What the HECK is a Tensor?!?

What the HECK is a Tensor?!?


Thanks to Brilliant for helping support this episode. Hmm, tensors. Holo Clone! What do mathematicians say about tensors? A rank-n tensor in m-dimensions is a mathematical object that has n indices and m-to-the-n components that obeys certain transformation rules. Pfff! We can do better than that! This episode was made possible by generous supporters on Patreon. Hey Crazies. If you’re like me, you find definitions in textbooks incredibly unsatisfying. In their defense, definitions like this one are correct, complete, and concise. We call them the three c’s. (Off Camera) No one calls them that! I call them that! Anyway, correctness is absolutely vital. It doesn’t matter how clear your explanation is if it’s wrong, but the other two are very flexible, so let’s see what we can do. By the end of the video, you should understand this definition with all its math speak. To get there though, we’re going to need a little context because that [BEEP] is abstract as [BEEP]. The word “tensor” actually comes from an old Latin word meaning “to stretch.” If you pull an object outward along its length, it experiences something called “tensile stress.” In response, it’s length increases. Which, you know, makes sense. Except, that’s not the only type of stress an object can experience. This cube could be stretched or compressed along any of the 3 spatial directions. Wouldn’t a vector be enough for that? First, a vector is a tensor and, second, there are 6 other stresses I haven’t mentioned yet. The cube could also be sheared along those directions. That’s 9 possible stresses. Yeah, but can’t we just add the forces together along each direction? No. No we can’t. Each of these forces makes the cube respond in a different way. We have to consider them all separately. These 9 different stresses are usually organized into a 3-by-3 matrix called the stress tensor. Quick Disclaimer: It’s not a tensor because I can write it as a matrix. Matrices and tensors are not the same thing. A matrix is just a convenient way to organize numbers sometimes. Writing the stress tensor like this, we can see it clearly has 9 components, but our definition mentioned two specific properties: Rank and Dimension. This cube is 3-dimensional, so any tensor describing it’s behavior will also be dimension-3. That’s why our stress tensor is organized into 3 rows and 3 columns. Each corresponds to a specific direction in 3-dimensional space. Seriously. It’s that easy. Rank is the amount of information you need to find a specific component. In this case, we only need a row and a column. That’s 2 pieces of information, so we say the tensor is rank-2. The stress tensor is rank-2 and dimension-3. Matrix notation is really convenient for rank-2 tensors of any dimension. With the electromagnetic field tensor, you still only need a row and column to find a component, so it’s rank-2. However, there 4 rows and 4 columns, which means this tensor is 4-dimensional. The electromagnetic field tensor is rank-2 and dimension-4. This notation starts to fall apart with higher rank tensors though. For example, a rank-3 tensor requires 3 pieces of information to find a component. While this is still technically a matrix, the math operations aren’t very obvious. It gets even worse with rank-4 tensors. This is interesting looking, but it’s not very useful. Honestly, the matrix notation is kind of like a security blanket anyway. It’s only there to make people feel more comfortable when they’re first learning about tensors. Then what are we supposed to use? Index notation! Rank-zero means you don’t need any information to find a component. That’s just a scalar. Boring! Rank-1 means we only need one piece of information to find a component. In other words, we only need one index. That’s just a vector. Maybe something like the velocity of a ball across a table. Rank-2 means we need two pieces of information or 2 indices. Traditionally, we use Latin letters for 2 and 3 dimensions and Greek letters for 4 dimensions, which helps make rank and dimension more obvious at a glance. Rank-3 means 3 indices, rank-4 means 4 indices, and so on. Ok fine, but what makes them tensors? How they transform! Humans have a decent intuition about velocity, so let’s start there. This ball could encounter some wind, which would slow it down, but that’s not the kind of transformation we’re talking about. We’re not talking about how the situation might change. We’re talking about how our coordinate system might change. To use physics on this scenario, we need to assign a coordinate system to it. Something like this. That’s just a tool though. We could just as easily have put the coordinates over here, or over here, or even over here. We could have rotated them like this, or like this. We could have even stretched or compressed either of the axes. But our choice of coordinates should have no effect on physical reality. None of these transformations will change the velocity of the ball. Wait, wouldn’t a rotation change the direction? No, it’s still moving to the right. But don’t the components change? Yes, but that’s just how we’re representing it, not what it actually is. This vector is a rank-1 dimension-2 tensor. It has two components, one for each of the dimensions. Any change in coordinates will change the values of those components. But