Tensor Calculus For Physics Ep. 12: Christoffel Symbols

[@AndrewDotsonvideos]

*Typo*: The second term containing the connection coefficients in the definition of the covariant derivative of a second rank tensor has a typo. In general, the rho mu should be a mu rho. What I wrote holds only for symmetric second rank tensors. Since we took covariant derivatives of the metric which is symmetric, nothing was affected in the derivation itself.

[@jamesgreenwood6997]

How do you evaluate the christoffel symbols in terms of the metric tensor coefficients if your manifold has torsion?

[@abrarfaiyaz6503]

next tensor video when?

[@xMudball12x]

@@jamesgreenwood6997 You must lose metric compatibility, since the connection coefficients are no longer symmetric about the bottom two indices, so simply write down every term on the board every time you take a covariant derivative of a second rank tensor.

[@Starshine1399]

You know COVID-19 is bad when all there is to do is going back to Tensor Calculus.

[@OtiumAbscondita]

COVID-19 more like COVariant-19

[@Starshine1399]

@@OtiumAbscondita Oh my god... 😂

[@vencicekov2864]

@@OtiumAbscondita lol

[@robertsteinbeiss8478]

oh no, don't do that, live is not just relativ

[@howtobeadebaser]

I literally and honestly got excited when I saw the notification. Now you have a blackboard, you have progressed in the physics field: time to learn Walter Lewin's dotted lines! You go, Andrew!

[@robinsonrajan8564]

I love how he just talks like we know everything he is talking about. Good job, Andrew😂

[@TheGDiaX]

He clearly says "Smart People" in the beginning of the video.

[@maureendotson4634]

I don’t think I’ve ever seen you work on a blackboard. Thanks for getting my account back! ~Mom

[@cylosgarage]

Legend

[@schrodingerscat7218]

At 14:12 you said "non-zero Christoffel symbols does not mean your space is curved". Finally, I see how to generate the Euler and centifugal "fictitious" forces. Your videos are gold!

[@Diego_Alcantara]

Engineer: *Now tell me, how does all of this helps me on my future*

[@UVB4U]

you are an engineer... so it does not :)

[@khageshtanwar5089]

your future is flat! so it won't.

[@Kraflyn]

are you engineering satellite stuff? Then it does! :D

[@soccerbels7947]

Fk i feel offended my future should be engineering :(

[@wkkong88]

My classmate: are you playing hangman?

[@RC32Smiths01]

Excited for this one man! Also please stay safe and sound from the virus my man, wish the best for ye!

[@katsojohnobotsang2484]

Guys make sure you really understand this, you're gonna need it to understand my General Theory of Relativity & to derive my 10 field equations.

[@Kraflyn]

it's actually nordstrom's theory... :D

[@nishronw9549]

yes, that is right; but can you give me a source link that gives the actual derivation of the field equations.

[@bondmode]

GR exam tomorrow and you the only one person to make a clear distinction between connection coefficients and the Christoffel symbols, thanks man

[@festusmaximus4111]

I've been learning tensor calculus in preparation for learning about special relativity and radiation at university this year. These videos help to consolidate the knowledge and practice that I did maybe two days ago, and help to clarify pieces left out by my textbook.

[@cristianfreddie]

Andrew- " you know what I'm saying " ... Me who studies intellectual property law and Computer engineering - "yes, I was never lost" I love the videos!! even if I understand 10% of the video ! Keep it up !!!

[@kalyani6466]

The series was so awesome that I actually binge-watched it, and eventually learnt a great deal of tensor calculus I've been struggling with! I just cant wait for the next videos to come up!!

[@prateekmishra2070]

I'm learing tensors through your videos. Please don't stop this series it has been very useful. Thaks a lot man. Keep up the good work buddy.

[@calebdelage3025]

Am I the only one who has no idea what he is talking about, but still loves listening and supporting him?

[@chedidkamal837]

I am a simple man, I see the first equation, I quit but leave the video playing so I can support Andrew

[@AndrewDotsonvideos]

Chedid Kamal :)

[@shayanmoosavi9139]

LOL😂😂

[@D_poppins]

Me: On youtube KZbin: We recommend tensors Me: What the hell is a tensor? Me, after watching video: What the hell is a tensor?

[@connorbrown5267]

By far my favourite lecture series atm

[@aghosh5447]

Thanks that felt like a introductory GR lecture, in an entertaining way! More of those ASAP!

[@djangogeek]

Been waiting on this so long I had to use Alex Flournoy as a substitute! Dr. Dotson is finally back giving lecture :D

[@AndrewDotsonvideos]

His lectures are awesome tho

[@DavidAspden]

Love the ending. It was a very important observation.

