Riemann geometry -- covariant derivative

[@dXoverdteqprogress]

For more details on this subject, you can download the first chapter of my book here: www.researchgate.net/publication/342330200_General_theory_of_relativity_for_undergraduates

[@mnazaal]

If Tarantino made a math lecture series he'd use this intro

[@crehenge2386]

yupp, It's that bad....

Blaxploitation movies all start like that

[@SquirrelASMR]

@@crehenge2386 lmao

[@aesio]

It is not Tarantino's music, It is from Bruce Lee films.

[@GovernmentAcid]

love this! blown away by the fact that someone can so clearly and concisely convey info about such a potentially difficult-to-convey subject as riemann geometry

[@dXoverdteqprogress]

Thank you.

[@rgudduu]

Exactly my feeling. Thanks a lot

[@davidwright8432]

If only there had been the internet in my undergrad days - and grad student! I always knew that my questions had answers. It was finding good guides that was the problem! Many thanks & I look forward to more.

[@romanemul1]

Im undergrad and i like it

[@henrytjernlund]

Where I went the Math library was small and had limited hours. So wish we had this back then.

[@physicsninja76]

Agreed. KZbin would have been a big help. Even a computer more powerful than an Apple IIe would have helped. lol

[@ahsdiecb]

I still remember the index cards and limited availability of books!! I am so happy for the change.

[@David-km2ie]

I live in a good time, plz share your experience and knowledge, we will pass it on.

[@jennariseley3161]

Thank you so much, this is the only place where I have found an actual explanation of the metric tensor g. The wikipedia article has paragraphs and paragraphs describing what it is used for but doesn't actually tell you what it is in a straightforward way!!

[@jacobvandijk6525]

Understanding is a very personal thing, my dear. Try to understand that! You seem to be satisfied with this explanation (and you get a pathetic heart for it, because he is looking for positive attention too), but others do like the wikipedia-page. Merry Christmas :-(

[@Moreoverover]

@@jacobvandijk6525 Wikipedia doesn't not use simple english, it is rigorously academic which makes understanding math inaccessible.

[@jacobvandijk6525]

@@Moreoverover That's right. Most academics (writting about more or less complex subjects at Wikipedia) can't explain things in a simple way. They grew up in their own intellectual environment. Good teaching is a profession on its own!

[@j.vonhogen9650]

@Jacob van Dijk - Why the snarky, condescending comment? The creator of this video did a nice job explaining a topic many people find elusive. I must say, your comment is definitely 'heartless' (pun intended!).

[@jacobvandijk6525]

@@j.vonhogen9650 The creator did a great job, but saying that this is the only place to find a good explanation of the metric tensor (see above) is just ridiculous. There are many places. Her reaction only shows that teaching is very personal! That's what I told her.

[@LydellAaron]

I found this video while listening to Leonard Susskind's lectures on General Relativity (Lecture 3) after realizing I was unfamiliar with Riemann geometry. Your explanations and illustrations were very clear and easy to follow. I had to pause, rewind, re-listen, section by section, so that I could slowly follow your arithmetic but you kept a good pace. Thank you so much for your time and explanation.

[@chasr1843]

WoW - Very Nice This is the first time I ever saw the covariant derivative and gammas derived without explicitly using the metric tensor. Working through this has strengthened my understanding a lot. I look forward to more. Thank You :)

[@dXoverdteqprogress]

I'm glad to hear my video was helpful. Thanks.

[@christophergreeley4150]

As a programmer who learned C++ one ten minute tutorial at a time who was good at explaining things from the fundamentals upwards and didn't wave their hand and say "poof you do X and get Y" this is EXACTLY what I needed in order to learn this. Thank you so much these are some of the best physics tutorials I have ever seen, I am studying general relativity ahead of time as a physics student because I am very interested in it! Thank you so much!

[@dXoverdteqprogress]

Thank you! Comments like this one make me want to continue to make more videos. Good luck with your studies!

[@rockapedra1130]

This is the first time I understood the covariant derivative fully. You make it so obvious. I’ve looked in many places for this and it is always so confused ! And your explanation is obviously correct from the simple identification of the covariant derivate term in the full partial derivative formula. Sheesh! I spent a lot of time on something that is now so simple! Thank you!!!!!!!

[@dXoverdteqprogress]

I'm glad my video was helpful to you. Cheers!