the physical nature of that vector remains the same. Wouldn’t any arrow do that? Actually, no. Take angular momentum for example. If we put our coordinates at the center of this circular orbit, the angular momentum points up. It’s steady and constant. But, if we shift the coordinates to the edge of the circle, that’s no longer the case. The angular momentum changes over time. It even goes to zero for a brief moment, which is ridiculous! That shouldn’t happen with a real physical thing, so we call angular momentum a false vector or pseudovector. It has a direction, so it masquerades as a vector, but it isn’t actually a vector. Velocity is a real vector. Angular momentum is not. It’s a pseudovector. If a real vector is zero in one set of coordinates, it must be zero in all of them. No exceptions. But doesn’t velocity go to zero if your coordinates move along with the moving thing? Yes, but that’s not a 3-dimensional transformation. It’s a 4-dimensional one. Which means you can’t use 3-dimensional vectors. This ball is moving relative to the table, but not relative to itself. Shifting to a steadily moving coordinate system is something we call a boost and it requires we include time as an additional axis. This ball may be moving through space, but it’s also moving through time. It has its own time axis. We call this spacetime and a boost is a just a 4-dimenional coordinate rotation. But, if we want to talk about velocity, we need a 4-dimensional velocity or 4-velocity, which is a rank-1 dimension-4 tensor. Don’t forget. This video is still about tensors. This ball’s 4-velocity is a real vector. It remains the same under these 4D rotations. Just like regular 3-velocity did under 3D rotations. You can’t work in 4-dimensions without using 4-dimensional tensors. Come on crazies! The same goes for things like the 3-dimensional magnetic field. Moving electric charge will generate a magnetic field, but only if you see the charge moving. If you’re moving along with it, the charge is stationary, which means no magnetic field. The magnetic field is not a real vector. It’s a pseudovector. That’s why we came up with the rank-2 electromagnetic tensor. It fixes this problem. It’s a real tensor. Unfortunately, a rank-2 tensor can’t be visualized as an arrow like a vector can, but it can be understood as a transformation between vectors. In fact, that’s exactly what this equation says about the EM tensor. It transforms the 4-velocity of a charged particle into a force. A moving charged particle inside of a force field experiences a force. That’s a little magical, isn’t it? Ok, let’s use the stress tensor instead. Fine! I’m a huge dice nerd. I love dice. The best ones are the platonic solids. Obviously. Let’s consider the 4-sided one: The tetrahedron. If we want to know the force on one of its surfaces, we just need to know its stress tensor. Maybe one of its surfaces is facing this way. If it’s experiencing stress described by this, then our surface is being nudged this way. The area vector is transformed into a force vector. So what’s a tensor? It’s a number or collection of similar numbers that maintains its meaning under transformations. If you make a different choice in coordinates, the components of the tensor will change, but in a way that conspires to keep the meaning of the tensor the same. This velocity vector is a rank-1 tensor that describes the motion of the ball, regardless of the coordinate choice. This stress tensor is a rank-2 tensor and describes how to get a force from area, regardless of the coordinate choice. If your number or collection of numbers doesn’t do that, then it’s not a tensor. It’s a false tensor or pseudotensor. Not being able to tell the difference can get you into some serious trouble, at least in the math. So got any questions about tensors? Please ask in the comments. If you’re looking for a deeper dive, check out the book I wrote. It’s got an entire chapter explaining tensors and it’s available in paperback and as an eBook. Thanks for liking and sharing this video. Don’t forget to subscribe if you’d like to keep up with us. And until next time, remember, it’s ok to be a little crazy. If investing in your STEM skills is your kind of new year’s resolution, you should check out Brilliant. Maybe you’re naturally curious or want to build your problem-solving skills or need to develop confidence in your analytical abilities. With Brilliant Premium, you can learn something new every day. Brilliant’s thought-provoking math, science, and computer science content helps guide you to mastery. by taking complex concepts and breaking them up into bite-sized understandable chunks. There’s a whole course on linear algebra which is where matrices are important. There’s even a course on 3D geometry where you can learn about platonic solids. Brilliant helps you achieve your goals in STEM, one small commitment-to-learning at a time. If this sounds like a service you’d like to use, go to brilliant dot org slash Science Asylum today. The first 200 subscribers will get 20% off an annual subscription. I really enjoyed some of the take-aways people got from my last video. Like that the Earth has only been around the galaxy 20 times or that the galaxy has only rotated somewhere between 50 and 60 times. Most of us would have expected those numbers to be bigger, you know? Anyway, thanks for watching.

