The video is incredible, pure gold! The exhibition is masterful! It is the best EVER introduction to contravariant and covariant components, and metric tensor (I use many books and resources). I was reading the "Covariant Physics (Moataz)" book and I started to get lost in chapter 1.3, it is an easy chapter where the contravariant and covariant components are introduced along with the metric tensor in cartesian coordinates (metric tensor is not mentioned in any case, and it is used as the Kronecker delta like a kind of magic term to convert between contravariant and covariant components) . Only after watching the video, I have truly understood the chapter. I am excited about these GR videos, I have decided to study them carefully one by one. I have already seen the first one. I am a graduate in Physics who, after a long time, went back to studying certain topics to fill many gaps that were left during my career. Thanks Professor Edward D Boyes for this precious resource.
Just beyond belief! Wonderfully expressed, I love how you subtly hammer home the points to guide people away from the restrictions of geometry learned from a black board! " Be very careful what you put into that mind for you will never get it out!".
Hi Eddie, Thank you for the videos. Everything (the content, the flow, the math, the questions, the voice, the presentation etc.) is fabulous. At time stamp 57:21 on LHS third entry's subscript needs to be 3. 🙂 Thank you once again, Mani
Your series are wonderful. If I had studied them when I first started, I would be much much further along. Thank you I wish I could figure how to put this at the top of the intro to GR recommendations. You have no peer!
Excellent series! Thank you! The contravariant and covariant components and the lowering indices made sense for the first time. Now I am all confused about the many places in which there seems to be a need for a dual vector space with a set of dual basis vectors, etc.
Small typographical error I suspect: for the matrix equation at around 50:46 , the right hand side is product of 1x2 and 2x2, which would produce a 1x2 row vector, but the left hand side is 2x1 column vector. I suppose the matrix of dot products should come first followed by the contravariant as a column vector.
Eddie, you made my day😊🎉! What a great video. I am looking forward to see the whole series. I have a question. Do you have a reference to a good GR book which contains problems and solutions for practise?
Thanks Mani - another one I hadn't spotted! I'll probably put a "correction" into the Video Description rather than upload a new corrected version (as that would re-start the counts etc from scratch). Putting a "correction" into the Video Description ought to put a correction notice at the relevant time point (according to TH-cam) ...... but I can't seem to get that to work at the moment. Thanks again anyway. Eddie
Beautiful explanations and examples. I've seen a lot of GR videos but this series is one of the best. I have one question though. Why are covariant vector components with lower indices presented on same axis when they belong to dual vectors basis which are different in such manner that each one is perpendicular to the original one with different index?
In any case, you can also have the covariant components with respect to the contravariant basis: th-cam.com/video/nNMY02udkHw/w-d-xo.htmlsi=l0YONDtcKOnksqLt&t=468
Thnks for the serie! One question: the proof @32:25 holds only if basis vectors are unit vector, otherwise general formula is V_n=gmnV_upper_n ? Tnhks, bye
![fellow fellow - พรุ่งนี้ไม่มีใครรู้ feat. INK WARUNTORN [OFFICIAL MV]](http://i.ytimg.com/vi/QP6JyjYx_W0/mqdefault.jpg)
What a gifted instructor. If you truly want to understand GR you must make this series a stop along your way. Thank you Eddie.
This is THE place to start studying GR. Then you can hit the books and other video lectures more profitably.
I finally got the geometric interpretation of covariant and contravariant components … thank you!!!!
The video is incredible, pure gold! The exhibition is masterful! It is the best EVER introduction to contravariant and covariant components, and metric tensor (I use many books and resources). I was reading the "Covariant Physics (Moataz)" book and I started to get lost in chapter 1.3, it is an easy chapter where the contravariant and covariant components are introduced along with the metric tensor in cartesian coordinates (metric tensor is not mentioned in any case, and it is used as the Kronecker delta like a kind of magic term to convert between contravariant and covariant components) . Only after watching the video, I have truly understood the chapter.
I am excited about these GR videos, I have decided to study them carefully one by one. I have already seen the first one. I am a graduate in Physics who, after a long time, went back to studying certain topics to fill many gaps that were left during my career. Thanks Professor Edward D Boyes for this precious resource.
A superb series of videos which progress in slow logical steps. Impressive!
Just beyond belief! Wonderfully expressed, I love how you subtly hammer home the points to guide people away from the restrictions of geometry learned from a black board! " Be very careful what you put into that mind for you will never get it out!".
Muhteşem anlatım. Çok değerli bilgiler. Teşekkürler.
Hi Eddie,
Thank you for the videos. Everything (the content, the flow, the math, the questions, the voice, the presentation etc.) is fabulous.
At time stamp 57:21 on LHS third entry's subscript needs to be 3. 🙂
Thank you once again,
Mani
Your series are wonderful. If I had studied them when I first started, I would be much much further along. Thank you
I wish I could figure how to put this at the top of the intro to GR recommendations. You have no peer!
Beautiful instruction! Most enjoyable to learn quite sophisticated topics
Excellent series! Thank you! The contravariant and covariant components and the lowering indices made sense for the first time. Now I am all confused about the many places in which there seems to be a need for a dual vector space with a set of dual basis vectors, etc.
fantastic professor
This is an excellent series. Thank you very much!
Excellent, thank you
This is a great exposition of the subject thank you!
Believe me, I have tried a lot of others ...
.
A Master explanation !!!
Thank you very much!
you make it very easy to understand thank you
Small typographical error I suspect: for the matrix equation at around 50:46 , the right hand side is product of 1x2 and 2x2, which would produce a 1x2 row vector, but the left hand side is 2x1 column vector.
I suppose the matrix of dot products should come first followed by the contravariant as a column vector.
Eddie, you made my day😊🎉! What a great video. I am looking forward to see the whole series.
I have a question. Do you have a reference to a good GR book which contains problems and solutions for practise?
Remember Co-Low-Pro
Co: Covariant components
Low: Use lower indices
Pro: Represent projections onto coordinate axes
THANKS !
Thanks Mani - another one I hadn't spotted! I'll probably put a "correction" into the Video Description rather than upload a new corrected version (as that would re-start the counts etc from scratch). Putting a "correction" into the Video Description ought to put a correction notice at the relevant time point (according to TH-cam) ...... but I can't seem to get that to work at the moment. Thanks again anyway. Eddie
Beautiful explanations and examples. I've seen a lot of GR videos but this series is one of the best. I have one question though. Why are covariant vector components with lower indices presented on same axis when they belong to dual vectors basis which are different in such manner that each one is perpendicular to the original one with different index?
You are right, it is probably a simplification to not introduce dual basis vectors
In any case, you can also have the covariant components with respect to the contravariant basis: th-cam.com/video/nNMY02udkHw/w-d-xo.htmlsi=l0YONDtcKOnksqLt&t=468
Great videos. Is there a playlist so that I can watch them sequentially? Thanks!
Thank You!
Should the basis vectors always come from the covariant bases?
1:00
Black background with light letters please, it make it easier to learn
second that!
1:06:52 Isn’t it a bit early to name g a tensor? We know :) that not every object with indexes is a tensor, right?
You has no explain it!!!!! SoNOLIKE
Thnks for the serie!
One question: the proof @32:25 holds only if basis vectors are unit vector, otherwise general formula is V_n=gmnV_upper_n ?
Tnhks, bye
Thank you very much!