*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.
I have a question (about pseudotensor) but it is about how to define 3D-spinning object ? I ask this question first time when I was in high school and when I try to find the answer of that question. I end up to tensor and I don't understand it. Now I find your video but still can't answer that question. However, your video make me know that it is about pseudovector or pseudotensor instead. To understand the question let use something (I called it spinning game), imagine you have a sphere and ring shape spinning machines. When you put a sphere in a ring and turn it on, The sphere spin like in this picture. sv1.picz.in.th/images/2020/01/17/R5AFJN.png (You can create new ring if you want without any size or number of ring but you have only one sphere.) You can do same thing to the smaller ring as well by spin it with bigger ring and you can use that spinning ring to spin any objects inside it as well. If there are more than one thing inside the ring you can choose to spin either each object or all objects. By this way, is there any way to define any spinning sphere result that you can get from this game ? How many number do we need to define ? The part of answer that I know is, If you do this only one time, any result can be define 3D-angular momentum pseudovector. (You can spin it in many direction but all of them can be define 3D-angular momentum pseudovector.) But the real question start to be complicate when you do this more than one time in different way.
Angular momentum can be tricky however if you use the language of Clifford algebra I find their description much easier, Angular momentum is then a bivector (which you can think of as an oriented plane or a plane with an intrinsic sense of clockwise or counterclockwise rotation.... this is analogous to a vector being a line with a sense of direction). This also allows you to define angular momentum in dimensions other than 3. Another advantage is that you don’t have to worry about the behaviors of pseudovectors vs vectors.
6:43 Finally! Someone who states explicitly that angular momentum is a pseudovector, something for which I get a lot of kickback (some of it insulting) whenever I say or write it. The very fact that the direction is arbitrarily assigned using a right-hand rule when a left-hand rule will work equally well should send up warning flags.
I’ve learned a little about geometric algebra. It deals with multivectors. When formulated in GA angular momentum is considered a bi-vector which is mathematically different from a vector and represents an oriented area rather than a vector. It uses a wedge product rather than a cross product. Which keeps the angular momentum in the plane of the product rather than perpendicular to it. Actually the cross product isn’t used at all in GA. It only works in three dimensions while products used in GA work in any number of dimensions. And pseudo vectors and pseudo scalars are easily identified as separate mathematical objects.
@Jamal Wills: Likewise, the magnetic field is 2-vector, not a vector. (I think naturally it'd be a 2-form and the electric field a 1-form, but by metric duality you can think of them as contravariant instead)
I think he must meant that _working_ with tensors is usually much more of a pain than working with vectors, because there’s more information to deal with. I mean, it’s also a lot easier to work with scalars than vectors, because it’s less information to deal with. But if you _need_ the additional information, using vectors and/tensors actually makes things easier. It’s just that, when we don’t all the information, ignoring some of it can save time and hassle. It’s why we often pretend electricity and magnetism are separate forces - the additional information of how they’re actually unified as one force isn’t always necessary or helpful so it’s easier to ignore it. Another example is why still use Newtonian physics, even though we know it misses some fundamental things about how the world works - 99% of the time that information doesn’t make any practical difference and taking it into account would cost so much time. Yet, sometimes we absolutely do need to take it into account. GPS satellites, for example, wouldn’t work properly if not for GR.
I learned some new things, but still don't quite understand. I guess I reached my limit. Is it possible to dumb it down more? Like I'm in 6th grade? :-( Thank you for your excellent videos!
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 RUclips. 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..
I'm really happy that you got your life straightened out and it's nice to hear that my videos helped. Your circuit room sounds awesome! Hobbies are a great distraction 👍
@@ScienceAsylum Can you please explain why the Magnetic field is not a real vector if the "4-velocity" is by simply applying the 4th dimension of time? I didn't really understood that part. Thanks otherwise your videos are awesome.
So 40 years ago when I got my undergraduate, Nobody could explain it like this. My study group after about a week finally figured it out for ourselves. We knew why it was needed and what it did, but the actual gearshift eluded us for a bit. This guy is real good!
Dude, your animations are beyond top notch. I don't think anyone else out there does science animations that are so attractive, informative, and easy to understand. They make some really complex concepts far easier to understand than they otherwise would be.
"A tensor is a number or collection of numbers that maintains its meaning under transformations." Thank you! That is a great sentence that I will repeat to myself 1000 times.
1:54 That right there did it! I never fully understood why are there so much individual types of forces applied to only 3 dimensions, and the way you showed it right there made it all clear, thank you!
@Daniel Braga well, everyone who watches at least pays attention, so there's that if nothing else. Puns aside, I'm happy to help out a talented educator, and the monthly live stream is cool.
You have sought out this knowledge. Most don't. That's what the tax money is for. Tax money works great for ensuring we live in a society with a decent minimum standard of education. If the masses are uneducated, we are all less well off.
What I love about your channel is that you actually get into things at a somewhat sophisticated level, even with math, while still being entertaining and explaining things really well... and having great visuals. Most channels meant to explain science will either be dry if they go into detail and sophisticated math, or not go into detail or sophisticated math if they have good pedagogy and visuals. I really can't overstate how awesome it is that you put it all together! This is an invaluable channel.
Great job. I had to master basic tensor algebra for General Relativity in college, and it took me a good month to sort of grasp what they were conceptually. You explained it better in 5 minutes than my college math professor did in a week. And he had the advantage of knowing his audience already knew vector calculus. .
By the way, this video prompted a discussion with my algebra student about tensors. She didn't understand the video because she hadn't learned about vectors yet, so we had a discussion about the basics of using vectors to represent physical quantities, then about how we can use different coordinate systems to represent the same things, and which one we use generally depends on what's most convenient in the context. Then we talked about fictitous forces and how centrifugal force is present if we set the origin with the frame of rotation but it disappears otherwise, and how that means it's not a tensor quantity (the explanation of centrifugal force is from your video about that, as that's part of what's made me understand it). That discussion ended up being a great learning experience for us both and it wouldn't have happened if we hadn't both watched this video. Keep up the good work Nick. I've learned so much about Physics from your channel!
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?
I just got the notification for this and I immediately shouted "YES! I've been waiting so long for this!" I was chatting with my algebra student who was very amused at my sudden excitement but immediately said she was going to watch it too when I said it was by The Science Asylum.