[@manpreetlakhanpal9720]

That intro though😂😂

[@aadika45]

KZbin recommended me this video after 2 years I learned general relativity. And now I have a little idea about covariant derivatives. I wish you were my GR lecturer.

[@biagiodevivo4968]

While taking a tensor analysis course I had a hard time finding these concepts explained concisely. Great video

[@paulhinrichsen8628]

BRILLIANTLY EXPLAINED !! So CLEAR ! Thank you for this beautifully clear explanation.

[@ginosuinoilporcoinvasivo8216]

my god, finally a good explanation, as a high school student who is trying to learn the math of general relativity by himself is really hard to find good material.

[@dXoverdteqprogress]

Thank you. I'm glad my video was helpful.

[@Therfgd]

Thanks a lot, I did really make progress in my math by understanding the covariant derivative with your explanation.

[@atzuras]

That's why the Flat Earth society do exists: they never got a grip on covariant derivatives and made parallel transport wromg all the way.

[@dXoverdteqprogress]

Haha, exellent :D

[@zdenekburian1366]

I don't believe in flat earth but I don't believe in riemann geometry either, it is only a idealistic construction without connection with reality, truth is surely in another, totally different direction, this probablly partially works inside a constrained set of assumptions, imagined ad hoc to tie in precisely to experiments, future scientist will laugh in your face thinking to this gigantic mass of crap

[@dXoverdteqprogress]

@@zdenekburian1366 Pane Buriane, na to co tvrdite treba nejake dokazy.

[@frankdimeglio8216]

@@dXoverdteqprogress Greetings. THE CLEAR, BALANCED, AND TOP DOWN MATHEMATICAL PROOF THAT ELECTROMAGNETISM/energy is gravity, AS E=MC2 IS clearly F=ma (ON BALANCE): The Earth (A PLANET) is a BALANCED MIDDLE DISTANCE manifestation, as the stars AND PLANETS are POINTS in the night sky; AS E=MC2 IS F=ma; AS ELECTROMAGNETISM/energy is gravity. Consider the speed of light (c) !!! TIME dilation ULTIMATELY proves ON BALANCE that E=MC2 IS F=ma, AS ELECTROMAGNETISM/energy is gravity. INDEED, TIME is NECESSARILY possible/potential AND actual IN BALANCE; AS ELECTROMAGNETISM/energy is gravity; AS gravity/acceleration involves BALANCED inertia/INERTIAL RESISTANCE; AS E=MC2 IS F=ma.(Accordingly, the rotation of WHAT IS THE MOON matches it's revolution.) E=MC2 IS clearly F=ma. This NECESSARILY represents, INVOLVES, AND DESCRIBES what is possible/potential AND actual IN BALANCE, AS ELECTROMAGNETISM/energy is gravity. (Gravity IS ELECTROMAGNETISM/energy.) Great !!! Carefully consider what is THE SUN AND what is A POINT in the night sky ON BALANCE. (Very importantly, outer "space" involves full inertia; AND it is fully invisible AND black.) E=MC2 IS F=ma ON BALANCE. The stars AND PLANETS are POINTS in the night sky. Think about the man who IS standing on what is THE EARTH/ground. Think about what is THE EYE. NOW, think about THE EYE that is actually in what is outer "space" comparatively. TIME would be INSTANTANEOUS of necessity, as it would basically stop. Consider the speed of light (c) !!! SO, we can now consider the experience of the man/EYE (along WITH the BALANCED MIDDLE DISTANCE in/of SPACE) who is NOW standing on what is THE EARTH/ground AS WELL. GRAVITATIONAL force/ENERGY IS proportional to (or BALANCED with/as) inertia/INERTIAL RESISTANCE, AS E=MC2 IS F=ma; AS ELECTROMAGNETISM/energy is gravity. Gravity/acceleration involves BALANCED inertia/INERTIAL RESISTANCE, AS ELECTROMAGNETISM/energy is gravity; AS E=MC2 IS F=ma. THE EARTH/ground is thus E=MC2 AND F=ma IN BALANCE, AS ELECTROMAGNETISM/energy is gravity. (The sky is BLUE, AND THE EARTH is ALSO BLUE.) The stars AND PLANETS are POINTS in the night sky. TIME DILATION ULTIMATELY proves ON BALANCE that ELECTROMAGNETISM/energy is gravity, AS E=MC2 IS F=ma; AS ELECTROMAGNETISM/energy is gravity. So, TIME is NECESSARILY possible/potential AND actual IN BALANCE; AS E=MC2 IS F=ma; AS ELECTROMAGNETISM/energy is gravity !!! It ALL CLEARLY makes perfect sense, AS BALANCE AND completeness go hand in hand. THE EYE AND TIME are to be compared on balance to/WITH what are THE SUN, THE EARTH/ground, AND what is A POINT in the night sky !!!! TIME dilation ULTIMATELY proves ON BALANCE that E=MC2 IS F=ma, AS ELECTROMAGNETISM/energy is gravity. (Consider what is THE EYE. Consider what is the MIDDLE DISTANCE in/of SPACE.) TIME is NECESSARILY possible/potential AND actual IN BALANCE, AS E=MC2 IS F=ma, AS ELECTROMAGNETISM/energy is gravity. "Mass"/ENERGY involves BALANCED inertia/INERTIAL RESISTANCE consistent with/as what is BALANCED electromagnetic/gravitational force/ENERGY, AS E=MC2 IS F=ma; AS ELECTROMAGNETISM/energy is gravity. SO, objects AND MEN fall at the SAME RATE (neglecting air resistance, of course); AS E=MC2 IS F=ma; AS ELECTROMAGNETISM/energy is gravity. It ALL CLEARLY makes perfect sense, AND BALANCE and completeness go hand in hand. Consider what is the BALANCED MIDDLE DISTANCE in/of SPACE. Gravity IS ELECTROMAGNETISM/energy. E=MC2 IS F=ma. Consider time ON BALANCE, AS the stars AND PLANETS are POINTS in the night sky. Time DILATION ULTIMATELY proves ON BALANCE that ELECTROMAGNETISM/energy is gravity, AS E=MC2 IS F=ma !!! SO, finally, consider what is the speed of light (c) AS WELL; AS E=MC2 IS CLEARLY F=ma ON BALANCE; AS ELECTROMAGNETISM/energy is gravity. Gravity IS ELECTROMAGNETISM/energy. (Energy has/involves GRAVITY, AND ENERGY has/involves inertia/INERTIAL RESISTANCE.) Again, it all CLEARLY makes perfect sense; AND BALANCE AND completeness go hand in hand. Carefully consider what is THE EARTH/ground AS WELL (on balance). TIME dilation ULTIMATELY proves ON BALANCE that E=MC2 IS F=ma, AS ELECTROMAGNETISM/energy is gravity. Gravity IS ELECTROMAGNETISM/energy. GREAT !!! By Frank DiMeglio

