@@eigenchris The textbooks also entirely neglect the importance of basis elements to begin with! That's a huge problem, but I understand why the do so. They do it for economy of notation. However, including basis elements seems to make everything else make sense so much sooner.
@@eigenchris writing covariant things as rows and multiplying by the matrix on the right and contravariant things as columns and multiplying by the matrix on the left makes a world of difference. I learned this from the Wikipedia article and thought it was a fantastic idea. However in this video it is even much better explained because you go through the trouble of actually drawing/writing every intermediate step.
Really cool explanation. For a long time I had understanding covariance and contravariance. It only tells how important it is to have a good teacher and a simple explanation to explain concept topics.
Well done - really well articulated. I've struggled with these concepts and looked at various texts and videos for many years until stumbling upon this explanation. Expounded so well, very gifted! Thanks
This is more like it- genuinely supper stuff! This shows how very cleary you understand the concept to have been able to make it comprehensible even to those who are not physics graduate. Thanks Chris.
This single video explains a concept that is so poorly explained in the majority of online tutorials I have seen and eluded me for years. Now it makes simple intuitive sense!. Thank you very much sir!
Spectacular presentation. I'm a self learner who has a Ph.D. in biology and I'm retired. Taught myself calculus since I've been retired and now want to go onward with Einstein. Been thinking over the last few months of which books to buy and what is on TH-cam. Ordered a book, etc. Noticed you have improved immensely and are stating you will also go back and redo parts of your series. Fantastic. I shall keep coming back if you keep up the good work. You have a good competitor with blue/brown. thanks much. I've found a good book and you!
I love your SR series. Top quality. I just want to suggest making the background black to make it eye friendly just like in khan academy videos. Thank you
I just started my grad studies in physics and I was really really struggling with the definition of covariant and contravariant. This gave me a complete understanding about the concept that any text book could not do. This is great and Thank you very much..!
I remember struggling a lot with it too. The first definitions I came across always used Jacobians and derivatives. It's very frustrating the concept isn't explained simply first.
Eugene Khutoryansky has a video explaining the covariant and contravariant a little different. He talks about two different ways of representing a vector. In one of them, the components do not change in the opposite way from that of the basis. This is called the covariant vector or covector. I think it ,makes more sense to me. Components that chance the same way as the basis are components of a covector while components that change the opposite way as the basis are components of a contravector, so their product will always be the same, as we expect from the inner product.
Yeah, I talk about covectors in video 106a of this series, when I introduce them to explain the doppler effect. I didn't bring them up this early in the series since I worried they would just confuse people, and they wouldn't have practical applications for another 15 videos or so.
Thanks. I'm just a guy with an engineering physics degree who thought it would be cool to learn general relativity, but quickly got depressed at how difficult and incomprehensible the math was. After searching for the best explanations possible and thinking about it for a while, I decided to make videos teaching GR how I would have wanted it taught to my past self.
Thank you so much for this amazing presentation , i am actually working on wormholes in my master thesis 2021 , and i always get angry because of my none understanding of mathematical tools, well i know very well how to use them but i always ask myself questions like why did they use tensor , why covariant and contravariiant vector, why this topology and why this geometry , i guess things are getting clear with your videos thank you so much !! I wanted to know if you are a physics professor or expert in the field ?!! i feel soo overwhelmed of not beeing able to understand things like you do :( , sometimes i think that general relativity is not made for my mind , But i am learning and i won't give up till i find answers to all my questions ! THANK YOU VERY MUCH SIR :D
wellnice video but what I think is missing from here (or does not go with your analogy) is that you can also have kontravariant basis vectors and covariant components. In general relativity for example you have the tangent plane that consists of all the base vectors d_x(P) in the Point P but you also have the kotangent plane wich is buid by all kontravariant base vektors d^y(P) in point P. they have to fullfill the formula =delt(y,x). So nice video for the very beginning but I dont think the analogy holds forever (sadly).
First of all thank you for doing all this. It's a massive help. I have a small question. Since the change-of-basis and change-of-component matrices are mutually inverse then they could simply be interchanged in the matrix products. Therefore, what's stopping us from saying that the column vector transforms according to the change-of-basis matrix thus making it covariant etc. ?