Didn't know lecturers saw these kinda videos. My Physics lecturer was watching a comedy show when I went to ask him something 🙄 Your students are lucky
My new favorite channel. I wanted to understand these concepts for several years now; almost a decade. Never before have I found such intuitive explanations of these ideas. Thank you! Now I just need to learn calculus of variations?
This video is really good. If the sheer amount of time I’ve spent trying to understand tensors is any indication, there’s a _lot_ to unpack in this topic and you did a really good job presenting it in a way that people without a lot of math background can understand. That’s not an easy thing to do, so good on you for doing it!
Exactly. Tensors is really math heavy topic. I am studying linear algebra right now using Lang' Introduction to Linear Algebra. Can you please suggest me a book on tensors?
@@laxminarayanbhandari855 I don't read too many textbooks. I mostly do online sources. If you just want the general mathematical definition of a tensor, it's a member of a vector space that satisfies the tensor space axioms (don't remember the axioms off the top of my head but you can Google them). If you want some intuition, I believe the math3ma website has some really good articles explaining what the formal definition actually means intuitively. There's also an article called something like "How to overcome tensorphobia" that's really good.
I just recently finished my undergrad in physics, and I swear everytime I asked someone about tensors, they just babbled the mathematical definition at me, makes me think about how much we all just take for granted and don't really understand. Great video!
But the mathematical definition is the simplest and the most intuitive one: a (covariant) tensor field is a field of multilinear forms. Period. For the mixed case: a p-covariant and q-contravariant tensor field is a field of q-vector-valued p-multilinear forms. A q-covector is just a (linear combination of) wedge of p vector fields. Multilinear algebra is not difficult, really, and it clarifies tensor fields a lot.
Wow, this is great! And I'm definitely going to have to watch this one three or four times for sure to absorb what you are saying here, not because it is not well said, but rather you are saying a lot, and saying things well worth rewatching to full get full jist of just what you're saying.
@The Science Asylum This is one of the best videos by you I've seen! When I was a physics undergrad, I had to take Mathematical Methods for Physicists. I passed the class just fine, but the section on Tensors (which we didn't spend too much time on) was confusing as hell with all the weird notations and transformations. If only they explained it intuitively as you have. Thank you! Also, I took general relativity and completely flopped at that! So hard! Would be cool if you did some videos on that.
@pyropulse What are you talking about?? When did I say I didn't know what a tensor was? I said we learned it in class, and it was difficult to understand. It's not straight forward. The math methods for physicists class covered a huge amount of math in just *one* semester. So much so that we were in a different topic literally every week! You try that out. The university realized how insane that was, and decided, after I already passed the class, to split the class into two semesters. So now it's math methods for physicists I and II. Unfortunately, that was too little too late for me. I don't know if you've ever studied physics at university, but at least at my school, tensors were not really necessary for any other classes except maybe small parts of special relativity and E&M. And in both cases, we just used matrices, not the confusing notations I was referring to.
@pyropulse it looks like that only when you've finally managed to understand it intuitively. Before that it is just a bunch of numbers with some magical formulas being applied to them. Same for integrals, derivatives, maybe even fractions, but i can not recall having trouble with understanding that
@@Lucky10279 Not really, unfortunately. Whenever I try to plan out video topics too far in advance, I never end up sticking to the plan anyway. I'm sure I'll get to it in 2020. I'll _probably_ get to it by summer. That's the best I can do right now. You know how long it took me to finish this video.
Oh my God dude this is SUCH A GOOD VIDEO!!!! It has been so hard to find a source that doesn't give the concise definition that basically gives no information. This was such a joy to find! I feel confident that I can come to understand what I currently don't but the analogies, and pictures provide the perfect context. I wasn't expecting to find something this comprehensive today. I will be recommending this to everyone in my Fluids class. Virginia Tech sophomore signing off.
If you want something more thorough, I have a whole chapter devoted to tensors in my book. Advanced Theoretical Physics (Paperback): www.lulu.com/shop/nick-lucid/advanced-theoretical-physics-a-historical-perspective/paperback/product-24250687.html Advanced Theoretical Physics (eBook): gumroad.com/l/ubSc
I used to be a structural engineer, and now I'm in computer science... Your explanation using stress tensors really "cemented" this in my brain. Thank you! :) :) I'm much less "stressed" now.
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!!
Wow, as a programmer, this make me realize that I've probably made rank >1 tensors before. Awesome video. Tensors are to vectors in the same way as tables are to lists, at least for rank 2. Neat!
A video about Geometric Algebra would be interesting as a follow up! In some ways they're the best type of multivector I heard.. I feel like you're the only guy that can do the topic justice!
@@thstroyur and Mikayla, but it always good to know there are people who've even heard of it. Not to mention, even understand it! The best one I've watched is by Mac T Borsch
6:20 you know... if you change coordinate system, and don't change axis along what you calculating momentum, you must consider it. bc even if coordinates' 0 is in the center of circle and you calculate from axis on the side of the path, that would be still changing and going 0 for a moment.
We need a part two on tensors. I still have the book on tensors and maths which I really wanted to avoid for the next year's course but after this video you changed my mind.
"Magnetism is a pseudo vector" aha that's one doubt my teacher never understood. Thanks man, now I understand electromagnetism so much better. Because as soon as you said angular momentum is a pseudo vector i was like then induced electromagnetic fields have a vector that is pseudo, i thought that couldn't be right but you explained so well. That was the only hard part for me. Thanks again so much man.
his video about tensors is the one who easily taught these mathematical and physical objects. And I can say that for sure because I've read several books and watched several videos on the subject!
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.
so my favourite RUclipsr (spoiler, Nick: it's you!) uses my favourite software (Blender!) to illustrate a topic that has both intrigued and interested me for years. it's like Christmas in January!
Fun video lesson to watch. I think that I can sense why this one had you a wee bit more nervous than usual. I also think you pulled it off brilliantly. Got a little D&D dicey there for a few moments, though. One of the best science channels on RUclips.
THANK YOU! As a programmer, I'm trying to understand tensors; I’m learning the math and computer science of tensors for applications with packages like TensorFlow (used for AI development). Breaking down each definition with visual aids helped me unclutter the confusion between what makes something a 2-rank tensor and how it can still be considered 3-dimensional.