[@jeremymarschke2314]

Hyped to see this series continue.... I’ve been struggling to connect all this tensor stuff back to the intuition I developed for cartesian/euclidean coordinates/spaces. Great presentation through the whole series. Suddenly GR self study seems much less insurmountable. Please keep them coming!

[@julianelesnioski27]

i love your videos, please release more of then, you are helping me a lot with my master degree.

[@anupkodlekere3633]

Oh so this is an actual lecture!? I thought this was one of those professor jokes blabbering some shit

[@gustavgadehebsgaard5727]

I have waited for this for so long!

[@rowanmakesfilms]

My favourite tensor is the StReSs TeNsOr!

[@MrCaptainFrosty]

I understood nothing but your enthusiasm made it fun!

[@zae9359]

Good video to watch during my (extended) break! ALSO I recently got accepted to the SULI internship at JLab in nuclear physics! I mentioned you and your videos in a couple essays about why I wanted to work there so thanks for the boost :)

[@AndrewDotsonvideos]

Congrats! I'll be there this summer assuming the 'rona virus doesn't cancel everything.

[@zae9359]

Andrew Dotson that’s awesome! Maybe I’ll see you there! Best of luck with your work over break!

[@thetheoreticalnerd7662]

I love watching these videos!!!

[@abhinovenagarajan.s7237]

I have been wanting to learn tensors for a long time. Recently I started my masters and I was glad I'd be getting to learn tensors. Lmao cut to my prof starting and finishing the material on tensor calc in a week and I spent the last two days binging this playlist a few times. I just finished a test and I hope I pass. I'm now convinced physics is basically masochism and I'm going to get myself some Tensor Boi merch from Papa Andrew. PS. In all seriousness thank you so much for this playlist! I really liked the motivations behind your derivations (esp this video).

[@damienthorne861]

The covariant derivative also involves taking the derivative of the basis vectors as you're in an arbitrary coordinate system, But it uses Christoffel symbols which are the components of the derivative of the basis vectors with respect to a particular coordinate direction and the covariant derivative will be 0 if a vector field or vector is propagated parallel to itself. It represents the difference between a vector that is propagated parallel to itself and one that isn't. And I believe also that a covector uses a minus sign for the christoffel symbol.

[@intuition9107]

I really enjoy your videos...excellent teacher

[@doodelay]

Hey Andrew I have a channel suggestion for you. Videos like this one where you explain high level physics to us as if we're PhD students is a good thing to have online, but is frankly impenetrable for most of the viewers. We have NOO idea what you're saying given the jargon and absence of explanation. However one thing that's lacking on youtube is physics teachers like yourself who are NOT AFRAID to dive into the maths of a physics problem. Physics youtube doesn't have a culture of solving random, but challenging physics problems like maths youtube does (Take Blackpenredpen or Flammable Maths for example). Physics youtube instead prioritizes telling us about big mysteries or the heroes of the past, it doesn't TEACH physics or solve interesting problems like "how fast would the sun cool if you submerged it in an infinite fresh water ocean?" Even International Physics Olympiad questions would be interesting to watch you work through. In short, I think you can pioneer these sorts of physics videos on youtube, and you can do it without dumbing your channel down to solving simple rope and pulley questions or staying surface level.

[@aryandhir7286]

One more great video 🤘🤩.... this is probably the best channel on youtube for me

[@jaimes5716]

I absolutly did not understand this

[@kiwi4731]

Me neither but I love his way of teaching

[@sansamman4619]

When i don't understand a video, i feel guilty because he always calls us smart people xd

[@HeyEveryoneHi]

@@sansamman4619 lol

[@Minuey]

I hate this about these kinds of videos, but I always tell myself I would understand it if I studied the terminology a little more and, of course, the content itself

[@sansamman4619]

@@Minuey These videos can be quite fun if you understand them. But the problem is that many people don't specialise in these topics so it's for a small audience, There are times when I have been a part of that small audience and i can tell you that they can be pretty fun! And you are right, it doesn't need a lot of effort to understand these kinds of videos, if you have a good background on calculus you could learn everything in this video in like a week, and it will be a good skill to just understand them, you don't know when you will need them. That's if you have free time however..

[@shanecoyle3676]

So without watching any tensor calc vids and being a first year in college I wanted to see how long until i got lost. Turns out about 5 seconds feelsbad. I tried. Keep up any content tho man I personally cant enjoy this but im sure many people do.

[@shanecoyle3676]

Oh he rehearted my comment very cool I edited it because I noticed a spelling mistake. Thanks andrew!

