Tensors for Beginners 16: Raising/Lowering Indexes (with motivation, sharp + flat operators)
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- Опубліковано 16 бер 2018
- Error:At 10:39, the last v~ component should have an upper k index instead of upper j index.
Error: At 11:14, the kronecker delta should have a j subscript, not an i subscript.
This is the last video in the Tensors for Beginners series. I'll be starting a Tensor Calculus series later in April.
ERROR at 14:13: It's true that "α = a·__ = g(a,__)" but the line after that should read "a = inverse-g(α,__)". The result of this expression is a tensor whose components have 1 upper-index, so they are vector components. Sorry!
Just wanted to attempt to express my gratitude for these videos. This was a topic I found inaccessible for years and with this series it feels entirely demystified. This series is really invaluable to me and to many others as well I’m sure. Thank you.
same. It's amazing how it that every other source makes tensors almost intentionally obtuse
I agree, these series of videos really makes things clear. Great work! thank you very much.
Yes thank you so much for these videos :D
Great Job!!! I took a course in Tensors and somehow made it out with a B-. Yet I can say I've learned more watching these videos than in said class. This series is a rare piece of gold on UA-cam!
Thanks. Glad they were useful!
I'm finishing my math degree, doing the final proyect, and it's thank to this and the other tensor series videos that I'm going to be able to do something to feel proud of. In 4 years no one ever explained this as nearly as quick, clear and simple as you did. Thank you so very much.
Glad they helped. Good luck on your project!
These videos are excellent. It teach us more than any vector or matrix class for more than 40 hours. Really appreciate what you have done here in just 18 small videos. I feel very grateful.
I just binged the whole series. And it resolved so much confusion around tensors, which in hindsight are probably a bit silly. I'm so glad you still create vids. I'm probably gonna watch the spinors series soon. I really enjoy your content and I know I'm not the only one. THANK YOU!!
Always glad to hear people still like these. I made them because I found tensors pretty confusing too.
Just completed watching all the 19 videos on Tensors. This has been the most educating video series I have ever watched on Tensors. It cleared a ton of issues for me. Thanks a lot for making these videos.
I never knew about the musical sharp and flat's usage before, we usually just did it the long way. I wish i had these when i was learning GR.
Thanks for the amazing series!
I keep coming back to this series and it hit me this time that I would NEVER have truly understood tensors studying any other material. You were instrumental in relieving all the frustration in trying so hard to get a grasp and intuition about what these powerful abstractions are, and the fear of being stuck forever on the sidelines and not being able to move forward with Einstein's theory of gravity. It's been said many times before in the comments but a big thank you for putting this together. Let there be light.
Thank you very much . It must have taken a tremendous number of hours to put this series together. It is extremely well presented and understandable. It fills so many holes in my understanding.and corrects many misunderstandings.
Thanks. It did take some time. I'm glad it's paying off for people.
Throughout my entire physics bachelor's programme, tensors were never properly introduced and we were expected to just pick up on them on the way. Now, thanks to this series which I've spent my entire day today on, I finally have a solid grasp. As a top comment phrased so well, the whole concept of tensors got demystified. And it could not have come at a better time, as I just started my bachelor thesis which essentially is about decomposing tensors. Thank you Chris, your efforts are highly appreciated!
This series was extremely helpful. You have a gift for explaining things. Thank you!
Thanks. I'm glad you found it useful.
@@eigenchris any idea where I can get some exercises for this?
@@eigenchris please upload the notes if possible.
@@shameer339 They're here: github.com/eigenchris/MathNotes/tree/master/TensorsForBeginners
@@eigenchris Thanks a lot!!! 🥳🥳🥳🥳🥳🥳🥳
Many thanks to these videos. I once attempted differential manifolds, but the lecturer made such a mess. Since then I have been afraid of tensors. This series start from really simple algebra and use lots of ways to make things not abstract any more, and totally eliminate my fear of tensor.
This is great ! I hope you don't give up. You should write a book on tensors and use these vids to support it. Strang made (likely) millions off MIT Algebra. He has a huge fan base. Its probably well worth your while to do this. Tensors are hard and I'm just trying to learn them for Householder Reflectors
I just discovered myself today that raising both indices of the metric tensor gives the inverse metric tensor, and raising one index of the metric tensor gives the Kronecker delta. I feel like a genius. Thank you for your amazing teaching Chris.
Thank you very much Chris! please do not stop doing these videos!
It was an excellent series, I'll recommend this channel to a couple of friends, so please, keep like this!
This series is great, very clear and informative and I think it is good of many levels. Thank you!
Incredible series of video! Your effort is greatly appreciated!
Excellent series: clear, well-presented, and concise. Thank you so much!
Whhhheeewwww. That was a long ride. Off to the tensor calculus series. Thank you so much Chris!
These videos provide the best explanation of tensors I’ve seen to date. Thanks for sharing your work on youtube!
Outstanding! Look forward to many more presentations on this topic!
I really appreciate all your effort to make this series. It has clarified the areas where I was consistently getting confused
Many thanks!
Please accept my gratitude!!
These video series are really outstanding!!
