because I couldn’t quite remember. I had an amazing college physics teacher, I will say.😊

[@RexGalilae]

<a href="#" class="seekto" data-time="735">12:15</a> It's me who should say "Thanks very much for your time and effort!!" ^_^

[@tintin924]

+Mohammed Zaid I was about to add that comment. But you did before me :-)

[@RexGalilae]

+Tin Tin Are you a fan of Tin Tin like me? Here's a high five!

[@tintin924]

high five bro! Big fan of Tin Tin

[@shanebenjamin7519]

This is the most amazingly clear and simple explanation of Tensors that I have ever seen. Well done sir.

[@arriamirorei1925]

This is the best explanation of what a tensor is that has ever been made. An absolute marvel, thank you very much!

[@aliquliyev6125]

It was a perfect example which explains everything can be simple in the right hands, fully depends on who teaches that. Thank mister, that is one of the best videos I have ever watched. We need teachers like you.

[@frankbholle]

Thanks Professor Fleisch for your work. I really appreciated your book "A Student's Guide to Vectors and Tensors".

[@colinjava8447]

I feel tenser after watching that, good video!

[@seandafny]

I always laugh at shit like this

[@ooloncolluphid7904]

So do I, but it's because he forgot the most important basis vector, the ass-hat.

[@seandafny]

+oolon colluphid dammit !! Lol !!!!

[@MusixPro4u]

Tenser? So you mean your tensor has been gaining some basis vectors in the y direction?

[@fredflintstone9657]

Smoke a doobie....you'll feel better.

[@josephwright8752]

The “Thanks for your time.” after so elegantly picking apart Vectors and Tensors was great! I’m checking out your book, A student's guide to vectors and tensors, from my School Library this afternoon. Thank you so much for taking the time to make this video!

[@brandontylerburt]

I could listen to Mr. Fleisch explain tensors all day long. 💙

[@arianmurillo5752]

I haven't seen such a beautiful explanation since a i learned what's an integral. What a work !!!

[@oliverbecerragonzalez7751]

Professor Fleisch. First of all I want to congratulate you for such a simple yet wonderful to explain what a tensor is. I wish I had a professor like you during my bachelor, because people like you, that devote to make complex concepts easy to understand to people are true Masters. I am looking forward in the future if you could upload more videos like this.

[@drgothmania]

I watch this video after 11 years, but I think people will continue to watch it after 21, 31, 41 years... Thank you for making the concepts so simple.

[@victordashmohapatra3546]

sir with all due respect i would like to thank you for bringing this up, it changed the way i see world.

[@BTC_DNA]

You just did a better job of explaining tensors in a 12 minute video than 4 years of Mechanical Engineering. Where were you twenty years ago?! lol Great job, Dan. Thank you. :-)

[@aghakhanieng]

One of the best intuitive descriptions of vectors and tensors I saw in my life!!! As an Engineer I never heard a better explanation.

[@jkgkjgkijk]

I'm an engineer, I stock shelves at Piggly wiggly

[@smolytchannel5062]

I don't know if it's just me but I'm completely lost at the ending statement where he says something about components and basis vectors

[@wearytrader535]

​@@smolytchannel5062 He said that all reference frames agree on the combination of basis vectors. This makes sense because this is exactly what defines a tensor: it is itself a combination of all possible reference frames. For instance, if you were shooting an arrow at my face, and we both set our reference frames so we would claim that "if the arrow approaches me, it is in the *positive* direction", you and I would contradict each other. I would claim that from your perspective, the arrow is negative in direction. Let's say two people next to us did the same thing except they claimed "if the arrow approaches me, it is in the *negative* direction". Again, they contradict one another. However, if we instead generalized the direction of the arrow as a Tensor and not a Vector, then all these reference frames would agree, by definition. Because the purpose of this tensor is to make all the reference frames agree. The reference frames that once contradicted one another are now simply members of a unified cube (in my example, I think the tensor would be rank 1 and a 2x2 plane...not sure though). I'll admit, everything I said just now might be wrong...this is my first exposure to tensors.

[@smolytchannel5062]

@@wearytrader535 huh, I thought the only thing that was conserved during a vector transformation, the invariant, was the scalar magnitude of the vector

[@wearytrader535]

@@smolytchannel5062 I'm not sure I knew what the heck I was talking about, now that I've actually learned what a tensor is in class...