Hello Sir Nick Lucid, i regularly watch your videos, and i get completely amazed by your knowledge and the way you explain the most difficult topics.... I just want you to create a short video on tesseracts and please explain something more about the 4th dimension.... I will be most thankful to you...
if you havent read Richard Abots "Flatland" start with that. After watching so many Avenger movies I didn't realize the importance of what the tesseracts is! If I understand it correctly it is a 4-dimensional object. Cool...
So, I was reading up on some artificial intelligence stuff for college and there were mentions of tensors, so I quickly googled it just to get the idea what it was, I read up on Wiki and was completly lost so I brushed it off since I didn't really NEED to know what tensors are to continue. Jump forward a few days and last night I dreamt about tensors the whole night, I almost couldn't sleep because of them for some reason. So I wake up and I'm like, "well, I guess I have to learn what tensors are", I google it and this time I see a "The Science Asylum" video on tensors, I just think to myself "Damn, I'm in luck" and move into the video.
Wow this video really helped understanding tensors. At Uni you are just dealing with its math and properties so you don't get to understand these stuff. Now it's actually making sense to me :D
For me, it started focused but then went off the rails in moving into areas making the subject far more complicated than it needed to be. But I appreciate this is your way of expressing yourself.
Derek is a much more pop-sci kinda creator, less detailed and more informally explained. Note I am in no way saying he's not as good, Veritasium is an amazing channel. Just a different audience.
Agree, you are doing what schools should. Thanks for explaining so well. You’re an amazing explainer. To me, that’s even better than a teacher. Explainers cut through the crap and make you understand.
8:44 That's only true in SI units, but spacetime tensors are usually written in c=1 units. 3:33 You can certainly make the k-axis forward instead of backward, though I don't think it matters. There's no reason the indices _should_ obey the right hand rule (like the spatial unit vectors). It might be good just for consistency though, so I'll give you this one.
Wow! Congratulation! Excellent explanation! It is simple and easy. So the next challenge, I guess, would be how to explain to students a concept of covariant and contra variant tensors without messing up lower and upper indices. It would be an interesting challenge.
im struggling with my higher level physics classes ( like I want to quit cuz I suck so bad at setting up problems). This video finally showed me what a tensor was, which everyone seems to talk about but no one attempts to explain. Thanks again for making me feel like I can understand complex topics. I wish my classes could be 1/16 as good
I think the last time I checked, maybe a year ago, your book was only eBook, but now it's also in paper form! Yay!!! Can't wait to get it sometime soon, even though I am currently not studying physics. I want to understand how math works better first, so I'm studying foundations of mathematics, while also pondering away at philosophy and philosophy of language. But once I feel content enough with myself, I'll get back at it! As always, great vids.
The Science Asylum Reminds of when i kept hearing that a vector was a member of a vector space, then, when I asked what that was, it was defined in terms of vectors. Then I found the list of vector space axioms, but it still took me a while to figure out how those axioms had anything to do with the usual geometric interpretation of a vector. Hard to believe that was only about a year ago.
@@ScienceAsylum but in the video they are also defined with their transformation properties, so what's the difference? These definitions always include the concrete form of transformations.
@@ScienceAsylum And you clearly understand the subject. Circular definitions reveal that the definition writer doesn't quite grasp the real meaning. Even though I had a difficult time following the video, my difficulty is only due to a lack of algebra skills and physics.
That just made my years of casual study of relativity make a lot more sense in an intuitive way. I just trusted the math. No one has explained it that way and frame dragging now makes sense in terms on non-euclidian geometry such that I could approximate a graph in my mind without math in constantly changing or accelerating coordinate spaces.
I remember asking my professor in detail about tensors. I didn't ever watched the video yet, but your explanation is way superior. Edit. I would really like to buy your book. I can't wait to get a brand new insight on physics. Also, do you plan adding some terminology about thermodynamics? I think it would be awesome to have it there.
So in that demonstration (6:39), in order to maintain conservation of angular momentum you must also consider the attracting body that's in the middle of the circle? and so it must have the greatest velocity (relative to the shown axis) exactly when the orbiting body passes through the axis (and it's contribution to total angular momentum is 0 as show)?
This channel is legit the best science channel I've ever seen. Maybe the best science videos period. It's like everything I wanted from Bill Nye before he became some old hasbeen who won't shut up about global warming. You deserve your own platform and I think you should seriously consider making online courses or something. The Khan academy guys got nothing compared to this
The t-shirt is evil! I can whole-heartedly recommend your book to anybody, not because I'm a fan of the channel. I genuinely find it useful and refer back to it all the time.
I now just realized why in programming the common notation to use for 'for loops' is 'i' and then 'j' when dealing with nested loops. You use said loops to go through arrays, which can be the same as vectors, which are tensors, and to access an element from the array you need to index it. The variables that we used to hold the index value as we go through the array with the 'for loop' are the freaking ranks! I mean, kinda, normally the implication in programming is that you would need a rank per dimension of the array, which is no really the case with tensors as I saw in the video, but still.
Actually I think it goes back to the days of FORTRAN. IIRC, the letters starting with "i" (maybe "i" through "n") were implicitly defined as integers so they were easy to quickly use as loop counters.
YES!!! FINALLY!!!! I waited such a long time for this😅😅😅 I've to watch it a few more times though, it's a lot of information to digest in one take. I have a question, how does the angular momentum change when you change the coordinate system? P.S : a new clone, nice :)
@@ScienceAsylum oh, I thought the "r" in the *L* = *r* × *p* formula was calculated from the axis of rotation. In every problem that I solved so far the axis of rotation was either the x, y or z-axis so I assumed that "r" was calculated from the axis of rotation. Now I understand my mistake. I need to study these concepts deeper and pay more attention to the definitions. Thanks nick :)
@@ScienceAsylum by the way, what physics textbook explains these concepts clearly? I have Halliday-Resnick but I don't think this book explains them in enough depth. I also need a book who isn't too technical because I get lost easily.
@@ScienceAsylum no it's not, it's a distance, relative to an axis you've chosen to calculate from, often it's axis of rotation. Coordinates system changes, the axis of rotation does not
I passed differential and integral calculus and linear algebra back in college but not multivariable calculus or differential equations. Also passed classical physics and E&M but not modern physics. I am still struggling with tensors! 😖 I'll watch the vid a few more times and see if I get it 🤓🤪🙃
I've watched TENS OR hundreds of your videos and your sidekick asks the right questions what do you call him Mini Lucid ? Great video you always get the like before I watch your videos ... keep up the great work going to watch video again
This is much better than the first chapter of Sean Carroll's Lecture Notes on General Relativity. Next video: Magnetic Reconnection in the context of magnetohydrodynamics?
