Conceptualizing the Christoffel Symbols: An Adventure in Curvilinear Coordinates

[@dialectphilosophy]

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.

[@DanielKRui]

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.

[@dialectphilosophy]

Thanks for watching and for the kind review!

[@-danR]

I've watched most of the dialekt videos. This is the first one that left me stumped within the first 2 minutes. I'm baffled by the kudos the comments seem to universally bestow. For "anthropomorphization" I think it would be better called zoomorphization, but in any case I find it a genuine distraction

[@ethandrood]

All stories are about people. That is true.

[@jakobr_]

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”

[@thenoobalmighty8790]

Yeah hilarious

[@RARa12812]

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

[@theastuteangler]

AI did most of it

[@HaramGuys]

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

[@dialectphilosophy]

Thank you! 😊

[@alhasibsifat6903]

Riemannian Geometry in grad school?

[@gaemr_o5147]

@@veil6666I learned it in 2nd!

[@radonkule1564]

it follows that i knew it when i wasn't even born into this coordinate space

[@jmcsquared18]

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.

[@ScienceClicEN]

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

[@dialectphilosophy]

beaucoup de nuits blanches, mon ami...

[@The-Martian73]

Je t'aime dude@@dialectphilosophy

[@JuBerryLive]

@@dialectphilosophy :O T'es francophone? Québec?

[@atismoke]

No way its scienceclic??????????????????????

[@D_0ktor]

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.

[@dialectphilosophy]

Thank you very much! 😊

[@alphadog6970]

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

[@tyedee7552]

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

[@lowersaxon]

Which has 256 ( two hundred fifty six) components. Yes, many of them can canceled, but you still will enjoy the ‚big picture‘. 😂

[@tyedee7552]

@@lowersaxon In 2D it's just 16 components, which is a bit more reasonable.

[@dialectphilosophy]

That's the plan! However, to temper your expectations, the Christoffel symbols and geodesic equation will require at least a couple more videos, so it'll be some time.

[@tyedee7552]

@@dialectphilosophy Sounds good, and as always when watching Dialect, I'm always staying tuned.

[@4pharaoh]

Subscribed. Beautiful job.

[@Lapester45]

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

[@GeoffryGifari]

Impressed by the amount of effort going to the animation

[@BobStock-n8h]

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”…

[@keithdow8327]

A one-form is just an oriented line segment. A two form is an oriented patch. A three form is an oriented volume.

[@BobStock-n8h]

@@keithdow8327 that's very helpful, thanks!

[@oni8337]

@@keithdow8327 I think they're joking

[@keithdow8327]

@@oni8337 Dude, For a physicist, forms end up in integrals where they represent a small patch of something. The orientation of the patch is important. For Gauss' law we care is the integral over the surface is about a vector pointing in or out, for each patch.

[@oni8337]

@@keithdow8327 Gauge Theory Gravity exists and it's written in geometric algebra. Im pretty sure multivectors and k-blades are nothing new to physicists

[@septimir5278]

I am studying differential geometry for GR and your videos, especially with the animations, are invaluable for arriving at an intuitive and clear understanding of these concepts. Many physics and math textbooks offer symbolic or proof explanations that are rather stiff and don't promote the intuition as clearly and easily as your videos. This is a serious contribution to higher math and physics and helps so much. Thank you!

[@surrealphysics]

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! ☺️

[@christopheriman4921]

Hey I was just talking to you earlier, didn't expect to see you here.

[@dialectphilosophy]

At twenty-four minutes, this video definitely tried our patience, surely you can relate 😂 but thank you and looking forward to checking out your podcasts soon!

[@Kralasaurusx]

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!

[@mankritsingh4058]

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⚡

[@MATT-ll2zf]

Dialect is back with a bang, thank you

[@001firebrand]

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

[@ahmadsharif991]

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!

[@golden_smaug]

I came hopeless and found gold, God bless you

[@dylwhs]

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. 🙌🏻🤩

[@dialectphilosophy]

Thank you for watching!