[@SreenivasanRS]

An amazing explanation, I never knew what Tensors were about until now and now everything just fits in the puzzle. Thank you!

[@iliketrains0pwned]

Thank you so much! I'm taking Advanced Aerodyamics and Aircraft Structures this semester, and it's my first time hearing about tensors. You really helped me intuitively understand what they are much more clearly than my professors could through mathematical proofs and linear algebra!

[@PavlinMavrodiev]

Yeah if you needed lamps shining on a table to understand what a projection is, you shouldn't be in these classes. Your professors did nothing wrong.

[@torzsmokus]

​@@PavlinMavrodiev I had some strange feeling like this guy promised to explain tensors (which I never learned), he explained vectors like I'm five, then changed to tensors and I was like “what? why? how?”

[@raymangoel5372]

I am very very grateful to you sir !! Never ever expected such a beautiful explanation to rather seemingly concrete mathematical idea . You are an inspiration to the generation of teachers!! ☺️🙏🏻

[@ukno918]

Revealing, especially after linear algebra. This stuff is so powerful.

[@yachalupson]

Thank you kindly Dan. 10 years on, and may you still have the same energy and enthusiasm you did then. Good health, you good soul!

[@addy7464]

Best explanation of tensors i have ever seen....... This is the one of the best examples of explaning a very difficult concept in a simple manner.... I cant thank you enough.

[@donmiller6485]

I subscribed when he used the light shadow to demonstrate the x axis, even though I couldn't see the shadow, what a great idea.

[@borg972]

<a href="#" class="seekto" data-time="0">0:00</a>-<a href="#" class="seekto" data-time="680">11:20</a> I feel like he's speaking to me as if I was 5 years old. <a href="#" class="seekto" data-time="680">11:20</a>-<a href="#" class="seekto" data-time="720">12:00</a> I can't understand what the hell is he talking about..

[@darkingsky]

yeah, that escaleted quickly

[@samyuen8234]

if u take physics course, u will learn theory of relativity and this mathematical tools are the core studying.

[@-danR]

He has probably been helping students with this sort of thing for decades, and the reason why a troubled student would come to him in the first place, is because he is utterly lost. Like a 5-year old in Walmart who has lost sight of his mother. As a former ESL instructor, I learned that there is no other mode of expository style. Now, that pertains to the first part. I'm not sure whether the first part is proving problematic on its own or not, but you just have to keep going through it. If you never took vectors at all, then it could be problematic. I don't know. The second part is simply explanation as to why tensors are so powerful. Look at all those unit vectors sticking this way and that. What a mess, and worse, for every (hypothetical) viewer standing around the table, they take on a different aspect; how are they going to agree on what they are looking at? But a different viewing-angle of a _number_ doesn't change the number. Consider: 5 3 9 8 If you stand up, or move your head from side to side, the numbers don't change. His rank 3 tensor, for example, could describe the forces on an atom in a crystal with a beam of light going through it. The atom will be affected by two forces, electric and magnetic, at right angles to each other. We could call those the x and y indices, and the resultant force experienced (by the atom itself) could be the z index. All this would be an impossible mess to visualize, and worse for a computer to do a 'simulation' with. A set of 27 plain old numbers fixes everything for everyone at any time. And then you can surmise the value of tensors in Einstein's general relativity; or more messily, GR near a neutron star. How do you describe everything that's going on with an atom in its vicinity? There is the space-time manifold, there is frame-dragging, there may be intense radiation from an accretion disk (from a cannibalized stellar companion), a magnetic field, who knows what else? It helps to just have a bunch of numbers, and indices accounting for all the needed parameters.

[@lPlanetarizado]

well, you have to learn every detail of what a vector is , then you can move to the second part

[@NickWayne100]

He's basically saying that a tensor in one coordinate system can be transformed to represent the exact same thing in any other coordinate system.

[@fuckedinthepool8956]

Always wished to have a teacher like you, I learned fast and easily, thank you

[@bloodybritbastard]

I think this man summed up my 2 months of math incomprehension at school in 12 minutes… AND GOT THE POINT ACROSS! Bravo sir!