Way nicer than other 'tensor' videos, yet I am still not sure that I understood what tensor is actually. However there are great animations that give some idea of the relation of 'tensor', coordinate system and it's 'property'. Some things are still confusing, but hey, we can re-watch the video again and again trying to catch these little moments where some light bulb will shine :)
"Rank is the number of pieces of information you need to identify a component." Thank you for this. Now I finally get what exactly rank actually refers to. I was thinking it meant the number of dimensions needed to represent a rank n tensor as an array.
"I was thinking it meant the number of dimensions needed to represent a rank n tensor as an array." It does tho. Like, a 2-D array means you need 2 numbers to locate any piece of information in the array. That's what dimensions fundamentally mean. How many numbers you need to locate/define something within the dimension
@@amarj9909 He did explain what the rank of a tensor is -- it's how many data values you need to locate each coordinate of a tensor. Scalars are rank 0 zero because they only have one coordinate value, so not additional information is needed. Vectors are rank 1 because they're just column matrices and so we only need one data point to indicate a specific coordinate value (that's a big oversimplification of what a vector actually is, but I don't want to get sidetracked), and matrices are rank 2 because they you need to reference both the row and the column number in order to locate a specific coordinate value. The rank of a matrix, on the other hand, is just the dimension of the rowspace and columnspace (they're always the same number.) The rowspace is the general solution to the system of equations represented by the rows of the matrix and the columnspace is just the rowspace of the transpose matrix. 3blue1brown has a really good video about the column and rowspaces of a matrix and its rank. I highly recommend it if you want an intuitive understanding of what the concepts mean geometrically.
*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.
want more examples of tensors! Especially if they are rank 3 or higher
You make me a day. Now I understand tensor.
I have a question (about pseudotensor) but it is about how to define 3D-spinning object ? I ask this question first time when I was in high school and when I try to find the answer of that question. I end up to tensor and I don't understand it. Now I find your video but still can't answer that question. However, your video make me know that it is about pseudovector or pseudotensor instead.
To understand the question let use something (I called it spinning game), imagine you have a sphere and ring shape spinning machines.
When you put a sphere in a ring and turn it on, The sphere spin like in this picture.
sv1.picz.in.th/images/2020/01/17/R5AFJN.png
(You can create new ring if you want without any size or number of ring but you have only one sphere.)
You can do same thing to the smaller ring as well by spin it with bigger ring and you can use that spinning ring to spin any objects inside it as well.
If there are more than one thing inside the ring you can choose to spin either each object or all objects.
By this way, is there any way to define any spinning sphere result that you can get from this game ? How many number do we need to define ?
The part of answer that I know is,
If you do this only one time, any result can be define 3D-angular momentum pseudovector. (You can spin it in many direction but all of them can be define 3D-angular momentum pseudovector.)
But the real question start to be complicate when you do this more than one time in different way.
Angular momentum can be tricky however if you use the language of Clifford algebra I find their description much easier, Angular momentum is then a bivector (which you can think of as an oriented plane or a plane with an intrinsic sense of clockwise or counterclockwise rotation.... this is analogous to a vector being a line with a sense of direction). This also allows you to define angular momentum in dimensions other than 3. Another advantage is that you don’t have to worry about the behaviors of pseudovectors vs vectors.
You could probably add a notation comment to those times in the video, directing people to look in the comments for these 2 clarifications.
6:43 Finally! Someone who states explicitly that angular momentum is a pseudovector, something for which I get a lot of kickback (some of it insulting) whenever I say or write it. The very fact that the direction is arbitrarily assigned using a right-hand rule when a left-hand rule will work equally well should send up warning flags.
I’ve learned a little about geometric algebra. It deals with multivectors. When formulated in GA angular momentum is considered a bi-vector which is mathematically different from a vector and represents an oriented area rather than a vector.
It uses a wedge product rather than a cross product. Which keeps the angular momentum in the plane of the product rather than perpendicular to it.
Actually the cross product isn’t used at all in GA. It only works in three dimensions while products used in GA work in any number of dimensions. And pseudo vectors and pseudo scalars are easily identified as separate mathematical objects.
it was forced to be a vector so yeah youre correct
@@random22453 Read my response to Jamal Willis.
@Jamal Wills: Likewise, the magnetic field is 2-vector, not a vector. (I think naturally it'd be a 2-form and the electric field a 1-form, but by metric duality you can think of them as contravariant instead)
@@wayneyadams I always suspected angular momentum; I knew it was up to no good. Pseudo-vector. I feel vindicated.
He's did it. He actually did it. Years of "you don't want to do tensors if you can avoid them" convinced me it would never happen. Heading in!
😂😂
I think he must meant that _working_ with tensors is usually much more of a pain than working with vectors, because there’s more information to deal with. I mean, it’s also a lot easier to work with scalars than vectors, because it’s less information to deal with. But if you _need_ the additional information, using vectors and/tensors actually makes things easier. It’s just that, when we don’t all the information, ignoring some of it can save time and hassle. It’s why we often pretend electricity and magnetism are separate forces - the additional information of how they’re actually unified as one force isn’t always necessary or helpful so it’s easier to ignore it.
Another example is why still use Newtonian physics, even though we know it misses some fundamental things about how the world works - 99% of the time that information doesn’t make any practical difference and taking it into account would cost so much time. Yet, sometimes we absolutely do need to take it into account. GPS satellites, for example, wouldn’t work properly if not for GR.
I learned some new things, but still don't quite understand. I guess I reached my limit. Is it possible to dumb it down more? Like I'm in 6th grade? :-( Thank you for your excellent videos!
Would you please explain them to me? I got so confused!
@@Lucky10279 Yes, exactly that.
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 RUclips. 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..
I'm really happy that you got your life straightened out and it's nice to hear that my videos helped. Your circuit room sounds awesome! Hobbies are a great distraction 👍
Are you still doing well ? Props to you if so. I'm jealous of your circuit room though lol
@@ScienceAsylum Can you please explain why the Magnetic field is not a real vector if the "4-velocity" is by simply applying the 4th dimension of time? I didn't really understood that part.