[@Blox117]

its just a bunch of greek symbols and shit

[@japalocoturbo]

Well didn't have time to watch this tensor calculus videos. But my university just cancelled classes because of COVID. Let's go.

[@SMILYHIPPO]

A very short way is to write the KE of a free particle, with a general metric g, as a lagrangian and find the EOM. These EOM will give you a geodesic equation and Christoffels symbols comes out naturally in this procedure. Only thing I have found bit tricky is that one has to adjust the indices to the wanted answer.

[@billway1227]

It's really *USEFUL* . Thank for sharing knowledge! Thug some concepts are still quite like nightmare for me. I do need to review all the notes taken from your video before. Thank you again! :)

[@possiblepilotdeviation5791]

Named after Dr. Chris Toffel of Norway who first studied these types of connections in the 1890s while working at University of Oslo.

[@_Nibi]

k

[@possiblepilotdeviation5791]

@@_Nibi K

[@frede1905]

Norway?? That is my country!!! 😀

[@possiblepilotdeviation5791]

@@frede1905 🎶🇳🇴🎶🇳🇴🎶🇳🇴🎶

[@frede1905]

@@possiblepilotdeviation5791 Are you a Norwegian too??

[@aryanbhatt5835]

Thanks for this whole series! You've been a great help! Quick question though: where'd you learn Tensor Calculus? Was it in a specific class or just on your own from the book?

[@deepanshubaisoya8232]

Hi Andrew thanks for the superb videos on intro to advanced topics of Tensor Calculus, i was wondering will you cover all the topics of "Tensor Calculus for Physics" in this video series

[@michaelmotorcycle1789]

Thanks again man. I CAN NOT WAIT till the nextepisode!!! 🤙explained crystal!!

[@michaelmotorcycle1789]

Been waiting my whole life to see this!!! 😆

[@HimalBudhathoki92]

Now i understand clearly. Sure it helps me in my maematical physics exam thanks man. Please make videos on Einstein tensor

[@pikminlord343]

Excellent video

[@taw3e8]

lol im stuck now on something like 9th one gonna watch this in few days probably but i know its awesome

[@tomkerruish2982]

Taking a covariant derivative is easy. Take the partial derivative, then place a tittle above the comma (transforming it into a semicolon), and voilà! Seriously, another way to derive the Christoffel symbols is using the calculus of variations to find the equation of a geodesic. It's been a long time since I worked through it, and it'd be hard to describe it in ASCII anyway, but perhaps you could do a future video on it.

[@Supercatzs]

Get a load of THIS guy

[@uditlal2588]

Hmmm yes I totally understand this while scraping a B+ in calc 2... yes of course. Hmmmmm

[@subscribetopewdiepie8746]

Really? Because this is sophomore second semester class and sometimes third year class at least in Mechanical engineering.

[@uditlal2588]

Subscribe to pewdiepie I’m a senior in high school who is planning on majoring in Econ so I won’t be doing any tensor calculus.

[@rewind4964]

Calc 2 is not hs, calc bc is still calculus 1

[@uditlal2588]

Rewind really? I think if I get a 3+ on the AP then I get credit for 1 and 2

[@uditlal2588]

Rewind well AP says I’m right, my friends who took the class and got 4s and 5s to skip college calc say I’m right, and google says I’m right...

[@nev8413]

You're the best, keep it up.✊👌

[@Jeevanmn]

Hey Andrew, could you make a video where you explain how you fund your PhD- On the tution, scholarships and daily expenses, etc?

[@thevegg3275]

Re the Christoffel symbols related to a curvilinear csys, the superscript on Gamma stands for the components of the displacement vector, whose displacement is a comparison of the original vector (prior to transport) to the vector's new position (after transport). Are these components wrt the A) curvilinear csys or B) wrt to an orthographic csys? I ask because its an easy task to find the components of a vector on an orthographic csys since there are many tick marks. But on most graphic representations of curvilinear csyss, it's just four curves (not broken down into smaller units such that one could measure the change in the displacement vect wrt to said curvilinear csys. To me, that means that the breakdown of a displacement vector must be graphically calculated using an orthograhic cys superimposede under the curvilinear one. Are these components wrt the A) curvilinear csys or B) wrt to an orthographic csys?

[@anishmoolchandani]

Just waiting for your next video....

[@Onegod40-v4h]

Please do next videos on tensors....this off time due to covid,I'm really free to read further on physics & so tensors

[@mariohenrico]

Awesome Video, thanks! ⚖︎☯︎

[@nostopit179]

I am in high school, I don’t understand this, I shouldn’t be here, and yet I am. Why? I don’t have the slightest idea, but I’m intrigued

[@harshsharma03]

Dr.Dot be out there hearting every comment lol.