You could do that, but then the "inverse matrix" would be the one changing the basis and the "original matrix" would be the one changing the components. This would correspond to another different change of coordinates, but opposite to the one you originally had. The vector components still transform opposite or contrary to the basis vectors, so they are still contravariant.
A question: the length of the pencil is NOT the same in the two coordinate systems. In the first video with the rotated system the length was invariant. How to reconcile this? Thanks
By "length", I don't mean a number. The number you get when you measure length depends on the units you use (1000mm = 1m = 0.001km, but these are all the same length). What I mean is, the pencil did not get longer or shorter, so it's length has not changed.
Good night. Do you know why the vector basis transform is covariant e isn't contravariant? I understand the components and basics transform are opposite, mas don't understand why basics transform is invariant and components are contravariant.
I was always under the understanding, without figuring out what they were, that in a Cartesian coordinate system covariant vectors and contravariant vectors were the same. This is why when we study vectors we never talked about covariant vectors and contravariant vectors. How does your example show this, or am I wrong.
I don't bring up covariant vectors ("covectors") in this video. To learn about them, you can watch my "Tensors for Beginners 4: What are Covectors?" video (link below). Covectors should be thought of as stacks, not arrows. th-cam.com/video/LNoQ_Q5JQMY/w-d-xo.html
You're right. I should have been more clear on that point. I should have said "Special Relativity" predicts it, just as FitzGerald predicted it in the aether theory.
Here's a list: basis vector = covariant vector component = contravariant basis covector = contravariant covector components = covariant I don't actually cover covectors until video 106a in this series. But you can skip ahead to that if you already understand Galilean and Lorentz transformations.
this name "covariant" confuses me when basis vectors grow, the components shrink since the components behave opposite to that of basis, the are called contravariant. that is clear but why do we call basis as covariant? covariant to what? Please explain
There end up being other things that "behave like basis vectors". Basis vectors are basicslly thr definition of "covariant", and other things that behave like them are also "covariant". You can skip ahead to the 106a video to see what I mean if you want.
Ok, after months of thinking on this, I have an explanation of covariant vs contravariant that no one has ever uttered, as far as I know. Here goes. Hold my beer! --- You combine contravariant (components times their regular basis vectors) tip to tail to reach the tip of the vector you're defining. But with covariant, you combine (covariant components times their dual basis vectors) tip to tail to get to the tip of the vector you're definging. Why does no one explain it like this? But my question is how is covarant components of dual basis vectors relate to the dot product? Please correct me if I'm wrong on the following... DOT PRODUCT: A (vector) dot B (vector) = a scalar quantity CONTRAVARIANT: described by the combination of contravariant (components times regular basis vectors) added tip to tail of A (vector) dot B (vector). COVARIANT: described by the combination of covariant (components times dual basis vectors) added tip to tail of A prime (vector) dot B prime (vector). QUESTION: If we dot product A prime (vector) with B prime (vector), does that scalar quantity equal A lower 1 prime times e upper 1 prime PLUS A lower 2 prime times e upper 2 prime? If so, arent we then saying that a scalr is equal to a vector???
3:00 - It feels here like you are describing contravariant vectors only, but you're just saying "vector." Maybe you'll patch it up somewhere later, but the strength you put behind "basis vectors go down, components go up" worries me a little. What if your vector was describing a power density, or any other quantity which has the length unit in the denominator?
Aren't the concepts of covariance and contravariance relative? Like, you define basis vectors as covariant and vector components contravariant, because one grows when the other shrinks, but it is also true that when the other grows, the one shrinks. Couldn't you have just as well have defined vector components as covariant and basis vectors as contravariant? I feel like I'm missing something here...
We say the behaviour of basis vectors is covariant by definition. We could have done things the opposite way, but this is the standard and "most obvious" definition to make.
At 3:12 you say basis vectors are covariance. I can disprove that. As I increase my components, my basis vectors decrease. To that means that basis vectors are contravarient I'm contravarient systems. Also basis vector are covariance in covariance systems. Furthermore, if I had designed this system r would equal the contravarient component times the contravarient basis vectors or r would equal the covarient basis vectors time the covarient component, thus eliminating so much confusion.