[@noahhysi8622]

Thank you Dan Fleisch! I'm on my way with learning Linear Algebra but I've never come across an intuitive explanation of a tensor until now!

[@dilipsaraftopcoach]

Thank you very much for a lucid explanation of tensors and for making it very graphic and so easy to grasp. You have a gift of making your readers understand!

[@BOBMAN1980]

I literally knew nothing about Tensors until this video popped up. (Learned about scalars and vectors a few weeks ago.) Managed to understand up till about min. 7. . .but the dude is so clear in his communication that it won't be a problem for me to watch again. Learn one more thing. Thanks!

[@navisingh9394]

11 years later this is still one of the best educational videos explaining tensors, I’m trying to understand how tensors are used in diffusion tensor imaging and this is the only video that’s made it make more sense, thank you

[@maidy199]

Meanwhile, a child is wondering where all his toys went

[@KiranMehtaPuneIndia]

Hilarious comment too :-)

[@ir2001]

😂😂😂

[@guitarttimman]

The narcissistic attitude of people these days was predicted in the book of revelations. You should read it, and what will happen to us if we don't humble ourselves. Learning is a very humbling experience. I would rather have a teacher who explains every detail than one who jabbers so fast that nobody can understand him or her. I can't believe the amount of disrespect I am seeing here. I remember seeing some of this when I was going to Purdue, and of course the disruptive ones just lost their money because they were banned from the campus. The problem with higher thinkers is that they sometimes become too naive and actually start thinking that everyone is benevolent, but that's how dangerous knowledge gets into the wrong hands. People who mock and are mean. This guy is too good for you.

[@BlackJar72]

Nah, he seems to be talking to a child ... owner of the toys? To the humorless moron: A little fun, playful kidding, is harmless and would be good for you, you should try developing a sense of humor.

[@guitarttimman]

I think he's an awesome teacher, and I just think it's kind of rude to mock someone who is sharing knowledge. I do have a sense of humor if it's not sadist.

[@himanshubhusanrath8036]

Sir, you are awesome. I can never think of such concepts to be explained in such a detailed and clean way! Thank you so much for your effort. Simply what a beautiful teaching style.

[@syncronic]

Dr. Daniel A. Fleisch deserves immense praise for his incredible talent in explaining the concept of vectors as the foundation for tensor compression. His clear and concise approach makes this complex topic easily understandable, allowing both students and professionals to grasp the intricacies of data compression with confidence. Dr. Fleisch's ability to break down complex ideas and present them in an accessible manner is truly impressive. His passion for the subject shines through in his engaging explanations, making it easy for anyone, regardless of their mathematical background, to follow along and grasp the essence of the topic. I am grateful for Dr. Fleisch's dedication and his exceptional ability to make the concept of vector-based tensor compression come alive. Dr. Daniel A. Fleisch is an outstanding educator when it comes to explaining the concept of vectors as the basis for tensor compression. His approach is both clear and concise, enabling a deep understanding of this challenging subject. I am truly amazed by his ability to make intuitive analogies and connect key points, making it accessible even to those without advanced mathematical knowledge. Dr. Fleisch's passion for the topic is contagious, and his unwavering dedication to making the content accessible for all is truly admirable. I am thankful for his contributions in simplifying such a complex concept and for his exceptional skills as an educator in this field.

[@alexanderschafer8979]

I watched at least 15 different videos in at least 10 attempts to understand what a tensor is. I have now understood the intuition. Thank you so much!!!

[@joshbbloom]

That was awesome. I've been a high school physics & astronomy teacher for 20 years and have a degree in Physics and Astronautical Engineering but never really had to work with tensors (beyond vectors). I've always known about them (but never worked with them) in the context of General Relativity. This is the first time I've seen it explained in a way that makes (mostly) total sense. If I ever teach an advanced Astrophysics course I will have to include this video. Thank you for taking the time to explain it so clearly!

[@debashishchakraborty776]

Thank you sir. Being a teacher myself, I am overwhelmed and deeply moved by your knowledge and style of presentation. I recommend this video not only for novice students but also for experienced teachers.

[@notn0t]

I teach continuum mechanics to graduate students and I still refer them to this video more than a decade after I first saw it. Dan, thank you!