Thanks otherwise your videos are awesome.
I kicked my alcohol habit and now I'm trying to figure out how to quit RUclips. Friend of mine had success with heroin.
Apparently Tensors give you the same amount of stress not matter what orientation you describe it from. :p
Indeed.
It works, I feel stressed.
I'm stressed and I'm describing the Tensor using a gay voice. Does that count as orientation?
lol
Guess I’m a tensor
Holy shit this made some sense
I like how no one came to say Hey MajorPrep just to poke a finger at you😁
Hi Zach, love your videos too!
So you learn here and make your own videos...
And got more subscribers than science asylum
Yeaah, it made me at least know why einstein is using such creatures when he talks about spacetime!
Nothing Holy ,soap&Water
So 40 years ago when I got my undergraduate, Nobody could explain it like this. My study group after about a week finally figured it out for ourselves. We knew why it was needed and what it did, but the actual gearshift eluded us for a bit. This guy is real good!
Dude, your animations are beyond top notch. I don't think anyone else out there does science animations that are so attractive, informative, and easy to understand. They make some really complex concepts far easier to understand than they otherwise would be.
Software engineer here, with a lot of newly found love for physics because of this channel. Thank you Nick!
I swear this is the most underrated Channel on RUclips
The ability that most life forms have to predict motion is pretty impressive and shows how important it is for survival.
"A tensor is a number or collection of numbers that maintains its meaning under transformations." Thank you! That is a great sentence that I will repeat to myself 1000 times.
I was stock with higher dimension tensors... And this really helped; Use: 'index notation, not matrix notation'
Thanks!
1:54 That right there did it! I never fully understood why are there so much individual types of forces applied to only 3 dimensions, and the way you showed it right there made it all clear, thank you!
You have educated me better than over a decade of physics teachers and you didn't get any tax money....
He was a teacher at a school, so I'm pretty sure he did get some tax money. Now we support him on patreon where he does great stuff for us.
@Daniel Braga well, everyone who watches at least pays attention, so there's that if nothing else. Puns aside, I'm happy to help out a talented educator, and the monthly live stream is cool.
You have sought out this knowledge. Most don't. That's what the tax money is for. Tax money works great for ensuring we live in a society with a decent minimum standard of education. If the masses are uneducated, we are all less well off.
Username checks out
I'm sure you were the best, most motivated, focused student in school. None of those teachers did anything at all, did they? All horrible, they are.
What I love about your channel is that you actually get into things at a somewhat sophisticated level, even with math, while still being entertaining and explaining things really well... and having great visuals. Most channels meant to explain science will either be dry if they go into detail and sophisticated math, or not go into detail or sophisticated math if they have good pedagogy and visuals.
I really can't overstate how awesome it is that you put it all together! This is an invaluable channel.
Thanks! 🤓
Okyk
Great job.
I had to master basic tensor algebra for General Relativity in college, and it took me a good month to sort of grasp what they were conceptually. You explained it better in 5 minutes than my college math professor did in a week. And he had the advantage of knowing his audience already knew vector calculus.
.
By the way, this video prompted a discussion with my algebra student about tensors. She didn't understand the video because she hadn't learned about vectors yet, so we had a discussion about the basics of using vectors to represent physical quantities, then about how we can use different coordinate systems to represent the same things, and which one we use generally depends on what's most convenient in the context. Then we talked about fictitous forces and how centrifugal force is present if we set the origin with the frame of rotation but it disappears otherwise, and how that means it's not a tensor quantity (the explanation of centrifugal force is from your video about that, as that's part of what's made me understand it). That discussion ended up being a great learning experience for us both and it wouldn't have happened if we hadn't both watched this video. Keep up the good work Nick. I've learned so much about Physics from your channel!
I wish there was such a channel back when I was at graduate school in the 90s. What an efficient way to refresh such concepts after so many years.
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?
I just got the notification for this and I immediately shouted "YES! I've been waiting so long for this!" I was chatting with my algebra student who was very amused at my sudden excitement but immediately said she was going to watch it too when I said it was by The Science Asylum.
I knew you'd be excited for this one. It's that conversation with you that got me through the details I needed to make this happen.
At first I read “I was cheating with math student...” 🤣
@@gustavoehler 😆
I'm really glad that conversation helped you understand the concept better. :)
Didn't know lecturers saw these kinda videos. My Physics lecturer was watching a comedy show when I went to ask him something 🙄
Your students are lucky
Q: who is more tense, Mister Tensor or Mister Tensor's son?
A: Mister Tensor's son, he is a little tensor.
Willy Sound Change the first tense to uptight and it might be perfect!
Thank You .
Well, that's *intense*
I'm sorry I'll leave ok plsdonthurtme
"Doctor, I need help. I'm a teepee. I'm a wigwam. ... I'm a teepee! I'm a wigwam! ..."
"Relax. You're two tents."
Da Hawk ahaha... that was my first thought when I read the OP
This video is shear genius.
This comment is genius
Yes but is the shear stress worth it. i mean it makes me really tense(or).
Ya
Hilarious, don't strain yourself.
Really?! Hahah I don’t mean to boost your ego but that joke is timeless.
My new favorite channel. I wanted to understand these concepts for several years now; almost a decade. Never before have I found such intuitive explanations of these ideas. Thank you!
Now I just need to learn calculus of variations?
This video is really good. If the sheer amount of time I’ve spent trying to understand tensors is any indication, there’s a _lot_ to unpack in this topic and you did a really good job presenting it in a way that people without a lot of math background can understand. That’s not an easy thing to do, so good on you for doing it!
Exactly. Tensors is really math heavy topic. I am studying linear algebra right now using Lang' Introduction to Linear Algebra. Can you please suggest me a book on tensors?
@@laxminarayanbhandari855 I don't read too many textbooks. I mostly do online sources. If you just want the general mathematical definition of a tensor, it's a member of a vector space that satisfies the tensor space axioms (don't remember the axioms off the top of my head but you can Google them). If you want some intuition, I believe the math3ma website has some really good articles explaining what the formal definition actually means intuitively. There's also an article called something like "How to overcome tensorphobia" that's really good.
@@Lucky10279 thanks for these
I want nothing but you as my teacher whom I can ask my countless question list.