Thank you for all your hard work! These videos are a lifesaver when trying to learn the language of Relativity and Quantum!
This analogy with music theory terminology is so cool! Great video, thanks for putting this together.
Bravo! I've wanted to learn this stuff for a couple years now, and I finally decided to dedicate some time to it. I made the right choice with this playlist. Thank you very much sir, excited to get started on your tensor calculus playlist. Exciting stuff.
Thank you, this was a great series. I enjoyed them because they got right to the point, concepts were explained very clearly.
Thank you so much for making this series, really really helpful!
thank you very much for this. i was sitting through the whole series in two sessions and it's the first time that i feel to really understand some of this tensor stuff! i guess, i'm now ready for the tensor calculus series... :-)
Thank you very much for making these, it has been extremely informative!
Thank you so much! Really liked this series.
Thank you very much, it was very helpful. It's difficult to find a good content like this one on the internet. Thank you again!!
Thanks a lot from Italy! With these video now I understood many concept that was obscure to me
WoW. After many years of not using tensors I found this a brilliant way to regain acquaintance. Thank you so much.
I've just got to the end of this series of videos. As others have said, these videos make a topic often opaquely presented much more transparent. And as others have said, I am very grateful (and impressed) and by the amount of time you must have put into thinking this through and then preparing and presenting the ideas in this format. Finally, I thought it was all very interesting stuff!
I'm now very much looking forward to watching your other playlists as well :-)
Thank you for making this series! I always used to get confused when my professor used Einstein notation, but it makes a lot more sense now. Tensors in general seem a lot less mystifying as well. Your sense of pacing and your visualizations are really helpful.
This series is excellent, and this video in particular is standout. Eigenchris explains in 15 minutes what many a professor have failed to convey for years.
I really liked your analogies with musical sharps and flats. I've never heard these before, and I'm not sure why, because they really help in the understanding.
I am really impressed with these videos. I found it extremely difficult to learn tensor analysis from books, but these videos make the subject much clearer.
I believe you should write a book..,it’ll be a bestseller! Thank you so much! You’re so generous to many of us!
YESSSSS!! I finally reached the end of the series. And boy was it worth it, if only for the 7:50 mark, where it is revealed the gij v^i(Ep)^i can be written as v_i(Ep)^j. I am self-studying GR for personal interest, and my physics and math background are limited to a few college courses many, MANY years ago. I understood the general concepts of co- and contravariant, but was getting confused on the details. NOW I get that v_i(Ep)^j actually includes the metric tensor, it’s just summed over.
So thank you for an awesome series. I have watched some of the videos in the TC series, but will starting it from the beginning to follow in sequence. I’m hoping as I return to the GR studies, so many things will be obvious now.
Just started studying differential geometry, and your videos are incredibly helpful. Your way of teaching things simply, quickly, intuitively and logically seems to be very effective. Thank you and kudos.
ty so much for your videos, I have been studying general relativity, and your classes have opened my understanding of its math's.
Thank you, amazing video series which finally opened world of tensors to me!
What a great series, your explanations are just so clear. Hopefully, many teachers understand this is how you describe complicated topics to students.
I would just add a numerical example video to finally understand the concepts !!
Thank you so much for sharing this video series with us!!! Blessings!!!
This is gold Chris. I want to thank you so much.
Your explanations were fantastic, thank you so much. As pretty much every comment in this section says, this topic seemed absolutely inaccessible, and your explanations made it very clear and simple. Thank you so much!
Thank you for making this playlist. It was all very clear. Thumbs up!
I finish your course for introduction for tensor in 4 days, this course for me is really useful because I have ever been learned about tensor algebra but my work is related a lot with General Relativity, Thanks for your jobs now I clearly understand about the definition of tensor in algebra approach. Next, I am going to with your tensor calculus course. Thank you again a lot!!
Thanks I wish most instructors would use this video as a foundation lectures to lay down the basics, unfortunately most get straight into Tensors or into Tensor Calculus making this subject very inaccessible. When I was introduced to Tensors as part of the Special and General relativity I just followed the rule without really appreciating the profound meaning. It took away from the physics especially General Relativity. So many thanks to you for help rediscover Tensors and appreciate the deep insight in them.
Thank you for your teaching. I suddenly decided to learn general relativity by myself, and I have struggled two weeks trying to understand the chapter of tensor analysis. Luckily I found this series last week. It is really a great pleasure learning from you.
i watched this whole playlist. Thank you so much; this was very helpful.
Brilliant series!!!
Godly playlist
Very very thank you for sharing such insightful way of tensors
Bravo. Thanks so much for this series.
Lot of respect and gratitude u have saved me many hours of reading...
Glad the upload time of the video is not on the April Fools' Day --- because, you know, those flats and sharps could make a good joke. Though very intuitive and convenient notation. And of course thanks to the author for the great course and all the effort --- it gave me real insight on the tensors!
It's impossible to learn from this series and don't say thanks. Really excellent hard work . Waiting for your book
It was a pleasure to learn the basic definitions related to the topic of tensors across these short 18 videos. I'm hesitating though if I should start watching through the calculus playlist, since learning tensors was merely a subject of curiosity in my case and not of necessity.
i have been confused about vector to covector correspondence since the first covector video in the series. thanks for clarifying that!
