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What it means is that we can get the components of the vector in a certain direction by doing the dot product of the vector with the basis vector in that direction. For example: V1(subscript 1, i.e. covariant component in direction 1)=V(vector)*e1(basis vector 1). [Where * is the dot product.]
I've spent so much time trying to find a simple explanation of covariant and contravariant vectors online, and in the first 3.5 minutes you've managed to out perform anything I've come across. A well deserved round of applause to you, Eugene! Keep up the great vids!
The title is misleading _almost_ to the point of clickbait. This video is an 'intuitive' explanation for those already familiar with tensors on a formal basis. It's a 'now I get it', or 'I never thought of tensors that way' for people who took tensor theory in university, etc. For a _genuine_ introduction for straight beginners, try Dan Fleisch' video. (I'm not Dan Fleisch, incidentally)
I think this is the first time I ever saw a video where the person explaining had any idea why they were called covariant and contravariant. Other explanations I've seen have been as bad as "covariant means indices downstairs; contravariant means indices upstars." Which doesn't actually explain the meaning of covariant and contravariant at all, of course, but is a description of a notational convention.
This channel is honestly top notch. Most resources are either too simplified to the point where they are not useful to someone who actually needs to learn this material, or they are so dense that a new learner gets lost in the details and misses the big picture. You do a great job at making the point clear (with the aid of amazing visuals) while also keeping everything accurate. Seriously, this is world class educational material. Get more famous!
The professors I had in the university while doing my Bachelors all failed to explain the concepts of covariant contravariant in an understandable manner. You have done what they have failed to do in less than 12 minutes! :D #RESPECT
The music is really, really, really distracting, classical music isn't really suitable as background music as its very structured, and often complex. Try using something more repetitive and 'boring'. 3blue1brown's way of doing it works very well.
Having a good instructor makes a night and day difference when learning more advanced subjects. Great video. Making the jump from just dealing with vectors to tensors trips up a good number of people.
FINALLY A HELPFUL VISUAL REPRESENTATION!! I’ve been stuck on intuiting covariant vectors for YEARS! I think I get it now, it’s the *components* of the vector that are really covariant or contravariant, not the invariant/intrinsic vector itself
Excellent introduction to tensors. It's funny how you could complete a whole masters or PhD and never see these any more than a 2d drawing of these mathematical objects, but then a video comes along and in under 12 minutes shows you what it took so long to wrap your head around to imagine.
Your channel should be promoted by some other famous channels, like Vsauce. Your videos are just too good. 3Blue1Brown got promoted this way. Maybe one day, this channel will as well.
totally agreed. what is missing here though, is the charisma of the speaker and aesthetic design, I guess, which makes alot of difference in this platform.
I've literally spent several years trying to understand tensors through self-studying to no avail. Your videos are the most intuitive and easy-to-understand way I've found and for the first time, I actually feel like I have a good understanding of tensors.
I was looking into tensors 3 days ago and couldn't wrap my head around them and your video nailed it for me! Thanks a lot! Let me see if you have one on Quaternions, your skills might just finally break the wall for me to grasp how they are beyond Axis/Angle rotation and why if the axis is not normalized with a quaternion I get a skewed transform! Keep up!
Really good video, you've done that people can visualize something which many professors didn't get in many years with their students and tried to explain as a teachers a visual concept with lots of usefuless words and few quality visualizations. Thanks
Thank you so much. I've been trying to get some sort of intuition for what a tensor is, and this is definitely the best video I've found to help me with that.
Absolutely phenomenal video, i really wish we had these to study 8 years ago. I finally understood the difference between co and contravariant .... before i just knew the definition
Omg, this channel is a Gold mine for upper division classes. Again thank you so much. You’re helping me with Quantum mechanics and Electrodynamics! Specially as a nonverbal visual learner this really helps!
Just a humble piece of advice: I think music should be more "subtle". Orchestral music is beautiful but I think it can "bother" a little when you try to concentrate on explanations. Of course: this is my point of view, of course.
Great video. I prefer defining a covariant vector via its dot product with the corresponding contravariant vector being an invariant. This is how Tullio Levi-Civita defined it in his famous book, used by Einstein in his 1917 GR paper.