[@captjack5961]

@@frankdimeglio8216 ✨🔥👏👏👁️👄👁️

[@jackwilliams1468]

This explanation of a metric tensor is incredible.

[@jamelbenahmed4788]

Good comment. I also agree

[@dariosilva85]

LOL. This is the greatest intro I have ever seen. Riemann entering the dragon.

[@satyareddy2791]

Thanks for the great video. If you run out of explaining these complicated concepts, please make a video of simpler stuff. I am doing my undergrad in Physics and i am very sure i will be searching for this video in couple of years as i look forward and try to understand General Relativity. GREAT VIDEO!

[@juleskurianmathew1989]

Really nailed it explaining what exactly are the covariant derivative and the christoffel symbol...

[@pinodomenico5520]

Truly excellent video. I watched this twice then I decided to stop it down and make a copy of the notes in full detail - that's when it really started to sink in. I also love your theatrical style of including an entertaining introduction and trailer. Brilliant. ♥ Now a sample problem where we might use the covariant derivative to calculate a result would make an interesting video. Nothing makes math as concrete as an applied problem...with real numbers and values. All the Best !

[@dXoverdteqprogress]

Thank you. You're very kind. I might make a few videos to show some examples in a near future.

[@MissTexZilla]

You just made my life so much easier. Thank you.

[@dXoverdteqprogress]

I'm glad. Cheers.

[@Saki630]

fuck man this is one of the best videos ever made for covariant derivative, linear algebra, and Einstein Summation. WELL DONE. This is better that google+Wikipedia+topology textbooks.

[@YualChiek]

This is really well explained. Thanks for putting this together.

