Conceptualizing the Christoffel Symbols: An Adventure in Curvilinear Coordinates 

There are two whoopsies at the 21:47 mark -- first the correct Christoffel symbol there is Gamma - r - theta - theta, as we are considering the change in the r-component of the theta basis vector transported in the theta direction. It is NOT Gamma - r - theta - r, as depicted. (Moral of the story: indices are confusing, so make sure to pay extra careful attention to them!) The second whoopsie is a subtler, but much bigger and more important one. When we shift the our path radially to the right in Polar Land, the radial and theta components of the new diagonally-directed basis vector both pick up a 1/r acceleration component. This means we have to shrink the theta component by 2/r in Polar land to continue on the path of the Cartesian Land geodesic. We failed to show and account for these extra Christoffel components coming into play in that scene (as we showed only the -r Christoffel component corrections), and so technically the purple geodesic path depicted in that scene is not the truthful one, though the 1/r components do become more and more negligible the further out from the radius, so the path shown is still a close approximation.

This is truly the quality content we need. Why the hell does this kind of stuff have only a few hundred likes. It should be in millions.

That red pill was dry and painful, but such is the way of the Cartesian Bear. He is a friend nonetheless.

The animations are super well done, and the "anthropomorphization" of different coordinate systems is surprisingly effective at teaching even this complicated Riemannian geometry. Teachers of young people are taught to keep in mind that the human mind responds best to human stories (stories about animals, like bears trapped in a matrix, still count as human stories). I'm glad that you are demonstrating that this principle remains true even for more advanced subjects.

I’m just gonna take a moment to appreciate the humor in this video. When I had to pause for a laugh break once I heard “polar bear” as a counterpart to “cartesian bear”

One of the most visually pleasing illustrations of what metric-compatible connections are that I've ever seen. Imo grad students should be using stuff like this. You're doing a great service by putting videos like this out there.

Riemannian Geometry was one of the hardest subject I studied in grad school. This is an amazing introduction to many important concepts

Amount of work the author has put into this is amazing.

Fantastic video! I don't understand how you produce these so fast and so well!

Staying tuned to see how the Christoffel symbols lead to the Riemann Tensor!

Cartesian Bear and Polar Bear literally kills me. Oh my god. I live for this. My life once again has purpose.

Man watching more of it, you really must have spent weeks on this. Thank you for your service

When I heard "Cartesian Bear" I thought "That sure is a seemingly random choice of animal, but alright" - but then came "Polar Bear" and I was like "AHH I GET IT" 🤣 Well played!

Impressed by the amount of effort going to the animation

How is it possible I've found this channel only now? I was working on a thesis in General relativity and you explain all of the concepts incredibly well. You're doing an excellent work of passing the information to the viewer as well as keeping their attention. The presentation, the information, all of it is just magnificent. As one educator to another, I tip my hat to you. Amazing content.

Dialect is back with a bang, thank you

Very clear and concise. In just two opening sentences you described what others can't in a book. 👏👏👏

This video is INSANELY well animated and explained. I remember struggling to visualise christoffel symbols in college, this would have been a massive boon. Hoping this blows up soon⚡

Best, most thorough (and patient!) explanation of this content!! So great! And also something I very much plan to point to when I get to trying to explain how small/infinitesimal games relate to the Levi-Civita connection, etc. 🙏💗 Thank you! ☺️

This is the best video to visualize the metric tensor ive seen so far

Just outstanding! E.B. Christoffel, G. Ricchi-Curbastro & T. Levi-Civita would be very proud of their enormous legacy! 💖

I’m a PhD physicist and struggled with GR but now thanks to this patient walkthrough I finally grasp Christoffel symbols! Thank you! Now if I could only get an intuition for a “one-form”…

This is the first of your videos I've seen, and I have to say, it is excellent. Thank you for your carefully explained and wonderfully illustrated video. 🙌🏻🤩

This and ScienceClic's excellent series on "The Mathematics of General Relativity" are my favorite intuitive explanations of Christoffel Symbols, Geodesics and the Metric Tensor. Can't wait to watch the next few videos in this series!

One of the easiest explanation of Christoffel symbol I ever had seen in the history of maths or physics............ Piece of mind... 🎉

Wow this was fantastic! Your visuals did a great job making the math feel intuitive. Looking forward to the subsequent videos!

Let's just appreciate what a smart bear Polar Bear is.

This is such good intuition for metric compatibility! Thanks

Other explanation are burried in math without a visual understanding. But your explanation are very intuitive. Many thanks

Where have you been this whole time? The best videos

I discovered your channel last year and only realized yesterday that you've been stepping up the pace of uploading. I stayed up late last night and caught up with six of your last videos - I'm so hooked. Cannot wait to see your next video!

This is incredible work. Both your explanation and animations are so well done and make this challenging topic approachable. Bravo!

Your videos are seriously top tier

only a person that truly and deeply understands a subject can present it with such elegance and clarity. very well done! it was a joy to watch and listen. i walked away with much better understanding of why these symbols exist in the first place. happily subscribed and looking forward to more excellent content.