I just recently finished my undergrad in physics, and I swear everytime I asked someone about tensors, they just babbled the mathematical definition at me, makes me think about how much we all just take for granted and don't really understand. Great video!
But the mathematical definition is the simplest and the most intuitive one: a (covariant) tensor field is a field of multilinear forms. Period.
For the mixed case: a p-covariant and q-contravariant tensor field is a field of q-vector-valued p-multilinear forms. A q-covector is just a (linear combination of) wedge of p vector fields.
Multilinear algebra is not difficult, really, and it clarifies tensor fields a lot.
Its crazy how simple so many of the most esoteric topics become when they are explained by someone who understands them well. Thanks!!!
Wow, this is great!
And I'm definitely going to have to watch this one three or four times for sure to absorb what you are saying here, not because it is not well said, but rather you are saying a lot, and saying things well worth rewatching to full get full jist of just what you're saying.
Better yet, I just bought a copy of your book.
...hmmmm I was going to say I'd need five to twenty watches but who am I kidding. Now where did I put that book?
Exactly. Especially when it's been 17 years of post Graduating cramming the shit out of concepts.
How can someone be so informative and witty at the same sapce-time, Hats off !!
@The Science Asylum This is one of the best videos by you I've seen! When I was a physics undergrad, I had to take Mathematical Methods for Physicists. I passed the class just fine, but the section on Tensors (which we didn't spend too much time on) was confusing as hell with all the weird notations and transformations. If only they explained it intuitively as you have. Thank you!
Also, I took general relativity and completely flopped at that! So hard! Would be cool if you did some videos on that.
Oh, don't worry.... GR is coming.
@pyropulse What are you talking about?? When did I say I didn't know what a tensor was? I said we learned it in class, and it was difficult to understand. It's not straight forward. The math methods for physicists class covered a huge amount of math in just *one* semester. So much so that we were in a different topic literally every week! You try that out. The university realized how insane that was, and decided, after I already passed the class, to split the class into two semesters. So now it's math methods for physicists I and II. Unfortunately, that was too little too late for me. I don't know if you've ever studied physics at university, but at least at my school, tensors were not really necessary for any other classes except maybe small parts of special relativity and E&M. And in both cases, we just used matrices, not the confusing notations I was referring to.
The Science Asylum That’s awesome! I’ll be excited when those come out. Any idea of the timeline?
@pyropulse it looks like that only when you've finally managed to understand it intuitively. Before that it is just a bunch of numbers with some magical formulas being applied to them. Same for integrals, derivatives, maybe even fractions, but i can not recall having trouble with understanding that
@@Lucky10279 Not really, unfortunately. Whenever I try to plan out video topics too far in advance, I never end up sticking to the plan anyway. I'm sure I'll get to it in 2020. I'll _probably_ get to it by summer. That's the best I can do right now. You know how long it took me to finish this video.
Oh my God dude this is SUCH A GOOD VIDEO!!!! It has been so hard to find a source that doesn't give the concise definition that basically gives no information. This was such a joy to find! I feel confident that I can come to understand what I currently don't but the analogies, and pictures provide the perfect context. I wasn't expecting to find something this comprehensive today. I will be recommending this to everyone in my Fluids class. Virginia Tech sophomore signing off.
If you want something more thorough, I have a whole chapter devoted to tensors in my book.
Advanced Theoretical Physics (Paperback):
www.lulu.com/shop/nick-lucid/advanced-theoretical-physics-a-historical-perspective/paperback/product-24250687.html
Advanced Theoretical Physics (eBook):
gumroad.com/l/ubSc
I used to be a structural engineer, and now I'm in computer science... Your explanation using stress tensors really "cemented" this in my brain. Thank you! :) :) I'm much less "stressed" now.
Glad I could help! 🤓
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!!
Wow, as a programmer, this make me realize that I've probably made rank >1 tensors before. Awesome video.
Tensors are to vectors in the same way as tables are to lists, at least for rank 2. Neat!
No, you're only linking it to linear algebra. Not tensor calculus. How do tables transform..
This means that folding tables can be well expressed in tensors correct?
Explaining complex things simply is an art that only masters of the field can handle. Incredible energy transformation. Thank you
A video about Geometric Algebra would be interesting as a follow up!
In some ways they're the best type of multivector I heard.. I feel like you're the only guy that can do the topic justice!
Sho Am Geometric algebra is awesome, but it’s such a huge topic that one video couldn’t begin to scratch the surface.
If Hestenes can't make it intuitive, I got no clue as to who ever will
@@Lucky10279 I feel like we need an abundance of short videos about an introduction to the topic!
@@thstroyur and Mikayla, but it always good to know there are people who've even heard of it. Not to mention, even understand it! The best one I've watched is by Mac T Borsch
6:20
you know... if you change coordinate system, and don't change axis along what you calculating momentum, you must consider it.
bc even if coordinates' 0 is in the center of circle and you calculate from axis on the side of the path, that would be still changing and going 0 for a moment.
In my bed, ready to sleep.
Things changed with one notification of crazy asylum.
How do you do, my fellow Indian?
@@nootums well, going to sleep again. Understanding tensors in dreams.
Same
Same
Same.
We need a part two on tensors.
I still have the book on tensors and maths which I really wanted to avoid for the next year's course but after this video you changed my mind.
I swear, this is the best channel on all of RUclips.
"Magnetism is a pseudo vector" aha that's one doubt my teacher never understood. Thanks man, now I understand electromagnetism so much better. Because as soon as you said angular momentum is a pseudo vector i was like then induced electromagnetic fields have a vector that is pseudo, i thought that couldn't be right but you explained so well. That was the only hard part for me.
Thanks again so much man.
Really like your way to explain complex stuff. Just got your advanced theoretical physics book. I love it - Thank you!
You’ve done it again! You’ve taken a very complex issue and made it understandable.
his video about tensors is the one who easily taught these mathematical and physical objects. And I can say that for sure because I've read several books and watched several videos on the subject!
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.
so my favourite RUclipsr (spoiler, Nick: it's you!) uses my favourite software (Blender!) to illustrate a topic that has both intrigued and interested me for years. it's like Christmas in January!
Dear Professor, this is the most helpful for me to understanding the real concept of tensor. I want some series of videos on coordinate systems also .
Fun video lesson to watch. I think that I can sense why this one had you a wee bit more nervous than usual. I also think you pulled it off brilliantly. Got a little D&D dicey there for a few moments, though.