[@Magnetron692]

Hi, thank you very much!! Best wishes from Germany, Ralf

[@tonk6812]

Intro music by brucelee in reimann geomtry...i think the creator is a big fan of lee and mathematics

[@davidprice1875]

This is a terrific description of the Covariant Derivative. Thoughtful, well constructed and easy to understand for a seemingly complex topic.

[@dXoverdteqprogress]

Thanks!

[@rgudduu]

6:35 "The new basis e1' & e2' can be expressed in terms of the old basis e1 & e2 by moving the new plane to the old point". How to move is not clear. There is possibility of some rotation of the plane while transporting it. Is that allowed? Is any random transport okay; just that the resulting components will depend on that?

[@LeonardoRiojaManrique-j5f]

It is a infinitesimal translation without rotation. Remember what a derivative is: it is an infinitesimal translation, and then we see how much the values changed in the normal and parallel directions. Even for a good old derivative of a single variable function y=f(x).

[@bahtree2385]

Thanks so much for the clear intuition of the covariant derivative as the components of a directional derivative that a person intrinsic to the space will actually réalisé is happening! Ive been researching tensor calc and I always wondered why you subtract or don’t include the normal component of the cov-derivative, now this makes so much more sense - it’s because the normal component isn’t noticeable to people living inside the space, so therefore it isn’t much use and doesn’t make sense when you get to higher dimensional geometries. (especially in 4D Minkowski spacetime - how are you supposed to imagine looking at that from an extrinsic perspective? You aren’t, just stick to intrinsic :)

[@Considerationhhh]

thank you very much for the clean and concise explanation!

[@g3452sgp]

At around 8:28, You presented graphical explanations on the mechanism of christoffel symbol change. Here you explained under k=1 ( which is x1 direction) case only. I want to see k=2 and k=3 cases as well, because space is 3 dimensional. This way we can figure out how each component of the christoffel symbol function on each case.

[@morgengabe1]

k is an index, indices parameterize elements of sets/spaces. The point was that adding k, introduced a third dimension as i and j both single dimensions, and comprise a 2 dimensional space when combined. The diagram is of a 3D unit tensor, so all parameters should be set to 1.

[@jaeimp]

Excellent material. I just wanted to note that after 04:27 the indices of the one-forms change from subscript to superscript in ds. I wonder if a note regarding the adoption of supra-indexation for covectors would be warranted.

[@dXoverdteqprogress]

Oh, it's not an index, it is ds squared. I should have made that more explicit. Thanks.

[@dXoverdteqprogress]

Oh, wait, I guess you meant the right hand side.

[@jaeimp]

dXoverdteqprogress Yes, the RHS, but I noticed that you explain it farther down the line.

[@alexiacorradini8472]

Thank you for the explanation, great video! Loved the intro as well :)

[@dXoverdteqprogress]

Thank you.

[@jacobvandijk6525]

@ 3:00 THERE IS AN INFINITE AMOUNT OF BASISVECTOR-PAIRS DEFINING A TANGENT-PLANE, AT EVERYPOINT OF THE SURFACE (OR MANIFOLD)!

[@NeilGirdhar]

Nice presentation. Thank you.

[@simon3863]

I think there is a typo at 4:32. Shouldn’t dx1 and dx2 be written with subscript? Edit: oh 5:58

[@kentdavidge6573]

It's satisfactory to see that other people are achieving results using the same notation and reasoning as I thought. Unfortunately my professor told me that representing the covariant derivative that way is not correct, i.e., you can't take the ordinary derivative of the basis vectors like was done in the video.

[@dXoverdteqprogress]

Your professor was wrong, obviously. In math, you can do whatever you want as long as it is logically consistent and useful. It's true that the definition of the covariant derivative in this video is not the most general, but if it get's you where you need to go, e. g. general relativity, than who cares?

[@Mouse-qm8wn]

Thanks for the great video ❤ Question: Why using Taylor expansion?

[@STohme]

Very nice and instructive video. Many thanks.

[@gguevaramu]

Where can we buy your whole book?

[@dXoverdteqprogress]

Thank you for the interest. I have not written the whole book yet. I intend to finish it by the middle of next year and then try to publish it.

[@archishmore2654]

thanx for such a good video ...could u make videos on ricchi tensor ,stress tensor

[@dXoverdteqprogress]

A video on the Ricci tensor is coming up in the next few days.

