The Meaning of the Metric Tensor

In the follow-up to our prior video, Demystifying the Metric Tensor, we continue to explore the physical and conceptual intuition behind the complex mathematical object known as The Metric Tensor, whose crucial role in the General Relativity, requisite to understanding the theory, tends to be easily obscured by abstract ideas and mathematics.
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Contents:
00:00 - Introduction
00:54 - Spacetime Cartography
02:30 - Maps / Coordinate Systems
04:00 - Bar Scales / Metrics
06:03 - Spacetime Distance
07:30 - Topological Transformations
08:39 - The 2D Metric
13:58 - The 3D Metric
18:39 - Conclusion

[@MrAguc01]

This is a fantastic video, with well explained concepts and perfectly illustrated! I have a PhD in Physics and a master in Mathematics but despite all this, I have never seen the tensors concept so well explained. Other videos are always focused in the matematical operations and interpretations but here you can see the how it relates to real world. You make it so simple in this video that I will recommend this to my students! Your series are a very welcome addition to explain the beuty of Physics and Math! Please keep the excellent work and I will keep helping you as well.Thanks!

[@dialectphilosophy]

Thank you so much for your support! Feedback like yours definitely keeps us motivated, and your generosity is equally appreciated. Hopefully it will help your students wrap their head around the physics a little more easily. More is coming, so stay tuned!

[@stevecrothers6585]

:-: Crothers, S.J. Proof that Einstein’s field equations are invalid: Exposition of the unimodular defect, Physics Essays, Volume 34: Pages 420-428, 2021. Received: July 8, 2021; Accepted: August 14, 2021; Published Online: September 17, 2021: vixra.org/pdf/2206.0082v1.pdf Abstract: Albert Einstein first presented his gravitational field equations in unimodular coordinates. In these coordinates, the field equations can be written explicitly in terms of the Einstein pseudotensor for the energy-momentum of the gravitational field. Since this pseudotensor produces, by contraction, a first-order intrinsic differential invariant, it violates the rules of pure mathematics. This is sufficient to prove that Einstein's unimodular field equations are invalid. Since the unimodular form must hold in the General Theory of Relativity, it follows that the latter is also unsound, lacking a proper mathematical foundation. www.physicsessays.org/browse-journal-2/product/1896-2-stephen-j-crothers-proof-that-einstein-s-field-equations.html Stephen J. Crothers: The General Theory of Relativity: Its Faulty Mathematical Foundations kzbin.info/www/bejne/ZnS2fWpmZ6-Gmrs

[@jrv634]

Wow! I rarely write any comments in KZbin but I must say the level of detail and simplicity with which you describe these concepts is astounding. Hope you get millions of subscribers soon, not only you deserve it but they would hugely benefit from this content.

[@TheMemesofDestruction]

^ This :-)

[@ihatethesensors]

I completely agree and I can't wait for the next one. The graphics were amazing! A difficult subject described simply - it's beautiful.

[@-danR]

Dialect has suddenly become not only the most illuminating school of introductory SR and GR, but a model _master-class_ on _how to explain SR and GR_ . And I speak as one who has watched dozens of youTube channels on the topic and 100's of hours, and many years. It is in danger of rendering channels like eigenchris, The Science Asylum, SciencClic, Eugene Khutoryansky practically marginalized, if not irrelevant. They had better stand up and take notice: This is how you actually explain things. If I have any objection at all, it's that the (excellent) ambient music could be reduced to, say, 2/3's volume.

[@TheMemesofDestruction]

@@-danR We’re all entangled. But yes this Channel’s communicated knowledge per minute value is extremely high! ^.^

[@squarerootof2]

Maybe if he introduced some cute cats mixed with some really stupid pranks and some tik-tok style hoes, he would reach a million subs...

[@alemorita92]

I did my undergraduate research and master’s on GR. Had I seen a video with this quality back then, I wonder how much faster I would have grasped concepts in GR. Fantastic video, explanatio, and animations!

[@HarpSeal]

Is this channel accurate on gr?

[@jaimemat3]

De eso se trata, de acelerar la curva de aprendizaje en modo autodidacta, buscando el mensaje que calce con nuestro mood del momento.

[@pushpendrayadav7069]

Where are you from?