![[Live] : ONE FIGHT NIGHT 24 “จาร์เร็ด vs กุสตาโว”](http://i.ytimg.com/vi/ENNA-fsT67c/mqdefault.jpg)
This is THE BEST presentation of covariance and contravariance that I’ve ever seen. No textbook does this even remotely as well.
Thanks. In my opinion, realizing basis vectors can be written as rows makes a world of difference in making sense of things.
@@eigenchris The textbooks also entirely neglect the importance of basis elements to begin with! That's a huge problem, but I understand why the do so. They do it for economy of notation. However, including basis elements seems to make everything else make sense so much sooner.
@@eigenchris writing covariant things as rows and multiplying by the matrix on the right and contravariant things as columns and multiplying by the matrix on the left makes a world of difference. I learned this from the Wikipedia article and thought it was a fantastic idea. However in this video it is even much better explained because you go through the trouble of actually drawing/writing every intermediate step.
Really cool explanation. For a long time I had understanding covariance and contravariance. It only tells how important it is to have a good teacher and a simple explanation to explain concept topics.
this is the best explanation of covariance and contra variance relations .
Intelligence is taking complex things and making them simple, well done!
Excellent. This concept is usually glossed over leaving students with massive confusion. Keep going.
@pyropulse Thank you for the correction.
Thank you so much for these lectures. You are a natural educator.
Well done - really well articulated. I've struggled with these concepts and looked at various texts and videos for many years until stumbling upon this explanation.
Expounded so well, very gifted!
Thanks
This is the best explanation on this topic that I have ever seen in my life and I had 5 years of Masters in Pure Mathematics.
EXCELLENT AND VERY CLEAR EXPLANATION WITH SUPERB GRAPHICS!!
This is a totally intuitive and comprehensive presentation of Covariance and Contravariance. Thank you so much eigenchris.
Your an outstanding teacher if anything good is going to come out of this pandemic maybe it will be you having more to to make these videos
Agreed. And I will have more time to watch them.
Yep.
This is more like it- genuinely supper stuff! This shows how very cleary you understand the concept to have been able to make it comprehensible even to those who are not physics graduate. Thanks Chris.
This is the best channel on Relativity
Dozens of videos and never a disappointing one!
wow. This is one of the best explanation of this vector concept. I had reviewed this issue from many sources and never get it until now. So thank you.
This single video explains a concept that is so poorly explained in the majority of online tutorials I have seen and eluded me for years. Now it makes simple intuitive sense!. Thank you very much sir!
This is the simplest/easiest explanation I have seen for covariance/contravariance, I wish more teachers used this video as an introduction
Spectacular presentation. I'm a self learner who has a Ph.D. in biology and I'm retired. Taught myself calculus since I've been retired and now want to go onward with Einstein. Been thinking over the last few months of which books to buy and what is on TH-cam. Ordered a book, etc. Noticed you have improved immensely and are stating you will also go back and redo parts of your series. Fantastic. I shall keep coming back if you keep up the good work. You have a good competitor with blue/brown. thanks much. I've found a good book and you!
Man. I love your videos.
Another lovely series!
Thank you very much for the clarity.
Excellent. Waiting for your next video. Watching your videos is full use of lockdown and to cope with this deadly covid 19 pandemic.
Thank you for this amazing video
Im loving this series so far, i really wanna learn general relativity and i feel like im on the right track
Great work, great explanation keep making videos!!!!!!
Love it!!
I love your SR series. Top quality. I just want to suggest making the background black to make it eye friendly just like in khan academy videos. Thank you
Thank you for your outstanding explanation!! Please keep going step by step until you reach the general relativity equation..
I just started my grad studies in physics and I was really really struggling with the definition of covariant and contravariant. This gave me a complete understanding about the concept that any text book could not do.
This is great and Thank you very much..!
I remember struggling a lot with it too. The first definitions I came across always used Jacobians and derivatives. It's very frustrating the concept isn't explained simply first.