[@milansekularac6196]

Very good video lesson. Thanks

[@guillermoaguilar5738]

The metric tensor in a flat space it can be represented as a delta of kronecker?

[@dXoverdteqprogress]

Only in cartesian coordinates. The metric takes into account not only curvature of space, but also nonlinear coordinates.

[@cwc1440]

Hi, I like the video as it explains a lot clearly. I just have one question I hope to get some clarifications related to the normal vector ni. Could you explain what is being summed over for the index i of the normal vector? Isn’t there only one normal vector locally on the surface? Thanks.

[@dXoverdteqprogress]

Thank you for the comment. I started out the video with a two-dimensional surface embedded in three dimensions, but the result I presented at the end is a generalization to any number of dimensions. I should have mentioned this in the video. Higher dimensional surfaces can have multiple normal vectors, so the summation is over all of them. For 2-d surfaces, as you said, there is only one.

[@cwc1440]

Thank you very much for the quick reply that clears everything up for me. Appreciate the videos and the explanations. Looking forward to more great videos from you. Cheers.

[@Paulo_Dirac]

That intro had me hooked up

[@RARa12812]

where do you explain the derivative of basis vector and get cristoff symbol on right

[@dXoverdteqprogress]

I do that in this video: kzbin.info/www/bejne/iourqX1tbNSpsKc&ab_channel=dXoverdteqprogress

[@massimoa2361]

Pure science . Great

[@thevegg3275]

Quick question please. The dashed line for Y2 is parallel to Y1 basis. The dashed line for Y1 is parallel to Y2 basis. Why is the dashed line for the point on the surface to the Y3 basis not parallel to the plane formed by the Y1 and Y2 bases?

[@dXoverdteqprogress]

It is parallel in fact. You can see this by bringing the dashed line for Y3 down to the plain -- it will coincide with the solid diagonal line in the plane.

[@thevegg3275]

Thank you!

[@anugrahmathewprasad172]

Just a small question bro.. Towards the end you said that if an entity is living on the surface and knows no other dimension, the component of the ordinary derivative he calculates is the covariant derivative. I feel that the entity must have knowledge of the 3rd dimension if he wants to take projections of new coordinates on old coordinates and calculate gamma... Am I missing something? Lovely video btw

[@dXoverdteqprogress]

I see your point. They wouldn't know about the projection. It is us, beings from the three-dimensional world that make the projection. The idea is that once we give the entities living on the 2-d surface the mathematics of intrinsic geometry, they don't have/need to worry about any possible extra dimensions. Of course, they could figure out the math themselves by imagining higher dimensions, much like we did with general relativity. Does this make sense. Cheers.

[@anugrahmathewprasad172]

@@dXoverdteqprogress sort of.. Is there a book which I can refer to, to read about this in detail?

[@dXoverdteqprogress]

You can try a book by Arfken. If you type into google "arfken mathematical methods for physicists pdf" you will get the pdf of the whole book for free. It has a chapter on tensors that's pretty good.

[@anugrahmathewprasad172]

@@dXoverdteqprogress yeah I've read that book a bit.. which edition are you suggesting, the older one or the newer?

[@dXoverdteqprogress]

I wish I could be more helpful but I don't know. I don't actually own a copy; I just remember it being helpful in grad school (that was probably the older edition). The one available online is the sixth edition and it seems pretty good. Good luck.

[@banerjeepradip7]

fantastic explanation!!

[@rogerdodger8415]

I remember learning this in high school, and it's been fond memories since. I think it was mathematical methods for physicists, by Afer, Afrun, or something like that.

[@rockrock1908]

High school? Covariant derivative in high school?

[@rogerdodger8415]

@@rockrock1908 Indeed. A private school in Austria, where my dad worked at the time.

[@rockrock1908]

@@rogerdodger8415 Wow, the school must have been very pricy, or for the geniuses... I dont know, this is too much for an ordinary high school. I mean this is multivariable calculus....

[@rogerdodger8415]

@@rockrock1908 It's a prep school for Andrew Wiles. Our schools here in the USA are a disaster. We even had a few Asians that were pre-teens! Here in the USA, it's not about how smart the student is, but rather a reflection on our abysmal teaching staff. Very few Americans attend top STEM institutes in other countries.