[@stevecrothers6585]

:: Crothers, S.J. Proof that Einstein’s field equations are invalid: Exposition of the unimodular defect, Physics Essays, Volume 34: Pages 420-428, 2021. Received: July 8, 2021; Accepted: August 14, 2021; Published Online: September 17, 2021: vixra.org/pdf/2206.0082v1.pdf Abstract: Albert Einstein first presented his gravitational field equations in unimodular coordinates. In these coordinates, the field equations can be written explicitly in terms of the Einstein pseudotensor for the energy-momentum of the gravitational field. Since this pseudotensor produces, by contraction, a first-order intrinsic differential invariant, it violates the rules of pure mathematics. This is sufficient to prove that Einstein's unimodular field equations are invalid. Since the unimodular form must hold in the General Theory of Relativity, it follows that the latter is also unsound, lacking a proper mathematical foundation. www.physicsessays.org/browse-journal-2/product/1896-2-stephen-j-crothers-proof-that-einstein-s-field-equations.html Stephen J. Crothers: The General Theory of Relativity: Its Faulty Mathematical Foundations kzbin.info/www/bejne/ZnS2fWpmZ6-Gmrs

[@Blake_47]

Hey I'm an undergraduate physics student and I've some questions

[@eigenchris]

Another great video on the metric. I've been through pages Schutz and MTW, desperately trying to understand things. It really helps to have a video like this to consult first.

[@stevecrothers6585]

Crothers, S.J. Proof that Einstein’s field equations are invalid: Exposition of the unimodular defect, Physics Essays, Volume 34: Pages 420-428, 2021. Received: July 8, 2021; Accepted: August 14, 2021; Published Online: September 17, 2021: vixra.org/pdf/2206.0082v1.pdf Abstract: Albert Einstein first presented his gravitational field equations in unimodular coordinates. In these coordinates, the field equations can be written explicitly in terms of the Einstein pseudotensor for the energy-momentum of the gravitational field. Since this pseudotensor produces, by contraction, a first-order intrinsic differential invariant, it violates the rules of pure mathematics. This is sufficient to prove that Einstein's unimodular field equations are invalid. Since the unimodular form must hold in the General Theory of Relativity, it follows that the latter is also unsound, lacking a proper mathematical foundation. www.physicsessays.org/browse-journal-2/product/1896-2-stephen-j-crothers-proof-that-einstein-s-field-equations.html Stephen J. Crothers: The General Theory of Relativity: Its Faulty Mathematical Foundations kzbin.info/www/bejne/ZnS2fWpmZ6-Gmrs

[@chyldstudios]

I'm astonished at the level of detail in this video. Well done!

[@worldwithinadream]

I have bachelor degree in physics and I find this metric intuition as clear as it could be. Better than in any book I read.

[@stevecrothers6585]

>>>

[@nemvus_]

The amount of this video that is 3D animated is impressive. I imagine it takes a lot of clever work to get the 3D shapes to behave that way in the video. Excellent quality content. Needs more views.

[@stevewhitt9109]

This was amazing. After years I finally get the intuition.

[@mikegale9757]

Beyond brilliant. You really got it down to the brass tacks. Testify!

[@jmchez]

When Professor Goodstein of Caltech got a grant to film his lectures and use the latest 3d animation back in 1983, he created one of the greatest teaching tools in physics still fantastically useful to this day, "The mechanical World and Beyond". I've been wondering for a while how much better the animations could be done with today's technology. Frankly, not that much better. but your videos do fill in the gaps in some areas and are a worthy successor to Goodstein.

[@HassaanFareed]

How can someone be so perfect in explaining such a difficult concept. Your order of explaining things is perfect for the proper understanding. And I can understand how much difficult it is to produce animations of this quality. Keep making such an amazing content for us ❤️

[@agrajyadav2951]

bhai kya gajab video banaya maja aa gayi, pasta khate khate dekha, metric tensor bhi samajh gaye aur swad bhi samjho double ho gaya, mauj kara di matlab. Adbhut bro keep it up!

[@nosnibor800]

Thanks that was well described - my God the amount of work you put into the graphics. The 4th dimension adds time. Then we need to understand how mass causes tensor to change. Thanks

[@elindauer]

The visualizations in these videos are amazing. The whole series is beautifully instructive. Great work.