100 thoughts on “What the HECK is a Tensor?!?”

  1. What is current? A scalar, vector or tensor?
    Since current has magnitude and direction it must be a vector but It doesn't follow vector rules. That means it is scalar.

  2. Tensors are fun but get complicated to the point of Einsteins' field equations when you pause for a coffee!!🤔🤣🤣🤣🤣

  3. A tensor is that mathematical object which, if you know what it is, you cannot explain it to someone else unless they also know what it is.

  4. I learnt rank of a matrix to be the dimension of its image so I was confused as first by your description of the rank

  5. You are awesome!!!!!!
    Can we imagine tensors as a planes….?
    Also,
    Recently I felt extremely dumb..because since childhood we played with magnets…we now even study their phenomenons BUT I STILL DON'T KNOW what is actually going between those 2 magnets while attraction or repulsion… And same for charges?! 🙁
    Nick pls help
    🙂

  6. I need evidence that a mathematician ever defined a tensor that way. Tensor spaces, in mathematics, are usually defined up to isomorphism by the universal property of transforming multilinear functions into linear functions.

  7. Can angular momentum be thought of as a velocity being continually acted on by a centripetal force? For example gravity for planetary systems, em force for atoms, weak / strong for subatomic etc. What happens when you get to fundamental particles – thats when sh** gets wavy right? But then how do you go from a wavy thing to a spinny thing…..

  8. 9:09 I'm 99.909% sure that's besides the point of this vid, but the upstairs and downstairs indexes here are mismatched. Also, as a bonus, since no mention of the metric was made here, I'll just throw in the mystical statement that the "area vector" that appears here is actually a "pseudovector" – i.e., the Hodge dual of some 2-form that happens to coincide with the volume element of the surface as it is imbedded in 3D space. You can tell that to your friends, and trick them into thinking you're terribly smart 😉

  9. This guy is like Richard Feynman, a great explainer. Some one only truly understands something if they can break it down to the simplest components, yet show the more complex idea.

  10. Clarification 1: At 6:06, I showed an animation where angular momentum goes from constant to variable in a coordinate translation. That is only one example of how a pseudovector can misbehave under transformations. They can also pick up signs under reflections. They can even pick up entire coefficients under rotations and changes to non-Cartesian coordinates. It all depends on the pseudovector and the transformation. The point is that real vectors don't do any of this.

    Clarification 2: At 2:06, I said "matrices and tensors are not the same thing." Matrices are just a way of organizing numbers and I mean it. It is true that all rank-2 tensors can be written as square matrices, but so can rank-2 pseudotensors. Vectors can be written as column (or row) matrices, but so can pseudovectors. Matrices are just a way of organizing numbers. Matrix operations do work the same way as tensor/pseudotensor operations, but only if you order the matrices correctly. With tensors/pseudotensors you don't have to worry about multiplication order. With matrices, you do. That's why using matrix notation is not recommended.

  11. Why am I thinking twelve when he says nine?. I don't know what makes me think twelve instead of nine?. Did I over analyze? .

  12. where've you been all this time /
    you just answered the question i had trying to understand why do we need three different vectors of tension and why they can not be added up. Well, now it seems so obvious to me that i might be ashamed, if i would care. But still!

  13. Wonderful, I am Pro-Logic and this video, is the way I think all the time!. Deep thoughts, I love it. no one around here where I live has the foggiest idea what I am talking about when I talk about science.

