To see subtitles in other languages: Click on the gear symbol under the video, then click on "subtitles." Then select the language (You may need to scroll up and down to see all the languages available). --To change subtitle appearance: Scroll to the top of the language selection window and click "options." In the options window you can, for example, choose a different font color and background color, and set the "background opacity" to 100% to help make the subtitles more readable. --To turn the subtitles "on" or "off" altogether: Click the "CC" button under the video. --If you believe that the translation in the subtitles can be improved, please send me an email.
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
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
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.
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.
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.
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.
You can help translate this video by adding subtitles in other languages. To add a translation, click on the following link: ua-cam.com/users/timedtext_video?v=CliW7kSxxWU&ref=share You will then be able to add translations for all the subtitles. You will also be able to provide a translation for the title of the video. Please remember to hit the submit button for both the title and for the subtitles, as they are submitted separately. Details about adding translations is available at support.google.com/youtube/answer/6054623?hl=en Thanks.
@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.
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
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.
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, 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!
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
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!
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!
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
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.
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 have to thank the producers of this videos. You make it easy to understand. Also the interpreters of this. Especially the Arabian. You helped me. Thanks.
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.
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!
OMG!!! This video explained some things that I have struggled with for years, despite reading so many things on tensors. Wow. Thank you. The Rossini background music is very appropriate.
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.
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!!!
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.
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 !
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")?
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.
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.
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.
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 ;-)
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!
As all videos you did before ,all of them are great. this motivates me to create a youtube channel and trying to express and present your videos into arabic to be easy for Arab students to touch , see , feel and understand the science
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.
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.
Awesome video and simple to understand narration. Thank you so much. OMG ! It took more than couple of decades for me to come across such a lucid and simple visual narrative that captures the essence of how a tensor is defined. This video is a vote in the plus column of why the internet and democratization of media such as this makes sense for mankind.
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).
This is the most intuitive way to understand tensors for beginners i hav found Understanding Tensors has to b done togethr with concepts of covariance and contravariance Thanks Keep up the good work
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.
Compairing with the abstract learning in math books and engineering courses, I believe this kind of videos in YTube present a different perspective on math learning, and are very helpful to clarify concepts and equations. Thank you.
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.
Yessss! Finally an explanation behind the terminology "covariant" and "contravariant". It's alien language like this that can really throw me off learning new topics in physics & math. MAHASIVE Props to you.
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.
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.
I'm not a mathematician nor have any qualifications but an avid interest in science since school days so when I read that Newtons theory ,instilled in me in my school days,was wrong I tried to understand it. Alas tensors came up this is the clearest explanation I have seen . From these superb series of explanations I have discovered that Acceleration produces time dilation and that is a major factor in Einstein's theory not the warping of space .
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It was a pleasure to translate this video to portuguese. Everyone should have the chance to learn a bit about tensor calculus.
Thanks. I appreciate the translation.
What were these made with?
@@no_one6749 This looks like OpenGL to me, or perhaps DirectX, probably programmed in C++.
Eugene, can you tell me the name of the song, please?
As a hobbyist mathematician you have no idea how valuable these videos are, please dont stop making them, you're helping people be smarter
Thanks. More videos are on their way.
@@EugeneKhutoryansky how dot product gives vector
@@AkhilKumar-ci6pb dot product doesn't give vector
@@tripp8833 but in video it is daid like that what does it mean then at 2:40
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!
Thanks.
Eugene's videos are great but I still don't understand tensors :D
Thanks I will!
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!
Thanks for the compliment.
100% true
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
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
If you like this video, you can help more people find it in their UA-cam search engine by clicking the like button, and writing a comment. Thanks.
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Physics Videos by Eugene Khutoryansky
Physics Videos by Eugene Khutoryansky
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.
Next time, don't add such music
Your combination of graphics, content and music is otherworldly 😊
Thanks for the compliments.
Chronicles of tensors: the musical
THe Wilhelm Tell Overture is hilarious as the proper covariant choice of music, you'll agree. A hidden dimension!
our great classical music adds so much drama to on otherwise sober topic
Eugene, your videos are absolutely incredible. Thank you for doing such a great job making things so well-explained and intuitive!
Thanks.
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.
Glad my videos are helpful. Thanks.
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.
Nah, this video could have done with a tad of Mars, Bringer of War if you ask me.
@8:36 the music makes it worth waiting through a 5 minute ad to hear the punch line. It was great!
Just a humble piece of advice: Use the mute button if you don't want to hear sound. I happen to enjoy the music...
Yes, music level was distracting. And no, mute would not work, since the explanation is accomplished via audio (duh).
"duh"?? That says it all...
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.