[@flmbray]

What an amazingly simple and straightforward explanation. Although I not a physicist, I do watch a lot of these types of videos, and I've seen explanations of the metric tensor before. But this is the first (and only) one that really made it clear. The analogy to old maps and how they depict a curved space in a stretched fashion for our benefit was absolutely perfect. It all feels so natural now that I'm sure I will retain this new knowledge. Thanks! You earned a subscribe from me!

[@yashasvikatara1244]

I cannot put it in words how amazed I am that you made such an intricate concept so easy to understand. I'm really greatful, thanks a lot ❤

[@Maderum]

Wow! The animations and simplicity of explanations on this video blew my mind! Well done sir, well done.

[@stevenschilizzi4104]

Fantastic presentation. I’m eager to see the sequel. Thanks for putting all this together for us.

[@bofink5377]

Thank You for the very clearly and straight forward explained Metrics and mapping. This clears many problems for me, trying to really understand the General Relativity and Gravitation in a more visual way, than just the plain raw mathematics.

[@MictheEagle]

Pure Gold! This is the school we all needed. Bravo! Keep it coming, and Thank you.

[@manymany1191]

I really like the pauses in narration at the right spots to give me a moment to think through what's just being said.

[@alamagordoingordo3047]

Great clarity. I can't wait for the next episode. Thank you!

[@erikquintanilla7705]

Amazing content. Love the clarity

[@JoEbY-X]

I have a degree in mathematics and studied some physics but never formally studied relativity, but decided to learn it myself. After finally thinking I had a grasp of special relativity, I tried to tackle general relativity but never quite got a grasp of what a metric tensor is the few times I tried. Your video is the most amazingly intuitive treatment I could possible imagine. I get it now! Thank you! Also, I conceptualize special relativity just as you did at <a href="#" class="seekto" data-time="376">6:16</a> with a light clock measuring time perpendicular to the velocity. But I never quite understood why the spacetime interval involves subtraction instead of addition. I can easily perceive time dilation and length contraction with a Minkowski diagram, but the image at <a href="#" class="seekto" data-time="376">6:16</a> and your description of the spacetime interval as simply being the static proper time FINALLY made it click!

[@dialectphilosophy]

Thanks for watching! We put that visual of the light clock in exactly for that reason, so we're glad to hear it clicked for you!

[@abnereliberganzahernandez6337]

Man so you are mathmatican AND didnt had a course con diferencial geometry? Imean I am computer scientist AND love math. But It Is interesting to see that no body fully understands everything

[@stevecrothers6585]

>>

[@jeanklein2198]

May I share my praise to whoever scripted and executed this wonderful video. Excellent quality content

[@jack.d7873]

Newly discovered your channel Dialect. And as someone who's familiar with Special Relativity wanting to understand General Relativity, your visual presentation along side clear, slowed foundational GR language is absolutely brilliant. I feel it in your video's, your goal and target audience is too teach this subject to people who have zero knowledge of GR. And as someone who feels like your target audience, I must say again, the ways you teach are mindbogglingly sensational! I'll give you a donation for your service to humanity. You've earned it.

[@dialectphilosophy]

Thanks for watching, we appreciate your support!

[@tanyachou4474]

One of the most underrated channel on KZbin! Thank you !

[@beln33]

excellent job, Dialect. I highly recommend watching this video after the one titled "Demystifying the metric tensor in GR". After watching both (and taking some notes), the importance of the metric tensor should be obvious to anyone interested in GR and cosmology. I'm looking forward to watch "part 3" with will go about the 10 component tensor for curved spacetime.

[@nooneinparticular3370]

I watched the video and I honestly don't understand how such a small channel can manage to have such amazing animations, great job! Anyways, as a mathematician with physicist friends, in my experience what happens is that they don't have the mathematics physically and historically motivated, and so they're just left confused by it all, and it's only in hindsight that they realize what all those maths were saying physically, so this channel is certainly plugging quite a glaring hole in that regard! As it regards to tensors in particular, the issue I see is that the average physicist will ask you "How should I interpret _a_ tensor?", but the problem is that there is, to my knowledge, no good explanation or physical motivator for _all_ tensors that would allow physicist to go "Ooh, okay!"; you essentially need to go type by type, and then explain what the whole dropping and raising indices magic is. So far as I know (Gravitation is on the mail and I'll start getting serious with GR once it arrives), the only books that give a decent explanation and motivation for such geometrical concepts are (an honourable mention might be Jürgen Josts edition of Riemann's "On the Hypothesis Which Lie at the Bases of Geometry", which is a short and sweet book) Lee's "Introduction to Smooth Manifolds" and "Introduction to Riemannian Manifolds"; the latter even has the occasional detail about pseudo-Riemannian/Lorentzian manifolds. Which segways into my last point: I know it's hard, but I really hope you guys make justice to the distinction between regular old Riemannian manifolds (which are very intuitive) and the Lorentzian spacetimes one deals with in GR (which are completely alien to the uninitiated), and in particular explain at length how the mathematicallty seemingly innocuous condition of changing the signature of the metric tensor so as to ensure everyone measures _c_ to be the same everywhere has in fact huge mathematical (and therefore physical) consequences.