  14. I tried so hard to understand tensors while writing my mechanical engineering dissertation. I don't know what it is but the concept just doesn't blend with my way of thinking. I'll have to watch this again 😒

  15. Finally!! This helped a lot with distinguishing 3D transforms vs 4D transforms, I’ve been trying to study tensors for a few weeks and this helped tie everything together!!

  16. I feel tense now. Why on Earth didn't teach me all this in primary school? It would have made school a lot less boring.

  17. This is a really good explanation, and for most of it I had no problems. I'm just wondering one thing: how do you rigorously define the "meaning" of a tensor? What property of the tensor isn't changing?

  18. isn't vector a rank 2 ?
    [ 2 3]
    4 5

    you need row and column to get a specific value.
    can you explain rank 1 with 2/3 D and rank 2 with 2/3D.
    Thanks

  19. So could you explain the Christoffel symbols next?

    (I got a headache list time I tried listening to Susskind trying that, despite him being a brilliant lecturer.)

  20. This is a very surprising video. I really thought you would describe variants and not tensors, like so many other YouTube videos on the subject. But you didn't. You actually gave a description of tensors (at least qualitatively and with good examples). Probably one of your best videos!

    An idea for a followup video would be to talk about curvilinear coordinates and the curvature tensor. Might be a longer video, for sure. But it's a worthy challenge.

  21. Somebody have this madman arrested, he just detonated a bomb in my frontal cortex! Holy fuck I've wanted to know for so long wtf tensors are, and believe me Wikipedia was not helpful. Thank you so much. The moment I saw this in my feed I knew you were gonna blow my mind.

  22. I'm still confused how to tensors or rather I would talk about how vectors are placed along a matrix, further I'm trying to have a little sense of tensors calculus and matrix used in general relativity ,I.e Einstein field equation, that is literally blowing my mind

  23. I think I'll have to re-watch this a few times. I was kind of getting it in the beginning, then about half way through my brain crashed.
    BTW, what's a tensor? 🙂

  24. So does the 4-velocity no longer go to 0 when moving along with the object? Does the time basis compensate for that by not changing, and only the spatial bases change?

  25. I also follow "PBS Space Time", "Because of Science" and others, but this stays my favourite channel about physic.

  26. So, for the purposes of this discussion, is torque not a tensor? Or is it treated as a vector sum of the various other stresses with a spatial displacement? Feel free to correct the terminology.

  27. Sorry dude but in my opinion the mathematical definition is much easier to understand. Trying to explain mathematical notions using physical examples is a very bad idea.

  28. I’ve been trying to get my head around tensors. This video was very helpful! Would you please consider a video explaining spinors?

  29. Tensorsts, I hav d to use t descrbe how me brain shear when wtching dis, write ?!🧠 ⛮ ✒✏🖌🖍🖊🖊🖋✒🔦

    Anyway, I watched your old video about solar panel, why don't we use Mg and S or O for the doping instead of Bo and P?

  30. I owe you a great deal of thanks man. I'll be 2 years clean tomorrow, and when I was going through the worst part of kicking my habit it was your videos that kept my mind off of the pain. I had lost everything. My wife, my kid, my home, even wrecked my car. I laid awake every night going clean out of my mind, and it was watching your videos that kept me distracted enough to be able to deal with it. Prior to grabbing your phone and jumping on youtube all night the anxiety of withdrawal would drive me to go out and "find" a way to get well in the morning.

    I absolutely love science and technology. In my 2 years clean I've built my "circuit room" which amounts to some electronics equipment, 3d printer, a computer, basically everything I can find to design and build the things I think up.

    I am forever grateful for people like you on YouTube. You have so much knowledge about the things that we are all curious about, and the drive to share what you know is a beautiful thing. Keep it up man, and thank you from a fellow crazy..

  31. As I understood sometimes the tensor is a vector and sometimes it's a transformation between vectors , is this correct ?

  32. Very good presentation. I have been studying the work of Gabriel Kron lately and this helps a lot. Cheers…

  33. This is the only explanation of tensors that I've heard that actually explains what the heck a tensor is. Congrats, and thank you! Seriously, been trying to understand his for years.

    As it turns out, I was making it more difficult than it actually is. It just goes to show how poor most explanations are of this topic.

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