One of the best science channels on RUclips.
THANK YOU! As a programmer, I'm trying to understand tensors; I’m learning the math and computer science of tensors for applications with packages like TensorFlow (used for AI development). Breaking down each definition with visual aids helped me unclutter the confusion between what makes something a 2-rank tensor and how it can still be considered 3-dimensional.
Glad I could help 👍
After all those years of watching RUclips videos about tensors, Jan 15 is when I could finally understand them.
HECK yeah
Hello Sir Nick Lucid, i regularly watch your videos, and i get completely amazed by your knowledge and the way you explain the most difficult topics....
I just want you to create a short video on tesseracts and please explain something more about the 4th dimension....
I will be most thankful to you...
if you havent read Richard Abots "Flatland" start with that. After watching so many Avenger movies I didn't realize the importance of what the tesseracts is! If I understand it correctly it is a 4-dimensional object. Cool...
So, I was reading up on some artificial intelligence stuff for college and there were mentions of tensors, so I quickly googled it just to get the idea what it was, I read up on Wiki and was completly lost so I brushed it off since I didn't really NEED to know what tensors are to continue. Jump forward a few days and last night I dreamt about tensors the whole night, I almost couldn't sleep because of them for some reason. So I wake up and I'm like, "well, I guess I have to learn what tensors are", I google it and this time I see a "The Science Asylum" video on tensors, I just think to myself "Damn, I'm in luck" and move into the video.
Clone: "What are we supposed to use?"
Nick: "Index notation!!!"
I don't think I've ever heard anyone that excited about notation before. I like it!
Wow this video really helped understanding tensors. At Uni you are just dealing with its math and properties so you don't get to understand these stuff. Now it's actually making sense to me :D
For me, it started focused but then went off the rails in moving into areas making the subject far more complicated than it needed to be. But I appreciate this is your way of expressing yourself.
I wish this was here before my tensor exam, from 1 video I understand more than from the whole semester
I have learned more watching this channel for half an hour than the total of what I learn from Veritasium.
I go a lot deeper than Derek does.
Derek is a much more pop-sci kinda creator, less detailed and more informally explained. Note I am in no way saying he's not as good, Veritasium is an amazing channel. Just a different audience.
You, 3blue1brown, Eugene Khutoryansky and many more are amazing, thanks for the videos!
You've put me in good company there in that list 🤓
Agree, you are doing what schools should. Thanks for explaining so well. You’re an amazing explainer.
To me, that’s even better than a teacher.
Explainers cut through the crap and make you understand.
8:44 first row we should divide by c 3:33 axis are in wrong places. K axis must point in the opposite direction. K=ixj right hand rule
8:44 That's only true in SI units, but spacetime tensors are usually written in c=1 units.
3:33 You can certainly make the k-axis forward instead of backward, though I don't think it matters. There's no reason the indices _should_ obey the right hand rule (like the spatial unit vectors). It might be good just for consistency though, so I'll give you this one.
Wow! Congratulation! Excellent explanation! It is simple and easy. So the next challenge, I guess, would be how to explain to students a concept of covariant and contra variant tensors without messing up lower and upper indices. It would be an interesting challenge.
Had studied tensor analysis about 23 years back. Out of touch after that. Great refreshment, . Many thanks.
Glad I could help 🙂
i literally have in my to-do list "look up what the heck is a tensor", and suddenly this video appeared! awesome!
im struggling with my higher level physics classes ( like I want to quit cuz I suck so bad at setting up problems). This video finally showed me what a tensor was, which everyone seems to talk about but no one attempts to explain.
Thanks again for making me feel like I can understand complex topics. I wish my classes could be 1/16 as good
Glad I could help a little bit. Good luck!
Those animations helped me a lot. Satisfied.
I think the last time I checked, maybe a year ago, your book was only eBook, but now it's also in paper form! Yay!!! Can't wait to get it sometime soon, even though I am currently not studying physics. I want to understand how math works better first, so I'm studying foundations of mathematics, while also pondering away at philosophy and philosophy of language. But once I feel content enough with myself, I'll get back at it! As always, great vids.
Thanks a lot Nick! I really needed simple explanation of tensors ....
Great explanation of tensors. Always had trouble connecting the tensors of mathematics and tensors of physics!!!
I'll tell you, it's a thing that transforms like a tensor.
Definitions like that are the reason I had to make this video.
@@ScienceAsylum "Circular definitions are definitions, that are circular." paraphrase or quote (don't remember) from Isaac Arthur.
The Science Asylum Reminds of when i kept hearing that a vector was a member of a vector space, then, when I asked what that was, it was defined in terms of vectors. Then I found the list of vector space axioms, but it still took me a while to figure out how those axioms had anything to do with the usual geometric interpretation of a vector. Hard to believe that was only about a year ago.
@@ScienceAsylum but in the video they are also defined with their transformation properties, so what's the difference? These definitions always include the concrete form of transformations.
@@ScienceAsylum And you clearly understand the subject. Circular definitions reveal that the definition writer doesn't quite grasp the real meaning. Even though I had a difficult time following the video, my difficulty is only due to a lack of algebra skills and physics.
I also follow "PBS Space Time", "Because of Science" and others, but this stays my favourite channel about physic.
If there is ever a RUclips University...please be the head of the Physics department.
By far, this is the best explanation on tensors around ! Bravo !
Great video!! I wished professors would explain as clear as this
Thanks! 🙂
That just made my years of casual study of relativity make a lot more sense in an intuitive way. I just trusted the math. No one has explained it that way and frame dragging now makes sense in terms on non-euclidian geometry such that I could approximate a graph in my mind without math in constantly changing or accelerating coordinate spaces.
I remember asking my professor in detail about tensors. I didn't ever watched the video yet, but your explanation is way superior.
Edit. I would really like to buy your book. I can't wait to get a brand new insight on physics.
Also, do you plan adding some terminology about thermodynamics? I think it would be awesome to have it there.
I will not be adding thermodynamics, or any other chapters for that matter. That book took _forever._ I'm not ready to be writing again.
So in that demonstration (6:39), in order to maintain conservation of angular momentum you must also consider the attracting body that's in the middle of the circle? and so it must have the greatest velocity (relative to the shown axis) exactly when the orbiting body passes through the axis (and it's contribution to total angular momentum is 0 as show)?
