I did my phd more than 30 yrs ago on fluid dynamics as an engineering student, and as you can imagine I needed to use a lot of tensor calculus in my formulations. Too bad great series like these weren't around then. I want to salute you for putting out your work on UA-cam, which I really enjoy. You are doing a great service for humanity by sharing your insights and knowledge. You certainly deserve a lot more subscribers. Nowadays I browse math/physics video lectures on UA-cam just for fun and see if it can fend off possible cognitive decline in the future.
Thanks for your kind and encouraging words. My goal in doing the series was to present the material in an orderly and straight forward way, as I wish someone had done it for me. Feedback like yours leads me to believe I have succeeded. BTW the series is gaining momentum. It is now drawing 2-3 new subscriptions and 10-20 hours viewing time every day. This is actually more than I expected to begin with. Who knew there was that much interest in such a specialized subject.
Thank you for creating this truly excellent series! I have acquired the book you recommended, downloaded all of the material you so generously provided, and will work my way through your entire course. I am completely self taught, calculus, linear algebra, etc., and in the process discovered I really love math, so I am really looking forward to this!
This is pure gold!!! A mesmerising series on an advanced subject presented clearly, patiently, systematically and in detail. I waited until well into video 66 before writing my comments. This series takes away all the mystique behind tensor calculus. I am very confident that when I start studying GR, mathematics will not be an obstacle to my learning. The teacher has my heartfelt gratitude for presenting these videos.
Brilliant series of lectures. By far the best on the Internet, although the series Math the Beautiful is already excellent. My only suggestion is to have a few more concrete examples such as outer multiplication and the position of the indeces (it's easy to get messed up as that product goes where) and tensor decomposition - most people understand a vector and the metric tensor but having a good grasp of what higher order tensors are is more complicated. The tensor decomposition should provide some insight. But A BIG THANK YOU FOR YOUR TENSOR SERIES. The pace is great and the visuals are alway clear.
My hat to you and thanks for this wonderful exposition of Tensor Calculus : you managed to make one of the most difficult topic of Calculus as clear and easy as possible... Along with a rare didactic quality, as opposed to many video courses and books where authors "loose" the student via an indigestible symbolism.
I'm in the process of applying for a Plasma physics internship at the the UKAEA, and the recommended reading is about curvilinear coordinates for modelling magnetic fields. This video is great for mulling over the mathematical notation that I'm wrestling with
Thanks for your feedback. I use several programs for my publications. For the drawing board, I use free software called FireAlpaca in conjunction with the XP-Pen drawing board. For demos, I use GeoGebra, a free web-based tool. For video recordings, I use ScreenPal, formerly Screencast-O-Matic. There is a free version, but I pay a subscription fee for extra features.
Moving those triangles around isn't really a proof. Also, Newton's _Principia_ was done in geometry.... without using a coordinate system. He was doing calculus with geometry. I think only two people on the planet even understand the mathematics as it was actually written in his published book. Pure geometry is awesome. I love this serious you made, it helped me a lot
David Hilbert was a great mathematician. He certainly understood and used tensor calculus in his work with general relativity, but I am unaware of any contribution he actually made to tensor calculus itself. If he did, i would love to know about it.
@@tensorcalculus822 Perhaps not directly to Tensor Calculus so I stand corrected. However, Grossman helped Einstein with setting up the mathematics of GR. But Grossman walked away. Einstein than struggled for 3 years. Einstein then met Hilbert and he was enthralled that someone understood what he was trying to do (historical letter). Hilbert solved the issue and sent it in a letter to Einstein who then held a conference to present his new GR. In the mean while Hilbert had published the solution. Einstein was angry that Hilbert was trying to take credit for his work. I'm not sure if the two really patched up their differences. I'm skeptical that the field of Tensors would be as important to physics as it is today had it not been for Hilbert, Grossman, and others. Einstein was a great physicist but he got a lot of help doing the math. If you see this: what software package do you use for graphing. And if I haven't said it, thank you *so* much for this series. P.S. I'm not a physicist or a mathematician, but more history of science. Still I deeply love the subjects themselves as well.
Thanks for your feedback. I concur with almost everything you said. David Hilbert did a lot of the heavy lifting where the math was concerned. Indeed, there were only a few people at the time who had the skills to do so. It is my understanding that Einstein was concerned that Hilbert might get credit for his work, but Hilbert was gracious enough to step aside and give Einstein full credit. He acknowledged that it was Einstein's intuitive ideas that gave rise to the subject. It's also worth noting that Einstein was never really comfortable with the geometric model of relativity. Indeed he had some pretty unkind words for Hermann Minkowski who first proposed the four-dimensional hyperbolic geometry of Minkowski space. If I recall correctly he said something like "The old man has restructured my work into a form where now even I don't understand it." It was Grossman who convinced him that this was exactly what he needed to make headway for his general theory. Even though he went with the idea, he was never really comfortable doing so. He was always quick to point out that relativity is about motion and forces, not about geometry. The software I use for the demos is a web-based tool called GeoGebra. It is free and publicly available at this site. www.geogebra.org/3d?lang=en
Absolutely outstanding series - by far the best, most comprehensive and clearest tutorial on Tensor Calculus on UA-cam. Thank you!
My pleasure. Glad you enjoyed it.