[@diarya5573]

Thank god. My Gen Rel be saveth by thou

[@mrlittlefinger1516]

Dear Andrew, please do it regularly on white board with the tip of marker flattened. Also the voice of yours is dim and I usually repeat the video many times to listen you properly. Please improve your mic. Waiting for next video.

[@jad1910]

Do you sometimes worry that getting a PhD from NMSU might hurt your chances at post-doc or faculty positions at reputable universities in future?

[@AndrewDotsonvideos]

jad1910 nah

[@jad1910]

Andrew Dotson for some reason, that’s a relief

[@aswinibanerjee6261]

Please make a video on eponymous transformations(if you can...)

[@TheNiTeMaR3]

Our electrodynamics teacher mentioned this covariant derivative very briefly.. I now see why there is no affine connection term for regular E&M since we work with the Minkowski metric tensor!

[@Kraflyn]

There is one, when you replace derivative "d" with "d-ieA", where "e" is charge and "A" is the electromagnetic potential vector. The "eA" term is a connection. For this, however, you need Cartan geometry, not Riemann geometry. And it's not a generally accepted point of view. This appears in, say, Dirac equation, and therefore in the entire quantum field theory.

[@TheNiTeMaR3]

Kraflyn Ahhh yeah I’ve seen the covariant derivative in the QED Lagrangian but never realised the gauge field term is a connection term. Why is Cartan geometry not an accepted view? I haven’t personally heard of it before :)

[@Kraflyn]

@@TheNiTeMaR3 Well, Cartan geometry is accepted, but it's not how the General Relativity is done. Cartan does everything on a flat background! This is exactly what quantum physics does as well. The result can be projected on any curved surface, and this way it's fully equivalent to Riemann approach. However, it can do more things that the Riemann geometry, such as, say, torsion. You need some preparation for studying it, though, whilst you need only calculus to attack GR.

[@Kraflyn]

@@TheNiTeMaR3 Oh, i misinterpreted your question, sorry XD The vector potential A has only one tensorial index, since it's a vector, whilst Christoffel symbols have 3 tensorial indices, so it's not at all clear at first how could A stand for a connection. However, this is possible within Cartan formalism, which is not very well spread. There is however a plentitude of literature where the yang-mills formalism is deduced completely just from the Cartan approach. Can google it easily. Cheers! o/

[@TheNiTeMaR3]

Kraflyn I had a look and this stuff is way beyond my current understanding so I’ll take your word for it 😄

[@Kraflyn]

dirac: general theory of relativity is a good book for this. Cartan approach is mind-blowing, though...

[@αηομαλψ]

I love tensors

[@Awesomeale01]

I don't even know what a tensor is... still watched the whole video.

[@alexismedina8907]

My differential geometry professor said that he broke the spine on his textbook when he was in school learning the "christ these are awful" symbols, fun times

[@jowadulkader9006]

thanks..are you learning general relativity lately?

[@GOTHICforLIFE1]

At this point i'm questioning whether we even speak the same language.. I understood legit about half of the words you said xD Guess physics isn't quite my area

[@arbabamin2719]

Why don't you do basic to advance calculus. That would be soooo awesome.

[@TheNachoesuncapo]

YES!

[@davidlenir7517]

Are you still doing online tutoring? I'm looking for someone to help me with QED as well as programming this summer.

[@mps2209]

Tensor calculus is all about fancy lingo....if understood you're good to go....... Attaboy!! ;)

[@giovannicarmona1050]

Chalk boards ❤🤘

[@user-ew2tk8mn2w]

Hey dude, I am 14 yr old and I have finished books like Haliday and Resnick and I want to study GR and Qm . Dude could you tell me which books to start cause I tried to start GR but I couldnt understand the math and the matrices. Could you tell me books or lectures that I could take online?BIg fan man

[@anikethpai9442]

Take care man.

[@NovaWarrior77]

YES. This is what I subscribed to this channel for. I'm just kidding I came for the jokes.

[@rossboyer2764]

Great job!.... at confusing the shit out of an undergrad physics student 😑

[@UVB4U]

why all the permutations are anti-cyclic? it's the same in my text book and I wonder about it...

[@Kraflyn]

it comes from calculus of forms, differential geometry. Forms are anti-symmetric so this transfers over. Metric is a 2-form, for example. It gets seriously more complicated though than the general relativity, since there you don't curve spacetime, instead you project in a flat euclidean plane. Quantum theory works in that spacetime, the flat one, and so connections are different than the ones in general relativity which works with curvature.