[@richardmathews82]

Just at the level I need! Thank you. Could you make you powerpoint slides available?

[@TheBigBangggggg]

After 6:49 How do I construct these projections?

[@dXoverdteqprogress]

I'm not sure what you mean by "construct". The picture was just a visual aid. To get the projections you need to know the metric tensor, from which you can work out the Christoffel symbol. The Christoffel symbol gives you the components of these projections. Does this help?

[@TheBigBangggggg]

Thanks! I am trying to understand General Relativity, but I have a hard time putting it all together. I have a question: Is the metric tensor something you put into Einstein's field-equation(s) or is it the result of these equations? Might be a dumm question, but I can't figure it out.

[@dXoverdteqprogress]

In fact, my goal with these videos is to get to general relativity, which should happen after two more videos. About your question: Einstein's field equations in free space can be expressed in terms of the metric tensor and its derivatives; they simply express the condition that a small volume defined by particles moving along a geodesic does not change. A technical way of putting this is that the Ricci tensor is zero. My next video will be on geodesics and the one after will cover Riemann and Ricci tensors.

[@TheBigBangggggg]

Great, I'll be watching for your videos. I think I understand what you wrote. Without matter there is no curvature. So in that case the metric is the MInkowski-metric. isn't it? So it's matter that determines the metric? Do you teach this stuff?

[@dXoverdteqprogress]

Yes, without matter or energy (or pressure or momentum), you can't have spacetime curvature. But what I meant by "free space" was a region of spacetime that does not contain matter/energy but there is matter/energy near by. It's analogous to the electric and magnetic fields in free space. Div E =0 (free space) vs Div E = charge density (not free space). No, I don't teach this stuff. I mostly make these videos as notes for my future self, a self that will have inevitably forgotten this material.

[@suryakr9199]

Very nice explanation..thank you! Is there a book you could suggest with such geometric explanations...with good intuition?

[@dXoverdteqprogress]

When I was a graduate student, I found this book very helpful: "Gravity: An Introduction to Einstein's General Relativity" by James B. Hartle.

[@superkarnal13]

Thank you. Contravariant derivatives are also surface intrinsic variables?

[@dXoverdteqprogress]

Yes, you can form a contravariant derivative simply by multiplying the covariant derivative D_j by the inverse metric g^ij and summing over j. You can watch my video "What the hell is a tensor, anyway" for more details.

[@wmwilliam67]

This is great video. Well done. Thank you.

[@vector8310]

You had me at the Enter The Dragon open.

[@georgelane3564]

I'm glad that someone else recognized the music.

[@mrnarason]

Do you recommend Sean Carroll's GR textbook

[@dXoverdteqprogress]

I only briefly read parts of it a few months ago. It seems like a good book for graduate students. Carroll is a great communicator, so yeah, if you want something higher level, I'd say it's a good book to follow.

[@Mr.Not_Sure]

8:39 Cross components (Г211dx1 and Г112dx1) look incorrect. Zero should have been subtracted, not 1.

[@jacobvandijk6525]

He can't handle criticism.

[@dXoverdteqprogress]

Yes, you're right. I had added annotations to point out the error, but KZbin has gotten rid of annotations recently, so there's no way for me to correct it now.

[@morgengabe1]

Referring to the closing statement: Could anybody tell me a bit about the derivatives you can calculate when you have other measuring apparatus, so to speak?

[@biagiodevivo4968]

morgengabe1 the covariant derivative is perhaps the most general derivative. There are others which apply in different contexts, such as an exterior derivative or lie derivative.

[@zoltankurti]

You can't compare your basis vectors at different points, not by "moving" them on top of each other. They lie in a completely different vectorspace. The covariant derivative is something you can use to compare these, and not the other way around: comparing vectors and defining the covariant derivative with it.

[@NoNameAtAll2]

what music did you use in intro?

[@dXoverdteqprogress]

It's from the movie "Enter the dragon" with Bruce Lee

[@StratosFair]

Wonderful video, thank you !

[@g3452sgp]

This is a excellent video on Riemann geometry. But I have some question. At 7:30, There is a symbol N(ki)(j) associated with normal vector nj. You didn't explain much about it. What is this. Is this same as christoffel symbol of different notation? And on the graphical explanation around 8:28, these are no normal vector components associated with N(ki)(j) showed up. I expect a full-brown explanation on this part coming up.