I have seen many brilliant professors in my PhD struggling to convey a concept. I do not know if you are an academician but I am sure that you have a bright-mind with profound insight in the topic. Your way of looking at things is so effortless and effective at the same time that it goes straight into the brain. Kudos
Excellent Eugene. Great explanation and visual of co-variant, contra-variant, and sub/super scripts. Nice to grasp the concepts and rules of the game. I have had two different physics instructors who couldn't explain what you have put so succinctly. I have also read many texts that convoluted such simple material. I look forward to watching more of your videos. Thank you so very much!
Despite this being a great animation (like the one about Fourier transforms, which is even much better) this video I feel an inconsistency lurking with regard to the statement that the dot product decomposition is covariant. Let's take the most simple example of three orthogonal basis vectors and an arbitrary vector (like the situation around 20 seconds in this video). Now all the components of this vector are the dot product (orthogonal projections) with (on) the basis vectors. So if you make the basis vectors x times longer (or shorter) and giving this new basis vector the value 1 the components of the vector become x times as short (or long). But because the components are the dot product with the basis vectors, also the dot product decomposition becomes x-times as short, and this result is passed on to the case where the basis vectors are not orthogonal. Look for example at the video at around 2:58, where it is said that if you make the basis vector twice as large the dot product becomes twice as large too, but the basis vector you make twice as large gets again the value 1 and the corresponding vector component becomes twice as small (like is explained earlier: if you make the base vectors twice as large, the vector's components get twice as small), so each of dot product of the vector components with the basis vectors becomes x times smaller (larger) if you make the basis vectors x times larger (smaller), hence contravariance. A good example of a covariant vector follows from the (x,y,z) vector. This is a contravariant vector, but the (1/x,1/y,1/z) vector is a covariant one. More concrete, the wavelength vector [which corresponds to (x,y,z)] is a contravariant vector while the wavenumber vector, the number of waves per unit length, is a covariant vector [which corresponds to (1/x,1/y,1/z)]. See Wikipedia's "Contravariant and covariant" article.
I am learning tensors by myself and this has been the most incredible explanation of covariant and contravariant components. Thanks for this work. It´s great!
You need to get your name out there. You should talk to other popular youtubers for support. Your videos are incredibly unique and informative, more people need to watch them. Professors should also be using your videos as to tool to teach students.
Thanks you for the exlanation. It helps me clear tons of mistaries! However, I am still a bit confused about the covariant component at 2:58. If the resultant vector remains constant and the base vectors are doubled in length, shouldn't the value of the components be decreased in order the result in the same vector? Please correct me if there's any misunderstanding.
The dot product between two vector is given by the product of the normes times the cosinus between the 2 vectors : |v1| * |v2| * cos If |v1| stays constant and |v2| double in length then the dot product is doubled : it's covariant.
Wonderful....! Just amazing.... Eugene... Your videos definitely make life easier for those who truely want to master physics and mathematical concepts.... Kudos for you efforts and pranams for the profound Knowledge that you are imparting through ur videos.!
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@Leonardo Ramìrez Aparicio. In my understanding, you can perform dot product and what you have are the componets of the vector IN ANOTHER BASIS, that is the dual basis.
This is a great video, thanks Eugene and Kira! I understand your description of contravariant vectors, and how a vector can be represented by a contravariant combination of basis vectors. It would be great if you could elaborate on how a vector can be represented by a combination of dot products of arbitrary basis vectors. Perhaps "dot product" needs to be defined first (and "angle")?
Beautiful, such a great topic served with clarity and with great music in the background that was expertly timed. I love how the introduction of the covariant vector is joined by a very intense and vigorous passage that later resolves to calm once explained. Delightful !
exactly rightly time for me too. whenever I have confusion in a particular topic, you are uploading a video in that topic exactly. Thank you very much.
Physics Videos by Eugene Khutoryansky in all seriousness, I have been searching for quite some time for a good intuitive demonstration of what a tensor actually IS, and what it "looks" like. I'm deeply grateful to you for at last providing a particularly helpful one - not that I'm at all surprised at the source, given your astounding track record for such things. Thank you once more, not only for this but for all of your different videos and the hard work that has clearly gone into them. They've helped me tremendously in my academic pursuits over the years, as I'm sure they've helped many others. You and others like you are an integral part of the future of modern education.
How they transform. A rank 2 tensor with two contravariant components transforms doubly contravariantly, which means the components get a lot smaller when the basis vectors get bigger. A rank 2 tensor with two covariant component gets a lot bigger when the vectors get bigger.