Just some ideas. I wonder if it would be possible to visualise Lagrangian Mechanics, or Hamiltonian Mechanics. Or Calculus of Variations.
Thanks. I will add those topics to my list of topics for future videos.
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Thanks.
What do you mean when you say that we can describe a vector in terms of its poin product with each of the base vectors?
@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.
"Suppose we multiplay one of the contravariant component of the V with one of the contravariant component of the P"
For what???
"Suppose we multiplay one of the co-variant component of the V with one of the contravariant component of the P as shown"
Why? And?
7:50 WHAAAAT???????????? For what?
You tube amazing because of the people like you who believe that knowledge should be free.
Thanks.
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
Finally someone could explain in a concise and clear manner what covariant and contravariant components are! Thanks a million!
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.
Thanks.
tiuk23 : I think PBS Space Time would be a good candidate for collaboration.
Duuude. I just promoted him on minutephysics.
true
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.
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.
Glad my video was helpful. Thanks.
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!
Thanks. I am glad my videos are helpful.
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
These videos are consistently enlightening. They should be part of curriculum. Well done!
Thanks.
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!
I simply can't understand why this topic in the books is so entangled and you just made up so easy!
The music makes this the most stress intense tensor video anime show I have ever seen in my life.
ua-cam.com/video/XQIbn27dOjE/v-deo.html 💐💐
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!
Glad it was helpful. Thanks for the compliment.
The music when you got to rank 3 made me laugh
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
Thanks for the compliments/
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.
Simply EXCELLENT. I never post comments on UA-cam but this deserves to be the TOP video in any search on the topic.
Thanks. I am glad you liked my video.
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 have to thank the producers of this videos.
You make it easy to understand.
Also the interpreters of this. Especially the Arabian. You helped me. Thanks.
أظن المترجم شخص مغربي لأنه يكتب الثاء تاء فكنت أقرأ "المؤثر" "موتر" فأعتقدها موتور ويمكنك فهمها ك"شد" ف ال " Tensor " يعتبر قوة سحب او شد
ua-cam.com/video/XQIbn27dOjE/v-deo.html 💐
I wish you had uploaded this when I was taking Continuum Mechanics!
How can you be so good at explaining complicated physics and mathematics??? You teach and induce passion for these subjects.
Thanks for the compliment.
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.
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!
Thanks for the compliment.
Great videos. I can't wait to see the one on Einstein's field equation
OMG!!! This video explained some things that I have struggled with for years, despite reading so many things on tensors. Wow. Thank you. The Rossini background music is very appropriate.
I am glad my video was helpful.
Wtf i was trying to find the answer for this on the net and this just popped out in my notifications
-crazy
Glad I made this this video just in time for you. :)
Physics Videos by Eugene Khutoryansky haha thx, always loved ur videos
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.
Google knows what you search. Google owns UA-cam. Makes sense
Great video. Now I got a clear concept about tensor. This is the best video in UA-cam to get a visualization of tensor physically.
Thanks for the compliment. I am glad my video was helpful.
I have never clicked on a notification this fast before.....
Me neither
Exactly!
Ok
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!!!
Thanks. I am glad my video was helpful.
THANK YOU SO MUCH, YOU ABSOLUTE SAGE AMONG MEN
Thanks for the compliment.
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.
The music selection at 8:24 for the rank 3 tensor is HILARIOUS!! 😂
Start at 8:14 and wait for it!
As always, the most excellent video!
Glad you liked my video.
This type of program appeals to my intelligent side. Thank you. Much appreciated.
Thanks. I am glad you liked my video.
I didn't even know this was a thing! Amazing 😀
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 !
Glad you liked my video. Thanks.
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")?
I was confused here: since dot product gives scalar but here it says the vector V can be represented by the dot products of basis vectors?
It has been six years, and this video is still the best video on explaining tensor!❤❤❤
Thanks for the compliment about my video.
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.
I second Prof. Grinfeld's series of lectures. They are fantastic, and he explains the subject very carefully and well.
This channel should be a standard thing to be studied in colleges and universities.
Thanks.
Your videos are so awesome.
Note. I've never used super index values as you showed, I alwais use sub indexes
Forever changed the way i look at maths. THANKS AWFULLY EUGENE. U R DOING A MILLION DOLLAR JOB ACTUALLY U DESERVE BETTER. GO AHEAD SIR
Thanks for the compliment and I am glad that my videos have been helpful. Thanks.
Hmm... I would like to see a video regarding the Navier-Stokes Equations... somewhere in the future.
I will add the Navier-Stokes Equations to my list of topics for future videos. Thanks.
No problem, and thanks.