[@gajanantholbare2654]

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

[@6Darkfalt9]

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

[@terminatoratrimoden1319]

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

[@thenoobalmighty8790]

Cartesian bear is a prick

[@chrisronin]

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.

[@dialectphilosophy]

thank you!

[@tonglu3699]

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!

[@dialectphilosophy]

That's very encouraging to hear :-) Thanks for your support, and for binge-watching 🤪

[@boriskogan666]

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.

[@liamhagan4434]

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

[@arushibali7784]

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 KZbin ❤ extreme respect for you man ❤️❤️

[@uwuifyingransomware]

I just finished a course in differential geometry and was frustrated because it’s such an awesome subject but the course moved so fast that I couldn’t really understand what was actually being done. Your videos are incredible! I feel like I’m finally understanding the things I have learnt. Thank you for your hard work :)

[@xbrlinfinity9399]

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.

[@dialectphilosophy]

Thank you for sharing 😌

[@rickywong9560]

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

[@David-c5u6r]

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

[@marenagbg]

Your videos are seriously top tier

[@elijahberegovsky8957]

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

[@philipm3173]

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

[@dialectphilosophy]

😅

[@gnagyusa]

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

[@watsisname]

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

[@tahmidurjahan8204]

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

[@riemann4151]

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

[@mastershooter64]

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

[@angelsachse9610]

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

[@hollmandanielquinterosalaz2900]

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

[@-_Nuke_-]

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 :)

[@u.v.s.5583]

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

[@Michigan_Wolverine]

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

[@scienc-ification2539]

great video! good job! Honestly, this platform is the best!

[@angrygary91298]

Completely astonishing! Thats new level content.

[@Voshchronos]

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

[@sriramfavouritesongs32]

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

[@VuongNguyen-mq7po]

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

[@menglongyouk167]

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

[@maccarrena]

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 :)

[@dialectphilosophy]

Thank you for watching 😌

[@owen7185]

Where have you been this whole time? The best videos

[@tanchienhao]

This is such good intuition for metric compatibility! Thanks

[@Dalroc]

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

[@HinanawiFurina]

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

[@Darthvanger]

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 KZbin.

[@jayp9158]

Wow, amazing video. I'm not amazingly versed at math and despite of that I understood this! You have talent

[@catwiener22]

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

[@dialectphilosophy]

That's awesome, we've probably read it...

[@Killerkraft975]

2 minutes in and the humour for the great knowledge is too great

[@darrennew8211]

Fantastic animations. It really helps the description.

[@StovallX]

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

[@Sho-ryu-kame]

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

[@hasanaljamea2569]

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 😅)

[@dgriffon]

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

[@tronconesgym]

great stuff... great ending to this round!

[@dialectphilosophy]

Thank you and stay tuned!

[@---Lola---]

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 🤗🥰

[@PhysicsWithoutMagic]

Yup! Though, this particular video seems kind of useless so far (I’m at minute 16)

[@greenappleisspicy]

@@PhysicsWithoutMagic This is useful for General Relativity and calculating geodesic in spacetime

[@---Lola---]

@@PhysicsWithoutMagic This video is far from useless to those who have just learned something new. ;)

[@PhysicsWithoutMagic]

@@greenappleisspicy I’m pretty sure everyone who needs to know how to do that could already do that, no?

[@PhysicsWithoutMagic]

@@---Lola--- to what use will you put what you’ve learned, if any?

[@MP-tp3nt]

You are the masters of visualisation!

[@tezlaf734]

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

[@Technoyote]

Wtf the production quality on this is insane

[@lucmccutcheon6703]

This is amazing thanks for making it so engaging

[@DevRajyaguru-lx8pi]

Oh my god, it's really beautiful explanation. I already liked the video in first half minute because of the idea of polar bear

[@pAq-vv3pf]

Your bear intuition was so humorous thank you😂

[@iiiiii-w8h]

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

[@danielcimarosti7579]

Nice ending at 23:11, it actually gave me the chills for a sec..