[@Snails6534]

2023 and I cannot tell you how much I needed this video. Visual representation at its finest. I’ve been trying to wrap my head around machine learning but could get tensors.

[@zejalt8608]

What a great teacher. The explanations are just so clear

[@kimareum29]

ZeJAlt it's crystal clear

[@thailander5572]

right! so.. what is tensor ?

[@atyantkumarhota]

Einstein was right. I couldn't feel the 12 mins, while the lectures of my university feel like decade.

[@XxCreateFlowxX]

What you mean like if you cant explain it simply, you dont understand it? I know what you mean about college lol

[@industrialdonut7681]

@@XxCreateFlowxX he's saying "it's all relative" relativity

[@CaptainVita]

I wished i had such a great tutor when i was studying EE and CS. Thank you so much for this great explanation Dan!

[@parthsomani2962]

Thank you, @Dan Fleisch, for such an easy and imaginative explanation for the tensors. Your way of simplifying complex concepts is truly remarkable and I appreciate the effort you put into making it easy for everyone to understand. Keep up the great work!

[@mjw120046]

Excellent, very well done. I took a few physics classes in college (25 years ago, m'kay) and bumped into tensors a few times then, but I couldn't have told you what they were or why they were important until just now.

[@benjamincordes207]

Dear Dan This video really helped me understand tensors for my course i solid state physics. I liked that you used physical explanations because they are very intuitive and easy to remember. I would love to see you make a video about eigenvalues/eigenvectors in a similar style. I know this has been mentioned before in the comments and it's another very important concept in physics and mathematics that a lot of students have problems with. PS. Please don't take rude comments seriously and I hope they don't discourage you from making more videos that would be such a shame. With half a million views you are going to get comments from some sad immature people. I would bet that 99.9 % of people watching this video found it helpful and are very grateful. (as reflected in the many views and likes)

[@brandonscheiber7866]

This video is a blessing, it shows the true potential of youtube for education

[@PlasmaCoolantLeak]

The mark of an expert is someone who can explain a difficult subject using simple objects. Dick Feynman did it with an o-ring and ice water, you're doing it with kids' toys. Well done, sir.

[@composer1663]

Very well done! My compliments! I am a nuclear physicist, and I love the skill with which you described these complex subjects in a way that is understandable to someone hearing them for the first time.

[@AnthonyFrancisJones]

Fantastic explanation - you can only explain in such a simple and engaging way if you really understand your subject! Many thanks for taking the time to make this - clearly lots of background preparation with all the props!

[@saboo480]

As a random guy that doesn't do any math day to day but finds it fascinating, I absolutely loved the care and and time you took to making this video. Thank you so much!

[@parithimathi]

மிக மிக அருமை ஐயா. மூன்று அச்சுகளுக்கான மாதிரி மிகவும் சிறப்பாக இருந்தது. டென்சர் குறித்த விளக்கத்திற்கும் நன்றி.

[@Alley00Cat]

The power of representing data as tensors is that you can do matrix algebra on them. If you program, you'll appreciate that this decreases computation time by orders of magnitude when compared with loops.

[@guitarttimman]

Yes you are correct. I'm wondering, do you know how conical forms work? I took Linear Algebra years ago, and I got an A in the class. I used to be up on all that stuff, but if you don't do it for a long time, then you can forget it. Eigenvectors and values was fun too.

[@iycgtptyarvg]

Very informative! When you said, "scalars are tensors of rank 0" it clicked for me. Thanks!

[@chris_sndw]

This is just a way to organize the numbers. Its like saying a letter is word of 0 degree its actually bullshit but it sounds smart. Hope I could help. 😀

[@raphaelmillion]

@@chris_sndw if it's helpful and makes sense it ain't bullshit

[@JorgetePanete]

@@chris_sndw it's*

[@dietfitnessweightlosstips2830]

Woow - Mr. Fleisch - I feel enlightened and empowered at the same time. You delivered a master pieces for the ages!

[@dennisleas8996]

Dan, you are a gem! Thank you so much for putting this together! If we get you a bigger table, can you explain quantum mechanics? Please?

[@nasekiller]

i think for that he would need a very small table.