That would be something to look forward to *excited*
Naturally, just as I start to learn about tensors, you release this. Thank youuuuuuu
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.
I also have this problem. To get the same vector, it seems you have to contra-vary in both cases, right?
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.
thanx alot you always focus on critical issues and help many people to understand in better way
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.
@planet42 THanks for clearing that up
I just cant understand who gives a dislike and why? These videos are gold for anyone trying to study math in an intuitive way
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 ;-)
It's done that way to let you absorb what they're saying.
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!
Glad to hear that my video was helpful. Thanks.
I love the video! However, the music is too good. It is really distracting.
As all videos you did before ,all of them are great.
this motivates me to create a youtube channel and trying to express and present your videos into arabic to be easy for Arab students to touch , see , feel and understand the science
the music is epic
Agreed
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.
At last a simple illustration on the difference between co variant and contravariant components along with associated indexing. Brilliant.
Thanks.
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.
Suppose we get a life, eh?
Awesome video and simple to understand narration. Thank you so much. OMG ! It took more than couple of decades for me to come across such a lucid and simple visual narrative that captures the essence of how a tensor is defined. This video is a vote in the plus column of why the internet and democratization of media such as this makes sense for mankind.
Thanks for the compliment about my video.
best explained
Thanks.
Explenation of rank 3 tensor *William Tell overture ensues* ayy lmao
Wow, the only video about tensors where I actually understand everything
so what is the difference between the 2nd rank tensors produced with covariant, contravarient and combination vectors?!?
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.
you are the best, Please don't stop. You are really making a difference.
Thanks. More videos are on their way.
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).
This is the most intuitive way to understand tensors for beginners i hav found
Understanding Tensors has to b done togethr with concepts of covariance and contravariance
Thanks
Keep up the good work
Thanks for the compliment.
I seem to miss the dot product knowledge to understand the story :-( Maybe a good idea for a future video?
I cover dot products in my video at ua-cam.com/video/h0NJK4mEIJU/v-deo.html
Thanks. I looked for it and failed to find it.
может стоит сделать сайт с нормальной навигацией по темам?
Thank you for the description of tensors, it’s one of the most intuitive I’ve seen
Thanks. I am glad you liked my video.
V and P for the Tensors,
Yes yes, I can sense their relationship, subliminally they will become one.
Incredible video. Never saw a better visualisation of tensor than this one.
Thanks for the compliment.
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.
The tensor made by multiplying the contravariant components to the dot products stays invariant, of course.
Compairing with the abstract learning in math books and engineering courses, I believe this kind of videos in YTube present a different perspective on math learning, and are very helpful to clarify concepts and equations. Thank you.
Thanks.
Very distracting music 😅
I like it a lot for the "slow" speed. It made the concept more understandable. Thanks!
Thanks.
The bleeding obvious, repeated over and over, under nut-cracking classical miuzak!
After days of research, I can finally appreciate this video. Thank you very much for making it.
Thanks.
Thank you for the clarity - the music does get in the way however, would you consider making it much softer or not having it at all?
ua-cam.com/video/XQIbn27dOjE/v-deo.html 💐
this is most beautiful video i ever found on youtube.. huge respect for the team who made it
Thanks for the compliment.
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.
This is the best explanation of tensors I have seen so far. Thanks for posting.
Thanks for the compliment about my explanation.
That background music is annoying...
No better explanation for a tensor has ever seen.Thank you
Thanks for the compliment about my explanation.
51 people accidentally clicked dislike.
Lily Winters it's probably due to the obnoxious music. the visuals are great, but the music is too loud and distracting.
I don't know why the mistake is increasing.
Yessss! Finally an explanation behind the terminology "covariant" and "contravariant". It's alien language like this that can really throw me off learning new topics in physics & math. MAHASIVE Props to you.
Thanks.
what is geometrically a dot product of two vectors ab? aside of the area |a|cosf x |b|cosf, what does it mean?
if we have two vectors a and b, I just can't get what is dot product from this perspective
I cover dot products in my video at ua-cam.com/video/h0NJK4mEIJU/v-deo.html
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.
Thank you!
I saw the taiwaness sub and it's very good for those who are Chinese to see the excellent video. Thank you, Vera Wu.
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.
Uses people as a means to an end. Not very reality based, in the Buberesque use of the word ethics..
you're really rude bro
I'm not a mathematician nor have any qualifications but an avid interest in science since school days so when I read that Newtons theory ,instilled in me in my school days,was wrong I tried to understand it. Alas tensors came up this is the clearest explanation I have seen . From these superb series of explanations I have discovered that Acceleration produces time dilation and that is a major factor in Einstein's theory not the warping of space .
the music is distracting