[@WetCatGG]

hey I wanted to ask some advice from you all, planning on starting a physics degree this year and thinking maybe I should extand it to 4 years instead of 3 years because I heared its really busy and there is no much time for yourself, I'm thikning about going for masters in physics/CS so I also need total grade of about 85-89 so this also a factor, any thoughts? (btw a year of uni here is 3K usd and I might get a scholarship for the 4th year also, so consider money not a big problem), ty!

[@bondmode]

come on man, can't wait to upgrade to lvl 1 Riemann tensor boi

[@hwangsaessi2335]

Hmm yes I understand some of these words. (I have never studied tensord before, but I get at least some of the language from linear algebra)

[@pancreasman6920]

In the first line you wrote T_rho my , but shouldnt ist be T_my rho ,or is it necessarily a symmetrical tensor?

[@Kraflyn]

no, not necessarily symmetric. Maybe he made an error. Gammas are necessarily symmetric in lower indices though, at least in general theory of relativity.

[@souravdey90024]

Sir is it complete course of tensor for physics?

[@Layefamax]

Hello sir., Wanna be a theoretical physics. Advice please......

[@andromedia9649]

I have a questio, what is this all about? When are you gonna use this kind of math?

[@Kraflyn]

curving spacetime

[@Ohmau33]

Physics, apparently

[@Thegeektoendallgeeks]

ey right when we just learned them

[@somewhatblankpaper1423]

Wait... Your university is not closed or have you filmed this prior to the closure?

[@AndrewDotsonvideos]

I have card access into the building

[@gregoritsen]

In which part of physics are tensors useful to?

[@Kraflyn]

stresses and strains and pressures are all tensors. Then, in all of relativity. And so on, everything relativity related, such as, say quantum electrodynamics. There you also have spinors beside tensors.

[@Eigenbros]

Shameful admission: to this day, I still don't know the difference between contravariant and covariant, but hey that's why im an experimentalist 😅. Also, I don't trust anything with that many indices.

[@mikhailmikhailov8781]

www.dpmms.cam.ac.uk/~wtg10/tensors3.html Read this. Covariance and contravariance are obvious, when one looks at tensors as multilinear transformations. We want to make sure that a tensor acts the same way in arbitrary coordinates, that is that it maps its inputs to the same outputs in any coordinates. For instance, if i have a vector B acting on A via the dot product and a*b=1, i want a*b=1 in any coordinates, to do that we will transform b in an inverse fashion. Voila!

[@Kraflyn]

as mikhail said, one transforms as basis vectors do, the other transforms "contra" to the basis. Let "e" be basis vectors, and let these all be orthogonal one to another and of unit length. You know, like i, j and k. Then, e*e=1 is their scalar product with themselves, e* is the "dual" of e. Vector e is a column vector, and e* is a row vector. Now, switch from e to some other vectors f. Let matrix M take you from e to f as follows: Me=f. Now, N is inverse of M if MN=1, and let e*N=f*. Look at what happens to e*e=1: it becomes f*f=e*NMe=e*e=1. So, you see, there are "duals": e* and e, then N is dual to M -- it's just the inverse matrix of M. So you see, in order to form scalar products, you need "duals". Basic scalar product is e*e=1. When e becomes longer, e* becomes shorter. When e rotates to the right, e* rotates to the left. e* transforms "contra" to the basis vector e. You need both "co" and "contra" to form products! Cheers :D

[@tw5718]

Presumably you are doing this for extra practice? Why does upstairs get + and downstairs get -?

[@AndrewDotsonvideos]

It just works out that way with what you need to add to the partial derivative to cancel the terms that spoil the tensor transformation rule

[@tw5718]

I think you can derive it by using the metric tensor to change the index on a vector. Thanks for the reply.

[@jonathandavis7019]

"you know what I'm talking about" Lol nope, not one bit

[@rohanmathew5728]

Where is part 13?

[@carlosalbertomarquezibarra2392]

I need the next video

[@PapaFlammy69]

Nothing m8

[@someguy9046]

I definitely learned something... I learned that I still hate tensors. But that's not exactly new

[@pianoten]

phd student in theoretical chemistry here. Thought I dont need the first 11 parts. I was horribly wrong edit: I dont need this stuff but I thought I would give it a go. Physics math differs so much from chemistry math....

[@HilbertXVI]

Haha you guys need more QFT than GR so it's completely understandable

[@daniels4209]

ah fits the rule again subscribe will be given.

[@nathanbranson3765]

Do you check the email in your about section still?

[@JesusSavedSolomon]

Cant wait to take physics in grade 11 lmao