Best course on Tensors.
Excellent series!
Thank you for your intro series; fast-forwarded thru Kronecker tensor product hope I never need that; tensor products seem easier to use as you explained so well. Great explanation of raising and lowering.
Thanks a lot for these videos. It will help to understand tensors. Thanks!
Nice playlist, easy to understand, thank you!
This series is a really excellent job and I really appreciate for your hard working for this series.
Now I feel I have found a way to understand general relativity theory.
This series is an invaluable present for me.
from South Korea
I'm glad you found it helpful. I am currently working on a relativity series and am just finishing special relativity, and am planning to move onto general relativity. I only release 1-2 videos per month though, so my pace is slow.
@@eigenchris You know, Schubert's symphony no.9 is , but it's a still masterpiece. Your job is outstanding whether you finish your plan or not.
Excellent series. Thanks.
Fantastic work in this series, obviously not as detailed as a textbook but much clearer. You have a really good linearity in all your videos. Thanks!
Thanks. I would hope things would be linear, given that we're talking about tensors...
eigenchris ahahah of course, but it’s not always like that as you might find in some lectures...
Thank you very much!! You saved my differential geometry class!!!!
An excellent presentation. As an amateur, I always wondered where the raising and lowering of indices came from (not to speak of the sharp and flat symbols I have encountered). Thanks.
Hi, just watched in one long sitdown the whole series. I just read Part 1, about tensors, of the book An Introduction to Tensors and Group Theory for Physicists (which I highly recommend) and, even though the book explains really well and with a lot of detail, I just got confused in many parts. Your series helped me understand those parts, and this last video about raising and lowering indexes was for me the cherry on top, because for me it was especially confusing this part in the book.
If you could make a video about contraction of tensors it would be great! I fell it is missing from this series. Anyway, congratulations for it!
Never undergone a full course on tensors being from an engineering background, however, used some ideas ( in continuum mechanics). I am really curious about learning more and more and your videos have been of great help. Thank you very much. I teach the theory of elasticity to graduate students and could use these videos for training my students.
You might be interested in my Tensor Calculus playlist, which is a sort of sequel to this one. It covers how to handle vector fields, covector fields, and how to change basis using the Jacobian (which is like the continuous version of the F and B matrices). Although you might only be interested in the first 10 videos or so (later videos start talking about metric fields and the math needed for general relativity).
Thanks Chris. I shared a link on my LinkedIn now that I finished the series.
Hi,
first of all, let me thank you for this series. I'm a second year student majoring in teaching mathematics and physics at the University of Szeged, Hungary, and the question of what exactly a tensor is has been haunting us for the last two semesters, and no one we asked gave us a satisfactory answer in either of the two departments. Your clear explanations really helped me actually understand a lot of things I learned in the last few months. The only thing better would have been if I found this series last September, but I believe that's entirely my fault.
So, because up to this point I have not really found a better explanation of this concept, especially in Hungarian, I would like to ask your permission for me writing a set of notes based on these videos, in Hungarian, for further reference and for helping out my peers with it. Of course, I would link the relevant parts of this series wherever applicable.
the music analogy out of nowhere at 13:30 got me 😁😆
Thanks eigenchris. Great job.
I thoroughly enjoyed learning about the existence of musical isomorphisms.
Cant express the gratitude I have for these videos since I hv difficulty in mathematics (also being a physicist student😅) ...A lot of thanks sir..👏👏👏
And finally I see even music symbols popping up in a tensor discussion.
Great tutorials, learnt a lot, many thanks!!😃
Bravo pour cette vidéo excellent travail encore bravo
Pure bliss
You are not merely explain, you tell us the story, that’s why they are so attractive.
Skimmed through the whole series but unfortunately, tensor is still a mystery to me. Probably this series have gone to the other extreme - pure mathematics. But I know this must be my problem and I'm excited this series helps to give me a glimpse into pure and higher level of mathematics which is what interests me to go deeper. Have saved this and will definitely come back and have another go at a slower pace. Thank you very much!
Great series, thx :)))
謝謝
Thanks for sharing this series.
When I learned PyTorch... , I got problems with the tensors.
😊
Thank you so much!
Thank you 🙏🏼!!
Hi Chris, I finished in two days this playlist. Now, I will watch "Tensor Calculus" and after "Relativity". 🗿🗿🍷🍷 Kkkkkkkk
Good video!
Thank you for these videos! and thank you specially speaking english so clearly haha (i don't understand english very well)
Thank you!
Thanks a lot!
What a ride
thank you sir
brilliant
Thank you very much for you lectures. I am trying to study on my own general relativity and I had to step back for tensor calculus. Your video are extremely useful to me. I have one question: at 12:14 you write D^{ab}g_{ax} = D_x^b, is this equivalent also to D^b_x? I was tempted to write the b contra-variant index before because from left to right you find the contra-variant index before the covariant index. More in general the position of the indices is important in the mixed tensors? If it is important what is the rule to write the order correctly in the contracted tensor? thank you!