[@dialectphilosophy]

Thanks for watching! The interpretation of what a tensor "is" was something that frustrated us for the longest time, and we couldn't get good answers on it. The discussion of what defines a tensor in the abstract, general sense is something we dropped from these videos altogether, because we finally realized it just wasn't necessary. What a tensor is or isn't in terms of physical meaningfulness varies depending on context. And yeah, the Lorentzian vs. Riemannian signature was a little too much to squeeze into this video. More forthcoming!

[@HesderOleh]

This comment got me thinking that some of the math youtube channels should organize to make explainer videos for math wikipedia pages. Wikipedia pages are really great in most of the sciences, having both high level explanations along with some easier to understand explanations, but most of the math wikipedia pages just assume you know everything about the concept and are just wanting to look up some small detail of definition.

[@nooneinparticular3370]

@@HesderOleh I know exactly what you mean, as I've found myself with the same problem too many times. I think the worst ones were either the page for "Spinors" or the one about the "Clebsch-Gordan coefficients"; the ones about the mathematics behind physics are the worst offenders in my experience.

[@umeshchandramakwana806]

Awesome ! Really a great clarity in explaining the complex concepts! 👍👍👍

[@brianyeh2695]

The effort required for this video is beyond words... Keep going man. I truly admire your great explanation. Other videos just focus on the math operations rather than giving intuitions for what they are doing. For example, you explain the shrinking of areas near the north and south pole by playing around the transformation matrix and squeezing the element in the animation at the same time! That's extremely helpful to understand!

[@nicolaecoman7188]

Incredible animations and explanations. keep up the good work!

[@jialixx]

This is amazing! Can't wait for your next video. Thank you for sharing.

[@cesararf]

Amazing explanation. Until now I didn't understand anything on metric tensors although I had heard of them. You deserve a massive thumbs up. 🎉

[@cesarjom]

Very well explained and demonstrated with visualizations!

[@png7]

These video came out just before I needed them, They are very detailed and explain a lot with out be confusing.

[@jimlbeaver]

Outstanding…thanks for the patient and diligent explanation

[@mickwilson99]

Bloody good start. One of my favourite analogues for tensors is playing games like Railway Life. The further you in the cab are from "events" (e.g. your destination at your scheduled arrival time) then shorter are your "kilometers" and the faster your "hours" pass. A great way for short-cutting the boredom of being stuck in proper-time-and-distance for a 500km run across Germany to France

[@jewulo]

Hello Dialect. Awesome work. Simply brilliant. Thank you very much for this.

[@Redant1Redant]

amazingly good explanation. I wish I had seen this 10 years ago. It would have clarified a lot. Thank you for this video.

[@oblivitus.]

You deserve a ton of veiws! Brilliant quality!

[@casaroli]

Thank you for finally making this intuitive. No textbook I’ve ever read has been capable remotely achieving this.

[@roberthayter157]

Absolutely stunning graphics. Brilliant explanations.

[@SuperMenders]

Wow the best Introduction to Differential Geometry, i have ever seen!

[@sonysantos]

Your videos are very good, both in content and visual effects. You deserve more subscribers!

[@matthewbusche4547]

I've been looking for a simple explanation of GR tensors for years. This was fantastic. I too hardly ever write comments, and only hit the like button once or twice a year. Here I've done both. Thanks so much.

[@justynpryce]

Really happy to see this upload I appreciate you guys doing this!!

[@Ri-bc5ff]

Great job! Kindly make a video explaining concept of induced metric ,intrinsic and extrinsic metric

[@zerozzable]

Your videos are beyond brilliant. Keep it up.