Whizzer191: Do you know an example for a field/application where the version with two contravariant components would be used instead of the other example? I can't think of a way where I'd use it over the other one.
Hi,Thanks for the video and the explanations.In the beginning of the video you say "if we double the length of the basis vectors, the dot product doubles" if V = (2, 0) in the basis e1 = (1, 0), e2 = (0, 1), V.e1 = 2 But if e1' = (2, 0), V in the new basis would be V = (1, 0), and V.e1' = 2 So why didn't you express V in the new basis for the dot product but you did it for the normal components of V ?
(First I thought "what a sensible explanation" ... then I realized I don't get the covariant case, having the impression it played out similar to the contravariant case ... but days later ...) (As of now, edited, my comment doesn't fit here as a comment on Faw Bri) I believe I understand now. Before, I was wrong in two points: 1) I did not fully understand the dot product. It goes like (V dot E = |V| |Ê| cos(angle V-Ê)). Having learned the dot product in the context of coordinate systems with orthonormal basis vectors (all basis vectors at right angle to each other and of UNIT length), I IGNORED the basis vector's magnitude as a factor (it used to be always 1, because of unit basis vectors). 2) Even though explicitly stated in the video, I still did not realize that the the new component equals in fact the dot product itself. Instead, I wrongly assumed the new component to be that multiple of the basis vector length that is equal in lenght to the projection of vector V onto that basis vector Ê (alike to the contravariant case, where the component is a multiple of the pertaining basis vector).
For DECADES I've searched for an explanation of tensors that's as simple as the one that you've presented here in less than 12 minutes. Thank you, thank you, thank you ! I am in your debt.
OMG, I can't believe I've been trying to figure out tensors, covariant/contravariant components, etc for so long, and it suddenly made complete sense. great work!
I highly recommend the videos by Prof. Pavel Grinfeld (MathTheBeautiful) for more on this subject, as well as his textbook, which focuses on geometrically intuitive approaches to this subject. Prof. Bernard Schutz's books are also excellent, though they require more mathematical maturity on the part of the reader.
The contra-variant components are shown graphically to be related to the vector's length but the co-variant components are not. It doesn't show how one could derive the vector from the co-variant basis vectors which can apparently be multiplied to any size without changing the vector they define. When the covariant components were increased or decreased, the vector was unchanged.
In the video you mention that the same rank 2 tensor composed of two vectors can be described as various combinations of covariant and contravariant components of those two vectors. My question is, are these different representations completely determined by each other? For example, if you have a rank 2 tensor T, which you know was composed by the covariant components of a vector P and the contravariant components of a vector V, can you tell what the representation would be if you wanted it to be composed of the _contravariant_ components of P and the _covariant_ components of V, instead? Even if you don't know the actual vectors P and V but only the tensor T? Another question is, all these representations composed from different combinations of "variances" of some component vectors P and V feel like they would all be 'nicely' related to each other. Kind of how different basis vectors give different different representations of the same vector. Do all these combinations form a nice structure, similar to how vectors are still vectors despite the choice of basis used to represent them, if any?
If you know the metric for the space, then you can determine the covariant components from the contravariant components, or the contravariant components from the covariant components. The metric for the space is defined by the metric tensor, which lets us know how to calculate the length of a vector, given the vector's covariant or contravariant components. I plan to cover the metric tensor in my next video.
I am a bit confused by your statement about the covariant components. If you double the length of your basis vector, the scalar product with the basis vector (so your covariant components) will be divided by 2 and not multiplied ? Or if you don't set the new length as the new unit but just multiply by 2, then the scalar product remain the same ? In my understanding of tensors, the contravariant basis (ie the covariant components) was defined by the invariance of the covariant-contravariant product, that is by the metric tensor. May you clarify this point for me please ? And keep up the good work !
Asterisque and others, I'm trying to clarify this for you. Let's take the magnitude of v-vector sqrt(136). This magnitude comes from a rectangular "box" with the sides 6, 6, and 8. This "chosen" vector makes the angles 1,2,3 with the three directions of the basis vectors e1, e2, and e3. If the length of all vectors in the basis is 1, then (v)dot(e1)=sqrt(136)*cos(angle1), (v)dot(e2)=sqrt(136)*cos(angle2), and (v)dot(e3)=sqrt(136)*cos(angle3). Now, let's increase the length of all vectors in the basis to 2. The new dot products will be: (v)dot(e1new)=2*sqrt(136)*cos(angle1), etc. The values of these "new" dots product are the doubles of the "old" ones because the angles do not change. The dot products are covariant. In the "old" basis, the contravariant components of the v-vector were (6,6,8) while in the "new" basis they will be (3,3,4). The length of the contravariant components decreases when the magnitudes of the vector-basis increases.