Eugene Khutoryansky has a video explaining the covariant and contravariant a little different. He talks about two different ways of representing a vector. In one of them, the components do not change in the opposite way from that of the basis. This is called the covariant vector or covector. I think it ,makes more sense to me. Components that chance the same way as the basis are components of a covector while components that change the opposite way as the basis are components of a contravector, so their product will always be the same, as we expect from the inner product.
Yeah, I talk about covectors in video 106a of this series, when I introduce them to explain the doppler effect. I didn't bring them up this early in the series since I worried they would just confuse people, and they wouldn't have practical applications for another 15 videos or so.
Who are you, Eigenchris? You're explanations are so good!!
Thanks. I'm just a guy with an engineering physics degree who thought it would be cool to learn general relativity, but quickly got depressed at how difficult and incomprehensible the math was. After searching for the best explanations possible and thinking about it for a while, I decided to make videos teaching GR how I would have wanted it taught to my past self.
Explicit, vivid, excellent and wonderful 👍😊
Excelente
Thank you so much for this amazing presentation , i am actually working on wormholes in my master thesis 2021 , and i always get angry because of my none understanding of mathematical tools, well i know very well how to use them but i always ask myself questions like why did they use tensor , why covariant and contravariiant vector, why this topology and why this geometry , i guess things are getting clear with your videos thank you so much !! I wanted to know if you are a physics professor or expert in the field ?!! i feel soo overwhelmed of not beeing able to understand things like you do :( , sometimes i think that general relativity is not made for my mind , But i am learning and i won't give up till i find answers to all my questions ! THANK YOU VERY MUCH SIR :D
a million thanks!!! Just bought you a few cups of coffee...
ty
wellnice video but what I think is missing from here (or does not go with your analogy) is that you can also have kontravariant basis vectors and covariant components. In general relativity for example you have the tangent plane that consists of all the base vectors d_x(P) in the Point P but you also have the kotangent plane wich is buid by all kontravariant base vektors d^y(P) in point P. they have to fullfill the formula =delt(y,x). So nice video for the very beginning but I dont think the analogy holds forever (sadly).
I cover covectors in thelativoty 106a if you want to try watching that.
I actually get it!
The lorentz transformation matrix you have is for (ct, x) and not for (t, x) ryt?
whoops. indeed it is.
if you choose the units for c to be 1, it should be fine
On 8'32'', on the Special Relativity matrix to the right, first row, it should be v/c^2 and not v/c. Otherwise there is a problem with the units.
First of all thank you for doing all this. It's a massive help. I have a small question. Since the change-of-basis and change-of-component matrices are mutually inverse then they could simply be interchanged in the matrix products. Therefore, what's stopping us from saying that the column vector transforms according to the change-of-basis matrix thus making it covariant etc. ?
You could do that, but then the "inverse matrix" would be the one changing the basis and the "original matrix" would be the one changing the components. This would correspond to another different change of coordinates, but opposite to the one you originally had. The vector components still transform opposite or contrary to the basis vectors, so they are still contravariant.
@@eigenchris thanks a lot!
Are you Canadian? I noticed the (now retired) Canadian penny.
Yup! (eh!)
Buddy please upload Relativity 103
What tools do you use for making your videos?
A question: the length of the pencil is NOT the same in the two coordinate systems. In the first video with the rotated system the length was invariant. How to reconcile this? Thanks
By "length", I don't mean a number. The number you get when you measure length depends on the units you use (1000mm = 1m = 0.001km, but these are all the same length). What I mean is, the pencil did not get longer or shorter, so it's length has not changed.
Good night. Do you know why the vector basis transform is covariant e isn't contravariant? I understand the components and basics transform are opposite, mas don't understand why basics transform is invariant and components are contravariant.
The Lorentz transformation matrix and its inverse yield a multiple of the identity matrix. Why is that?
Are you talking about multiplying a lorentz matrix and an inverse lorentz matrix? This should give the identity matrix, without a scalar multiple.
@@eigenchris Ah, that's my bad. I forgot a property of gamma and beta. Sorry about that! This series is amazing!