[@anandbavkar8572]

Excellent Work!! I wouldn’t be surprised if this leads to true understanding. Thanks a lot for making these videos.

[@1three7]

I love your videos so much. I'm not someone with enough math background to keep up with most physics stuff. But I also hate how unhelpful it is to just get over simplified analogies. It's hard to strike a balance of being clear to lay people but also being accurate and describing it with the underlying math in an intuitive way. Your channel is one that excels at that in my opinion. The animations really help keep things clear. Plus I love the music too! Thanks for putting the effort in on these. It's greatly appreciated!

[@thorntontarr2894]

Unlike you, I find the music annoying and distracting.

[@DennisMHenderson]

@@thorntontarr2894 I foun/d it, ‘Hard; 2bete [0ff] 2it-2nite

[@Minstorm34]

Insane quality from a

[@tianchenzheng7464]

This is amazing, can't wait to see more of your videos

[@menglongyouk167]

Well-explained. Thank you.

[@DickHoskins]

Amazing video with respect to content and presentation. Graphics are superb and totally support the math. Thank you!

[@PamSesheta]

Beautiful! I’ve always wondered about that 4D mapping bit, was hoping for like, iunno a guess even. I could watch a video of a shadow of the 4D mapping with some “guesses” at what that mapping might be. Thanks for reminding me about that strange old diff geometry book I was working through for fun.

[@crabcrab2024]

I am very impressed by your videos! Great Job of visualization and explanation. UR going to be a success.

[@williamolenchenko5772]

Fantastic explanations and animations.

[@hexeract9541]

i literally have a project due on this in two days, thank you so much

[@muttleycrew]

Excellent presentation. Thanks uploader.

[@93katwal]

thankyou for such a wonderful explanation

[@BravoZulu2023]

Excellent⋯get me to know and further understand more about metric tensor. THX.

[@TranquilSeaOfMath]

Great presentation! Well animated and explained.

[@AlokKumar-ym8bl]

🎉 excellent information..thank you sir for making this highly informative article about space and time.

[@cykkm]

Dialect, you're a genius of a teacher!!! Holy Great Guacamole with vuvuzela on a Pogo stick, this is the most visual explanation of the metric tensor that I have ever seen! And I am not only a physicist by education; I had a full GR course with applied diffgeo in my grad school (following the MTW book) 25 years ago. The _abstraction_ of a tensor in my head has always been a row of columns of rows of columns of etc. (rows are covariant, living in a dual space, columns are your normal contravariant vectors); the metric you can slice either way. This is how it's easy to connect the tensor product with the normal linear algebra dot products, but that's only another, if a more familiar abstraction. Can you maybe eventually develop similarly simple visuals for tensor calculus? I'm not even working in physics any more, but I love GR so much for its concise elegance, but it's still a mathematical, not a visual elegance. But I'm ultimately a visual thinker, and I never looked at the spacetime like that!

[@dialectphilosophy]

Thanks for watching and we appreciate the compliments! In terms of how to define a tensor in the abstract and general sense, there seem to be a variety of ways to go about it and mostly they all confuse us 😂. When (and if) we understand the mathematics more fully, we may very well consider a series simply about tensor calculus in-itself. Of course, we tend to be more interested in the philosophy of mathematics, namely when its justified to apply certain calculuses, so you're more likely to see a series skewed towards that.

[@-danR]

This _alone_ : 0:35 - 0:38 is one of the most expressive and illuminating 3 seconds in the entire history of GR pedagogy. And I speak as one who has watched 100's of hours of youTube explanatory videos on the subject. Now granted one would want some background material to understand what that elegant bit of choreography is describing. But one could dispense with those hundreds of other hours (scienceAsylum et al.) and replace them with the rest of Dialect's classes in Relativity, which are also master-classes in _how to explain Relativity_ .

[@massimoacerbis8138]

One of the best video on general relativity ever I have watched that many on the topics Highly recommended

[@sumanthakur1519]

Really such a good explanation hats off sir keep it up

[@inverse_of_zero]

wow, this is an incredible video, with the concepts so well explained and illustrated! i have three mathematics degrees, with majors in physics and mathematics, and i have studied general relativity and differential geometry. but despite all this, i never fully grasped the concept of the metric tensor until you explained it so simply in this video. thank you! i will recommend this to my students; if only i had seen this when *I* was a student myself!