Man, words can't express how thankful I am for that insight… I've been trying to get an intuition of this sort on tensors since I first tried to study them. Always been a fan of yours, now more than ever. Keep up with your excellent work! ^^
By the way, I'm now doing my masters in mathematics and I'm used to making some math gifs in Maple (nothing as huge as you usually present us with, just some examples, but I really enjoy doing so ^^). I don't know which software you use for your animations, but if you ever need (or accept) any help, I'm here for you. Thank you once again for sharing your knowledge!
The 3D animations are what's really great about this video. Such things are necessary for a subject like Tensors in my opinion, and these 3D animations are very clean and accurate. Great job!
covariant vector.. im little confused when the value of dot products doubles along the doubling of basis' length, isnt the vector( white one. or V vector as you wrote) should expressed in basis which is before doubled? notice me if what my comment is imperceptible.
You’ve got it the other way around, the vector always stays the same, that is a given; it is independent of the basis. When we change the basis vectors keeping the white one constant, it’s dot product changes in the same direction. Like, 1 kg and 1000g are the same mass, but are expressed differently here.
@@h2ogun26 check out this series: ru-vid.com/group/PLRlVmXqzHjUQARA37r4Qw3SHPqVXgqO6c or if you really want to know, why the del is used: ru-vid.com/group/PLRlVmXqzHjUQHEx63ZFxV-0Ortgf-rpJo
This is by far the best explanation about tensors that I could find. This has helped me tremendously for my general relativity class. Thank you so much!!!
The William Tell Overture. I grew up with a classical music compilation CD (one of those various “Greatest Hits of the Classics” compilations). Though I *first* encountered the first two movements in old cartoons (there used to be a lot more classical music in cartoons), and had occasionally heard bits of the last movement in the context of The Lone Ranger.
You can think of a dot b as being the length of the projection of vector a in the direction of b "stretched" |b| times. Or the length of the projection of b in the direction of a multiplied by |a|, it will give the same answer. In physics this can be thought of as the work of a along the displacement b, in maths it is simply vector projection, or as you said, an area.
Excellent explanation to Tensors, your animation is on completely different level n so is your explanation. This video really helped me to clear my all doubts regarding tensors. I simply loved it. Thank you so much :)
Suppose we just shove some numbers together in some particular order. Not going to say *why*, but hey... at least they're swaying constantly. Suppose we then claim this to be intuitive.
Information density is a bit low, even when on 2x speed. The constant movement of the "3d objects" is a bit unnecessary. I still hit that like button, because the matter discussed is quite abstract and the explanation splendid! Well done ;-)
This is cool. I didn't realize it but tensors are used in backpropagation. When you multiply the activation vector for a layer with the derivative vector of the error over the net inputs to the layer, you get a tensor with the derivative of the error with respect to each weight (using tensor product as described in the video). This tensor is then used to train the network. I am glad I found this video because I knew what I needed to solve this problem, but I didn't know it was actually a tensor
Really? Einstein's field equations in the next video? You're going to skip over raising and lowering indices (which I really wanted to see), special relativity, curvature, the Riemann tensor, the stress energy tensor, and go straight into Einstein's field equation?
I already covered both Special and General Relativity in many of my earlier videos. I plan to cover raising and lowering indices, curvature, the Reimann tensor, and the stress energy tensor all in my next video. Thanks.
Dear Eugene, you are truly a great teacher! I'm spreading your channel to all the people who like me are interested in these topics.I wanted to ask: are you planning to make a video about the quaternions? It would be great to finally have a clear one like what you did for the tensors. Thanks for all your work
It's sad the amount of people who saw one of your videos and subscribed, but didn't continue to watch your new videos. You deserve way more views per video than you are getting on average, especially considering how many people thought you were worth a sub.
Wow, you must have a God given talent for teaching. You've simplified it so that a high school student with a decent algebra 2 or a pre-calculus background would get it on a first go. Thank you very much!