[@mehdimabed4125]

I love your videos ! The content and the art style are just perfect !! For this one I have 2 questions : 1) It is said that polar bear must change its interpretation of its coordinate system to better match reality : how do we know that cartesian bear lives in reality ? Does cartesian coordinate system play a special role ? 2) I never fully understood parallel transport... How do we define "the same orientation" precisely ? A parallel transport in polar coordinate will not translate in parallel transport in cartesian coordinate so how's the judge about the preservation of orientation ? Thanks again ! :)

[@OneLine122]

1) I think it's the cliffhanger. You think all the time you are cartesian bear, but in reality are polar bear. The special role is to be able to tell the shortest path. Just like the plane example. If you just draw a line on the cartesian map, you would not get the shortest path. If you just draw a line on polar bear map, you won't get the shortest path. 2) You calculate a 90 degree angle from the starting point and keep it that way. It works in cartesian space and is how Newton mechanics work. That's what he showed here, it's not from polar bear space. I suppose there would be a way but it's not shown.

[@dialectphilosophy]

Thanks for watching! To best address your questions: 1) In the context of General Relativity, there is never a coordinate system we can construct which will correspond to the "real" picture -- this is because, just like there is no 2D map we can draw of the earth which will not distort areas and angles of the earth's surface, there is no flat coordinate system we can choose which will not distort the true areas and angles of the spacetime manifold. Our "Meaning of the Metric Tensor" and "Spacetime Metric" video address this topic more completely. Now however, we can also apply this thinking to the Special Relativity case as well. In Special Relativity, the spacetime manifold is NOT curved, yet this does not necessarily mean we have drawn up a correct mapping of it. One then is compelled to ask, just as you have done yourself, how does Polar Bear or Cartesian Bear know that they live in reality? More on that coming in future videos... 2) The definition of parallel transport mathematically is a little more involved, and we plan on tackling it in more depth when we go to curved surfaces. Here in this video, the "same orientation" was defined in a global sense, relative to Cartesian space. Once you move to curved surfaces, these vectors begin to live in so-called "tangent spaces". The tangent spaces still lives in the real world though, being tangent to the manifold, so parallel-transport always refers to a process that is carried out in the "real-world".

[@robertcairone3619]

I was hoping to see these points addressed. Glad I didn't make any snarky comments about the colors of the pills first. They're both in 'a' matrix. 'The' matrix is just what you call the other one.@@dialectphilosophy

[@mehdimabed4125]

@@dialectphilosophy Thanks for the answer ! And all the work ! :) I'm very curious of the more rigorous definition of parallel transport, because it's one of the thing that is at the core of my non-understanding of general relativity I think (of curved space in general), so I'll stay tuned ;)

[@cauemorenokersuldecastroca2917]

As a visual thinker, this finally allowed me to understand Christofel symbola

[@kxs7267]

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

[@misterlau5246]

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!!!! 👋👋👋👋👋

[@at7388]

This was the most sophisticated polar bear I ever encountered.

[@alejrandom6592]

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

[@yagvtt]

Fantastic as always, thanks so much for those videos !

[@dialectphilosophy]

Thank you for watching 😌

[@David-c5u6r]

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

[@iamtraditi4075]

Holy cow this is a good watch. Thanks :)

[@manuelpena3988]

your videos before were already priceless. But the amount of work in this one is impressive

[@al7aroDos]

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!

[@alessandrotatullo5220]

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

[@Goku17yen]

crazy high quality! great job

[@jwalker216]

Cool. I had done stuff with polar coords in college, but either forgotten this or never was taught it. No tensors before I dropped my math.

[@RicardoSanchez-zf4km]

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

[@sashaguzman7792]

commenting because I think this is amazing !

[@ankitthealchemist]

Wow just amazing!! Nice work man!!

[@abhijeetsunny7629]

Keep giving me knowledge ❤

[@smbogan]

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

[@ВикторияКунчева-б6м]

Very useful video!❤

[@ImpShimadon]

After 25 minutes, you completely destroyed me when saying: we haven't scratched the surface of ...😂

[@FOCUSEDJUSTFOCUSED]

AMAZING EXPLANATION 👏🏻

[@hqs9585]

Great video, thanks.