[@dialectphilosophy]

Thank you so much for your support! We're really glad the video helped and hopefully if you check back in a few months we'll be on to the connection coefficients and curvature tensor!

[@a.andacaydn9736]

Another amazing video! I cannot thank you enough

[@coolwinder]

First time i dvel into relativity, but just amazing!! You are amazing, thank you for this pace of work!

[@dfkjbdfondfngg]

Awesome animations!

[@JeanSarfati]

Totally pedagogical. Bravo !

[@user-ug6zi2gg4m]

三D動畫真的很不錯 也相當易懂的內容 持續更新肯定可以得到更多的定閱 而且也很少人會做出如此精細的科普視頻。

[@Alpasonic]

Thank you. It was extremely satisfying

[@Mouse-qm8wn]

Master piece. Thank you so much. I finally get it 😉👍❤️

[@user-ph5in5fi1q]

Just awesome. We lost the essence of physics behind those equations and maths. No one describes it visually like you did. Thanks a lot.

[@stevecrothers6585]

;;;;Crothers, S.J. Proof that Einstein’s field equations are invalid: Exposition of the unimodular defect, Physics Essays, Volume 34: Pages 420-428, 2021. Received: July 8, 2021; Accepted: August 14, 2021; Published Online: September 17, 2021: vixra.org/pdf/2206.0082v1.pdf Abstract: Albert Einstein first presented his gravitational field equations in unimodular coordinates. In these coordinates, the field equations can be written explicitly in terms of the Einstein pseudotensor for the energy-momentum of the gravitational field. Since this pseudotensor produces, by contraction, a first-order intrinsic differential invariant, it violates the rules of pure mathematics. This is sufficient to prove that Einstein's unimodular field equations are invalid. Since the unimodular form must hold in the General Theory of Relativity, it follows that the latter is also unsound, lacking a proper mathematical foundation. www.physicsessays.org/browse-journal-2/product/1896-2-stephen-j-crothers-proof-that-einstein-s-field-equations.html Stephen J. Crothers: The General Theory of Relativity: Its Faulty Mathematical Foundations kzbin.info/www/bejne/ZnS2fWpmZ6-Gmrs

[@johnbest7135]

Great communications. Gladly received.

[@deltalima6703]

This better not be clickbait. Could not resist the title

[@winsomehax]

Yeah this is a great explanatory video. Really well done.

[@Wielorybkek]

Such a good explenation! I never trully understood this concept even tho I studies it at the uni.

[@jzblue345]

I'm so glad I found this channel.

[@johnathos]

Really wonderful! Please, keep going!!

[@Markoul11]

This video is a masterpiece! Bravo!!

[@themerrydancers]

I can’t wait for the next episode!

[@abulkhondker3556]

This is a superb video. The 3D animation is out of the world (pun intended). I cannot wait to watch more videos like this.

[@geraltofrivia9424]

Another masterpiece. Thank you very much for what you do.

[@GrandMarshalGarithos]

Wow, this is superb. Thank you.

[@misnik1986]

I have just discovered your channel, this is an outstanding video, waiting for next

[@Burkard42]

Wonderful video. THANKS!

[@stevedavey9435]

THANK YOU SO MUCH FOR MAKING THIS VIDEO!!!!!!!!!!

[@BravoZulu2023]

Excellent sharing on metric tensor and the maping⋯ those gives more explaination on it⋯

[@shebo96]

This is a great explanation ❤❤

[@tanvirfarhan5585]

Besttt..!!!!!!! i really really love your contents, i appreciate your effort. You deserve a lot 🔥🔥

[@zacmilne9423]

Such a well made and educational video. Subscribed!

[@sinekonata]

Nah, thank you. So much. I can't believe we now get this much better of an idea just because someone on the internet decided to put big effort into visualising, explaining and then writing in video what he understands. We're pretty lucky. This kind of shit used to be only accessible if you had direct written correspondence with a physics master, waiting forever for his letter to come back or travelling to his university.

[@brianhay4024]

The clearest explanation I've seen.

[@GizmoMaltese]

Best explanation of the metric tensor ever

[@samuelbrooks7455]

Thanks for sharing! Tough subject matter

[@lucasf.v.n.4197]

Best explanation and animation

[@aymenzaki6786]

very intuitive you are really a genius great work continue