[@morgengabe1]

The N term is another tensor, which as I understand, is being used at 7:30 as a map between pairs of indices, and then as a map from pairs of indices with respective thirds at 8:28. I suppose it would be the same class of object as Z but it is not for your planar vector's symmetries ({e1, ..., en}), but instead for those of their planes' respective normal vectors. That said, I'm no expert on the matter.

[@Zxv975]

He does (very briefly) explain that symbol at exactly 6:52. It's the component of the vector that lies normal to the plane (hence the symbol N). In answer to your question, yes it is a Christoffel symbol with different notation, but it is associated with a very specific choice of direction that does not lie along the surface being considered (the n direction). This is why the covariant derivative is considered everything *except* the N term. Since the N term is directed perpendicularly to the surface, it in a direction inaccessible to those restricted to a surface. Hence, the covariant derivative is "all you have" if you're stuck on a curved surface.

[@veronicanoordzee6440]

I don't like the visible link between the surface and the y-axes. It confuses me. There is no outside to spacetime. So you never can peak from the surface to the y-axes.

[@chiragkshatriya9486]

Nicely explained

[@TheBigBangggggg]

Nice video. Thanks! All these calculations have been done in space. Does it change in GR's spacetime?

[@dXoverdteqprogress]

No, it does not. In GR the metric can be negative, which is a strange concept geometrically speaking, but all the concept developed for 2-d surfaces seem to work fine in GR.

[@TheBigBangggggg]

Okay, thanks again.

[@givepeaceachance940]

Thanks good video- idea though: one could have explained how in physics the derivative of velocity is the rate of acceleration- the explanation about the rate of change of a surface wasn't made explicit enough with reference to derivatives

[@ermalfeleqi2004]

Hi, very nice video indeed. Could you please tell me how have you drawn those figures? I mean what kind of software have you used?

[@dXoverdteqprogress]

Thanks! For this video I used LaTex to generate the equations and then copied/pasted them into PowerPoint. But in my other videos I generated the equations in PowerPoint directly -- it's much faster that way. Cheers.

[@dXoverdteqprogress]

Oh, and the drawings were done in PowerPoint directly.

[@stancartmankenny]

Is it right to say the metric tensor is a dot product? If you are talking about an individual component, it is not a dot product, it's just a scalar product. If you are talking about the whole thing, it is a tensor product, isn't it? That's my understanding, but I am definitely not sure.

[@tarcisiomarques8784]

Thank you professor !

[@daniyalahsen5707]

AOA Man, here is a BIG question HOW DO YOU MAKE THESE PRESENTATIONS? Please answer asap.

[@dXoverdteqprogress]

If you mean what software I used etc, I first make slides in Power Point, then I import them to Windows Movie Maker where I add my pre-recorded voice.

[@naeembudhwani1141]

Superb visuals

[@doodelay]

that intro is why i come to the internet for these things lol

[@stevenmellemans7215]

Position vector?

[@theronsosachavez2757]

Beautiful video. I really felt in love once again with Differential Geometry. But I have I question dude. Maybe you are not familiarized with the: 'covariant derivative' of the Standard Model(SM) ( if you do, it would be perfect for me. Becouse, you will be able to understand my question faster). In the standard model, we work with this so named covariant derivative, wich carry out all the fields emergent of the Yang Mills transformations. You should be able to find its structure on Internet, it could be the covariant derivative of the group SU(2)xU(1) or that one of the group SU(3). With all this I mean. Does this SM derivative have a geometrical meaning? I mean, Is there associated a 'special' manifold, in the way we can construct a covariant derivative with this form?

[@dXoverdteqprogress]

Thanks for your comment. I'm probably not the best person to respond, but here I go anyway. In, for example, general relativity the covariant derivative has a geometric meaning because the theory itself is geometric. It's the "covariance" of the covariant derivative that connects all theories though. In GR we can form scalars (invariants) from covariant derivatives of tensors. In, for example, quantum electrodynamics the invariant is the Lagrangian, so there we also need a form of a covariant derivative to preserve the Lagrangian under a gauge transformation. But you probably know all this already. My point is that the geometric meaning in GR is probably coincidental, because I don't see it in other field theories. I suppose it's possible to visualize field theories as living on some manifold, but the manifold itself would probably not be physical. Cheers.