My grandma is a tensor, every situation is a bit tenser when she arrives.
Same with Mom.
Not sure if the appropriate response is rim-shot, womp womp or sad trombone. =(
@@beansnrice321Maybe a "da dun da tssssssss"
Like the comparison. Makes understanding tensors in free space easier to understand. Photon, graviton, proton,.. 👀
You made me chuckle out loud...
This channel is legit the best science channel I've ever seen. Maybe the best science videos period. It's like everything I wanted from Bill Nye before he became some old hasbeen who won't shut up about global warming. You deserve your own platform and I think you should seriously consider making online courses or something. The Khan academy guys got nothing compared to this
finally a new video.
been waiting for a long time....
after watching:
i am still a bit confused.
The BEST concise explanation that dispels the mystery of tensors
The t-shirt is evil! I can whole-heartedly recommend your book to anybody, not because I'm a fan of the channel. I genuinely find it useful and refer back to it all the time.
As someone who is currently writing his bachelor's thesis in physics... Thank you for finally making me understand what a tensor is.
Glad I could help and good luck 👍
I now just realized why in programming the common notation to use for 'for loops' is 'i' and then 'j' when dealing with nested loops. You use said loops to go through arrays, which can be the same as vectors, which are tensors, and to access an element from the array you need to index it. The variables that we used to hold the index value as we go through the array with the 'for loop' are the freaking ranks!
I mean, kinda, normally the implication in programming is that you would need a rank per dimension of the array, which is no really the case with tensors as I saw in the video, but still.
That was how I was visualizing it too
Actually I think it goes back to the days of FORTRAN. IIRC, the letters starting with "i" (maybe "i" through "n") were implicitly defined as integers so they were easy to quickly use as loop counters.
I thought "i" stood for "iteration". With nested "j" loops, your point makes more sense.
Thanks!
How did I actually understand this? You sir, are a genius.
But how'd you know that? Maybe _you_ are the genius here...
Finally , a video about something I had trouble with in physics !! Thank you !!!
YES!!! FINALLY!!!! I waited such a long time for this😅😅😅
I've to watch it a few more times though, it's a lot of information to digest in one take. I have a question, how does the angular momentum change when you change the coordinate system?
P.S : a new clone, nice :)
Angular momentum is defined as position times linear momentum. Since position depends on coordinate system, so does angular momentum.
@@ScienceAsylum oh, I thought the "r" in the *L* = *r* × *p* formula was calculated from the axis of rotation. In every problem that I solved so far the axis of rotation was either the x, y or z-axis so I assumed that "r" was calculated from the axis of rotation. Now I understand my mistake. I need to study these concepts deeper and pay more attention to the definitions. Thanks nick :)
@@ScienceAsylum by the way, what physics textbook explains these concepts clearly? I have Halliday-Resnick but I don't think this book explains them in enough depth. I also need a book who isn't too technical because I get lost easily.
@@ScienceAsylum no it's not, it's a distance, relative to an axis you've chosen to calculate from, often it's axis of rotation. Coordinates system changes, the axis of rotation does not
Thank you, you cleared my those doubts which otherwise would have taken so many hours to understand via text books, thank you from heart....☺️
What IS a tensor?
_I'm assuming you're going to tell us in this video_
I am! (finally)
I passed differential and integral calculus and linear algebra back in college but not multivariable calculus or differential equations. Also passed classical physics and E&M but not modern physics.
I am still struggling with tensors! 😖
I'll watch the vid a few more times and see if I get it 🤓🤪🙃
Here 2?
I've watched TENS OR hundreds of your videos and your sidekick asks the right questions what do you call him Mini Lucid ? Great video you always get the like before I watch your videos ... keep up the great work going to watch video again
You're a fantastic youtuber. Nobody else has explained Tensors as easily as you have. Great job.
Amazing explanation!!!! That's the intuitive vision that every professor should transmit to students before formalize the issue.
Aaaaasooom
As always entertaining and educational
Thank you for the great vid!
This is much better than the first chapter of Sean Carroll's Lecture Notes on General Relativity.
Next video: Magnetic Reconnection in the context of magnetohydrodynamics?
Wow!! You're not afraid to take on the big dragons. Best effort I've ever seen. 😀
Way nicer than other 'tensor' videos, yet I am still not sure that I understood what tensor is actually.
However there are great animations that give some idea of the relation of 'tensor', coordinate system and it's 'property'.
Some things are still confusing, but hey, we can re-watch the video again and again trying to catch these little moments where some light bulb will shine :)
"Rank is the number of pieces of information you need to identify a component."
Thank you for this. Now I finally get what exactly rank actually refers to. I was thinking it meant the number of dimensions needed to represent a rank n tensor as an array.
"I was thinking it meant the number of dimensions needed to represent a rank n tensor as an array." It does tho. Like, a 2-D array means you need 2 numbers to locate any piece of information in the array. That's what dimensions fundamentally mean. How many numbers you need to locate/define something within the dimension
@@feynstein1004 True. The way he defined it is easier to understand though.
@Mikayla Eckel Cifrese Whatever helps you understand :)
Hi, Can u explain what is rank really describes in matrix and tensors. I am searching for answers pls clarify me. Thanks
@@amarj9909 He did explain what the rank of a tensor is -- it's how many data values you need to locate each coordinate of a tensor. Scalars are rank 0 zero because they only have one coordinate value, so not additional information is needed. Vectors are rank 1 because they're just column matrices and so we only need one data point to indicate a specific coordinate value (that's a big oversimplification of what a vector actually is, but I don't want to get sidetracked), and matrices are rank 2 because they you need to reference both the row and the column number in order to locate a specific coordinate value.
The rank of a matrix, on the other hand, is just the dimension of the rowspace and columnspace (they're always the same number.) The rowspace is the general solution to the system of equations represented by the rows of the matrix and the columnspace is just the rowspace of the transpose matrix. 3blue1brown has a really good video about the column and rowspaces of a matrix and its rank. I highly recommend it if you want an intuitive understanding of what the concepts mean geometrically.
you are mind-blowing. Can you plz put a video on Bessel and Hermite function and spin-orbit coupling.
Holy sh*t, he actually did a video on tensors.
A very well communicated and comprehensive explanation. I now know what a tensor is, and why they're useful.