You are the masters of visualisation!

I can see the love that's been put into this. Truly inspiring! I am learning so much from these videos, the world that we occupy is absolutely beutiful and you are proving this to me in every single video! Keep it up :)

This is damn good. Nothing less expected from dialect. 👌👌👌👌 and polar bear in the matrix is just amazing😂😂

Dude, this had to have taken forever to make. Thanks

Dang finally understand the metric tensor and it was thanks to a bear pun animation. Bravo sir!

By far the greatest explanation I've encountered. Bravo!!!

Great visualization. Thank you. Looking forward to your next installment

the animations are absolute next level wow

Even having already learnt this before, this makes it so much more visual! Kudos!

Never had I thought that a polar bear with an existencial crisis would be the gateway to understanding relativity

Absolutely hilarious and beautiful video! Amazing presentation of fundamental differential geometry!

This is one of the hardest concepts I came across and this is explained in the easiest way ever. Hatts of to you for providing this quality content on RU-vid ❤ extreme respect for you man ❤️❤️

This is the best explanation of the metric tensor I've ever seen.

Holy shit bro your channel is godsend, clear animation and just enough amount of math for me to finally intuitively understand the physics rather than just the tedious algebra

Your educational videos are the best I've ever had the pleasure of learning. I constantly share your vids.

Completely astonishing! Thats new level content.

What a great step by step explanation. Thank you!

This is a fantastic explanation! Thanks for taking the time to explain this so clearly.

cutest video on general relativity 🐻‍❄️🐻 I want to give these bears a non euclidean hug

I'm happy that polar beard could find its way into the real world the same way I finally understood this concept: Thanks tou you!

Awesome video, please continue with your great work!! Many young and early scientist will be very grateful.

This is such high quality content. Thank you for spending so much time putting this together!

this is turning a sphere inside out level content here. and i mean that in full appreciation. the draw is the memeability, but the educational content is legit. here’s hoping this flourishes in the ytp space.

What a wonderful explanation of parallel transport. This work is truly a wonder

When I watch, listen and read this over and over again I get so much new information each time. My emotions are a little of inadequacy on my part, but mainly of amazement and feeling very lucky to have come across this, knowing I could never have understood it otherwise. Such quality, ingenuity and exposition. That's just my long-winded way of expressing my thanks to you and acknowledging your incredible work. I wish I could have thanked other you-tubers for similar reasons, but I write to you in the moment of emotion and particular opportunity.

Fantastic animations. It really helps the description.

Hats off, I would really like this kind of video to be produced more frequently.

yes yes yes back with more

Came for differential geometry, stayed for polar bear having an existential crisis 2:38

This video is amazing. Great explanation coupled with great visualizations.

This is gold for people who are somewhat interested in this stuff

Amazing! :) You make it so simple ❤ And I love the matrix spin :) Patiently waiting for the next videos, I want to see the real world! :) Simply the best and the greatest channel on RU-vid.

commenting because I think this is amazing !

Excelent video as usual. Thank you for your great work.

crazy high quality! great job

I wish this video existed in my 4th year of studying physics

Holy shit. Now THIS is some insane production value. Incredible video.

Wtf the production quality on this is insane

This was the most sophisticated polar bear I ever encountered.

The content quality is great and of course I am really grateful for explaining such a complex topic in a clear way but what really impressed my is the kinda old-school animation style, it fits so well and just makes me want to sit and watch... So relaxing. The cartesian bear and polar bear idea is 10/10 :)

As someone that likes concepts explained geometrically, this was extremely helpful! I wish I had something like this earlier!

Holy shit how have I never foud this channel before. This is golden content right here!

Such great content, thanks for posting ‼️

I ❤Differential geometry

Fantastic as always, thanks so much for those videos !

AMAZING EXPLANATION 👏🏻

I absolutely love the content of this creator. Who else get happy/excited when they see he has uploaded a new video? 🥰 it's like a gift 🤗🥰

Tnank you so much!. It’s really a great work. Please more video like this!! I enjoy and appreciate your job

So beautiful!

it's such a pity that this video didn't make it in time to attend SoME3, it will definitely win a prize!😢

I didn't watch this one before. And the L With a top to bottom rotation for christoffel's.. It's very good so far 👋👋👋👋👋 excellent graphical works of yours 👋👋👋👋 If this is going where it is most likely, with these great animations this will be a great way for ppl to learn!!!! 👋👋👋👋👋

Amazing, thank you.

Holy cow this is a good watch. Thanks :)

Great video, thank you!

This video is amazing!

Well done. I wrote an article over spherical coordinates and Christoffel symbols and it’s on Wikipedia cited to me.

Fantastic, thank you. Making more sense than my dimly remembered uni lectures...

Somehow I needed this video

Great video, thanks.

This is amazing dang

Nice presentation

Hype! Knowing the topics ahead, you will be excited, too!!! I like how he started with the most feared topic, which is the Christoffel Symbols, its entry barrier to many who study Tensor calculus. ( this why some dub them chirs-woful symbols 😅)

simply fantastic

Great video!

Excellent content

great stuff... great ending to this round!