I did my phd more than 30 yrs ago on fluid dynamics as an engineering student, and as you can imagine I needed to use a lot of tensor calculus in my formulations. Too bad great series like these weren't around then. I want to salute you for putting out your work on UA-cam, which I really enjoy. You are doing a great service for humanity by sharing your insights and knowledge. You certainly deserve a lot more subscribers. Nowadays I browse math/physics video lectures on UA-cam just for fun and see if it can fend off possible cognitive decline in the future.
Thanks for your kind and encouraging words. My goal in doing the series was to present the material in an orderly and straight forward way, as I wish someone had done it for me. Feedback like yours leads me to believe I have succeeded.
BTW the series is gaining momentum. It is now drawing 2-3 new subscriptions and 10-20 hours viewing time every day. This is actually more than I expected to begin with. Who knew there was that much interest in such a specialized subject.
Thank you for creating this truly excellent series! I have acquired the book you recommended, downloaded all of the material you so generously provided, and will work my way through your entire course. I am completely self taught, calculus, linear algebra, etc., and in the process discovered I really love math, so I am really looking forward to this!
This is pure gold!!! A mesmerising series on an advanced subject presented clearly, patiently, systematically and in detail. I waited until well into video 66 before writing my comments. This series takes away all the mystique behind tensor calculus. I am very confident that when I start studying GR, mathematics will not be an obstacle to my learning. The teacher has my heartfelt gratitude for presenting these videos.
Brilliant series of lectures. By far the best on the Internet, although the series Math the Beautiful is already excellent.
My only suggestion is to have a few more concrete examples such as outer multiplication and the position of the indeces (it's easy to get messed up as that product goes where) and tensor decomposition - most people understand a vector and the metric tensor but having a good grasp of what higher order tensors are is more complicated. The tensor decomposition should provide some insight.
But A BIG THANK YOU FOR YOUR TENSOR SERIES. The pace is great and the visuals are alway clear.
My hat to you and thanks for this wonderful exposition of Tensor Calculus : you managed to make one of the most difficult topic of Calculus as clear and easy as possible... Along with a rare didactic quality, as opposed to many video courses and books where authors "loose" the student via an indigestible symbolism.
Thanks for your kind words. My goal with these videos was to present the material the way I wish someone had taught it to me.
That is a very fine Tensor course. Thanks for posting Prof. and congratulations for the good job.
Thank you. I loved putting it together. I always welcome constructive criticism and comments.
@@tensorcalculus822 great!!
I'm in the process of applying for a Plasma physics internship at the the UKAEA, and the recommended reading is about curvilinear coordinates for modelling magnetic fields. This video is great for mulling over the mathematical notation that I'm wrestling with
Thank you very much for these wonderful videos.
I want to know what program you are working with
Thanks for your feedback.
I use several programs for my publications.
For the drawing board, I use free software called FireAlpaca in conjunction with the XP-Pen drawing board.
For demos, I use GeoGebra, a free web-based tool.
For video recordings, I use ScreenPal, formerly Screencast-O-Matic. There is a free version, but I pay a subscription fee for extra features.
@@tensorcalculus822 Thank you very much. I follow your channel with great interest
Moving those triangles around isn't really a proof. Also, Newton's _Principia_ was done in geometry.... without using a coordinate system. He was doing calculus with geometry. I think only two people on the planet even understand the mathematics as it was actually written in his published book. Pure geometry is awesome.
I love this serious you made, it helped me a lot
Thank you!
From which country you are sir ??
The United States. I live in the state of Virginia.
Surprised you left out Hilbert.
David Hilbert was a great mathematician. He certainly understood and used tensor calculus in his work with general relativity, but I am unaware of any contribution he actually made to tensor calculus itself. If he did, i would love to know about it.
@@tensorcalculus822 Perhaps not directly to Tensor Calculus so I stand corrected. However, Grossman helped Einstein with setting up the mathematics of GR. But Grossman walked away. Einstein than struggled for 3 years.
Einstein then met Hilbert and he was enthralled that someone understood what he was trying to do (historical letter). Hilbert solved the issue and sent it in a letter to Einstein who then held a conference to present his new GR. In the mean while Hilbert had published the solution. Einstein was angry that Hilbert was trying to take credit for his work. I'm not sure if the two really patched up their differences. I'm skeptical that the field of Tensors would be as important to physics as it is today had it not been for Hilbert, Grossman, and others. Einstein was a great physicist but he got a lot of help doing the math.
If you see this: what software package do you use for graphing. And if I haven't said it, thank you *so* much for this series.
P.S.
I'm not a physicist or a mathematician, but more history of science. Still I deeply love the subjects themselves as well.
Thanks for your feedback. I concur with almost everything you said. David Hilbert did a lot of the heavy lifting where the math was concerned. Indeed, there were only a few people at the time who had the skills to do so. It is my understanding that Einstein was concerned that Hilbert might get credit for his work, but Hilbert was gracious enough to step aside and give Einstein full credit. He acknowledged that it was Einstein's intuitive ideas that gave rise to the subject.
It's also worth noting that Einstein was never really comfortable with the geometric model of relativity. Indeed he had some pretty unkind words for Hermann Minkowski who first proposed the four-dimensional hyperbolic geometry of Minkowski space. If I recall correctly he said something like "The old man has restructured my work into a form where now even I don't understand it."
It was Grossman who convinced him that this was exactly what he needed to make headway for his general theory. Even though he went with the idea, he was never really comfortable doing so. He was always quick to point out that relativity is about motion and forces, not about geometry.
The software I use for the demos is a web-based tool called GeoGebra. It is free and publicly available at this site.
www.geogebra.org/3d?lang=en
Looks like the shapes when I enter my password 4033
love