[@theronsosachavez2757]

I see, that's some sad, because I was expecting this covariant derivative would give us a better idea of quantum world. However, thanks again and very nice video dude c:

[@theronsosachavez2757]

I see, that's some sad, because I was expecting this covariant derivative would give us a better idea of quantum world. However, thanks again and very nice video dude c:

[@Kazami101]

Why do we take the dot product of ds?? What does that actually mean, intuitively?

[@dXoverdteqprogress]

The dot product of a vector with itself gives the square magnitude of that vector.

[@no-one-in-particular]

@@dXoverdteqprogress And ds is not the magnitude of the vector ds

[@giuseppefasanella5446]

Very nice and instructive video. Thank you for it. I have a question about your slide on the partial derivative of a coordinate vector. You have a term related to the change in the orthogonal direction w.r.t the local tangent plane (the contraction term containing n_j in minute 7:40). I haven't seen this term in other notes I am consulting. See here, equation (5.6) contains only the Christoffel symbols and no reference to the normal direction: www.roma1.infn.it/teongrav/VALERIA/TEACHING/RELATIVITA_GENERALE/AA2013_14/dispense.pdf Could you help me in figuring out the apparent clash in these two equations? Cheers

[@dXoverdteqprogress]

Yes, most sources do not talk about the fact that a derivative of a basis on a curved surface will generate a normal vector (by the way, in my video the index "j" in "n_j" should only be 1, since in 2-d there is only one normal vector). There are various reasons for that, one of them being that you don't necessarily need a curved surface embedded in a larger space to have a curved space(time). Personally, I prefer geometric arguments and I have seen some sources using geometric arguments too, e. g. a book on general relativity by James Hertle, and also videos by Eigenchris. Hope this helps.

[@giuseppefasanella5446]

@@dXoverdteqprogress Thank you for the reply. It is in fact more clear now. I will also have a look at the sources you are citing.

[@AlessandroZir]

omg, u frightened me!!

[@javaandclanguagetutorials7721]

Thank You Sir

[@archishmore2654]

in 7:32 on LHS there should be total derivative

[@dXoverdteqprogress]

Why? We are taking a derivative with respect to one of the variables, e. g. the k th variable, so it is partial by definition.

[@Phyziacom]

Thank You

[@Smoothcurveup52]

Please upload more videos sir

[@duycuongnguyen227]

Excellent!

[@dilshodbekbardiev3660]

Which book you were using ?

[@dXoverdteqprogress]

I wasn't using any particular book, but you can find some elements of my explanations in this book: "Gravity: An Introduction to Einstein's General Relativity" by James Hartle.

[@michaellewis7861]

The metric tensor isn’t continuous.

[@declanwk1]

thanks for the excellent video. For others like myself trying to understand General Relativity I recommend the "General Relativity Step by Step" youtube videos by Trin Tragula. They are numbered GRSS 000, GRSS 001 .... and not only teach you GR but also a whole way to approach math.

[@dXoverdteqprogress]

Thank you! I will definitely check out the videos you recommended.

[@ILikeWeatherGuy]

lost me at 7:44

[@debendragurung3033]

Is this how method of parameterization began

[@sajumathew2299]

Where is RICCI tensor? This was Brilliant

[@pinodomenico5520]

Somewhere (not in this video) he mentions that it is the Trace (diagonal elements) of the Reimann curvature tensor.

[@alikarimi-langroodi5402]

Excellant. thank you

[@NEWDAWNrealizingself]

THANKS !

[@Anonymous-s7j1y]

References you used?

[@math11235]

are there all chapters

[@dXoverdteqprogress]

Just the one on the mathematical background (for now).

[@math11235]

@@dXoverdteqprogress thanks.

[@grandpaobvious]

You said "upper case" and "lower case" when you should have said "superscript" and "subscript," respectively.

[@emmetthume4702]

Fantastic!!

[@tarcisiomarques8784]

Ótimo vídeo, muito esclarecedor.Grato!

[@mohammadal-laqta2999]

see this intro, instant like and subscribe.

[@TheRiquelmeONE]

tyvm, very insightfull

[@ericsu4667]

Space-time geometry was proposed by a mathematician in 1905 and has been proved to be impractical by a physicist in 2021. Detail in sites.google.com/view/physics-news/gravitation