I was always under the understanding, without figuring out what they were, that in a Cartesian coordinate system covariant vectors and contravariant vectors were the same. This is why when we study vectors we never talked about covariant vectors and contravariant vectors. How does your example show this, or am I wrong.
I don't bring up covariant vectors ("covectors") in this video. To learn about them, you can watch my "Tensors for Beginners 4: What are Covectors?" video (link below). Covectors should be thought of as stacks, not arrows. th-cam.com/video/LNoQ_Q5JQMY/w-d-xo.html
@@eigenchris Ok. Thanks. I'll do that.
8:44 Can you say that Einstein “predicted” Lorentz contraction when it fact it was deduced _before_ Einstein came up with Special Relativity?
You're right. I should have been more clear on that point. I should have said "Special Relativity" predicts it, just as FitzGerald predicted it in the aether theory.
Didn’t Lorentz derive it entirely from Maxwell’s equations?
🥰
It make me confused why normal basis is covariant. Is it true that covector is covariant since it's components change covariantly with its own basis?
Here's a list:
basis vector = covariant
vector component = contravariant
basis covector = contravariant
covector components = covariant
I don't actually cover covectors until video 106a in this series. But you can skip ahead to that if you already understand Galilean and Lorentz transformations.
this name "covariant" confuses me
when basis vectors grow, the components shrink
since the components behave opposite to that of basis, the are called contravariant. that is clear
but why do we call basis as covariant? covariant to what?
Please explain
There end up being other things that "behave like basis vectors". Basis vectors are basicslly thr definition of "covariant", and other things that behave like them are also "covariant". You can skip ahead to the 106a video to see what I mean if you want.
Ok, after months of thinking on this, I have an explanation of covariant vs contravariant that no one has ever uttered, as far as I know. Here goes. Hold my beer!
---
You combine contravariant (components times their regular basis vectors) tip to tail to reach the tip of the vector you're defining.
But with covariant, you combine (covariant components times their dual basis vectors) tip to tail to get to the tip of the vector you're definging.
Why does no one explain it like this?
But my question is how is covarant components of dual basis vectors relate to the dot product? Please correct me if I'm wrong on the following...
DOT PRODUCT: A (vector) dot B (vector) = a scalar quantity
CONTRAVARIANT: described by the combination of contravariant (components times regular basis vectors) added tip to tail of
A (vector) dot B (vector).
COVARIANT: described by the combination of covariant (components times dual basis vectors) added tip to tail of
A prime (vector) dot B prime (vector).
QUESTION:
If we dot product A prime (vector) with B prime (vector), does that scalar quantity equal
A lower 1 prime times e upper 1 prime PLUS A lower 2 prime times e upper 2 prime?
If so, arent we then saying that a scalr is equal to a vector???
3:00 - It feels here like you are describing contravariant vectors only, but you're just saying "vector." Maybe you'll patch it up somewhere later, but the strength you put behind "basis vectors go down, components go up" worries me a little. What if your vector was describing a power density, or any other quantity which has the length unit in the denominator?
I don't introduce "dual vectors" / "covectors" / "covariant vectors" until the 106a video. I didn't feel it was necessary early on in the series.
@@eigenchris Ah, that explains it. Ok - fair enough. Concern retracted!
Aren't the concepts of covariance and contravariance relative? Like, you define basis vectors as covariant and vector components contravariant, because one grows when the other shrinks, but it is also true that when the other grows, the one shrinks. Couldn't you have just as well have defined vector components as covariant and basis vectors as contravariant? I feel like I'm missing something here...
We say the behaviour of basis vectors is covariant by definition. We could have done things the opposite way, but this is the standard and "most obvious" definition to make.
@@eigenchris Gotcha, thanks for clarifying.
At 3:12 you say basis vectors are covariance. I can disprove that. As I increase my components, my basis vectors decrease. To that means that basis vectors are contravarient I'm contravarient systems. Also basis vector are covariance in covariance systems.
Furthermore, if I had designed this system r would equal the contravarient component times the contravarient basis vectors or r would equal the covarient basis vectors time the covarient component, thus eliminating so much confusion.