Riemann geometry -- covariant derivative 

A Great work✨🥀🥀🌈🌈

@@freethinker5670 Thank you!

Nice, well written

@@dXoverdteqprogress What is E=MC2 is consistent with TIME AND what is gravity. (TIME is thoroughly consistent with what is gravity ON/IN BALANCE.) WHAT IS E=MC2 is dimensionally consistent. TIME is NECESSARILY possible/potential AND actual ON/IN BALANCE, AS ELECTROMAGNETISM/energy is CLEARLY AND NECESSARILY proven to be gravity (ON/IN BALANCE) !!! WHAT IS E=MC2 is taken directly from F=ma. Consider what is the man (AND THE EYE ON BALANCE) who IS standing on what is THE EARTH/ground. Touch AND feeling BLEND, AS ELECTROMAGNETISM/energy is CLEARLY AND NECESSARILY proven to be gravity (ON/IN BALANCE). CLEAR water comes from what is THE EYE. INDEED, consider what is (essentially and necessarily) BALANCED BODILY/VISUAL EXPERIENCE !!! Lava IS orange, AND it is even blood red. The hottest flame is blue. The hottest lava is yellow. LOOK upwards, ON BALANCE, at what is the TRANSLUCENT AND BLUE sky !! The orange (AND setting) Sun IS the SAME SIZE as what is THE EYE !! NOW, consider what is the fully illuminated (AND setting/WHITE) MOON ON BALANCE. (BALANCE AND completeness go hand in hand.) WHAT IS E=MC2 is taken directly from F=ma, AS TIME is NECESSARILY possible/potential AND actual ON/IN BALANCE; AS ELECTROMAGNETISM/energy is CLEARLY AND NECESSARILY proven to be gravity (ON/IN BALANCE); AS the rotation of WHAT IS THE MOON matches the revolution. WHAT IS E=MC2 is taken directly from F=ma. Gravity AND ELECTROMAGNETISM/energy are linked AND BALANCED opposites ON BALANCE, as the stars AND PLANETS are POINTS in the night sky. Consider TIME AND time dilation ON BALANCE. c squared CLEARLY represents a dimension of SPACE ON BALANCE. WHAT IS GRAVITY is, ON BALANCE, an INTERACTION that cannot be shielded or blocked (ON BALANCE) !!! E=MC2 is consistent with/AS WHAT IS GRAVITY, AS TIME is NECESSARILY possible/potential AND actual ON/IN BALANCE; AS ELECTROMAGNETISM/energy is CLEARLY AND NECESSARILY proven to be gravity (ON/IN BALANCE); AS the rotation of WHAT IS THE MOON matches the revolution. Magnificent. Notice that the curvature or shape of said Moon matches that of what is THE EARTH/ground (that is, given what is a CLEAR horizon, of course.) The diameter of WHAT IS THE MOON IS about ONE QUARTER that of WHAT IS THE EARTH/ground ON BALANCE. Excellent !!! It ALL CLEARLY makes perfect sense ON BALANCE. Consistent WITH WHAT IS TIME, WHAT IS E=MC2 IS GRAVITY ON BALANCE. Finally, the average ocean tide is about 6 feet; AND said Sun manifests or forms at what is EYE LEVEL/BODY HEIGHT. The tidal range on the open ocean is about 3 feet. Notice, what is THE EARTH is ALSO BLUE ON BALANCE. Outstanding. Again, ON BALANCE, consider what is the fully illuminated (AND setting/WHITE) MOON ON BALANCE. The BULK DENSITY of WHAT IS THE MOON is comparable to that of (volcanic) basaltic lavas on THE EARTH/ground. The surface of WHAT IS THE MOON is chiefly composed of pumice. Excellent. ELECTROMAGNETISM/energy is CLEARLY AND NECESSARILY proven to be gravity (ON/IN BALANCE), AS the rotation of WHAT IS THE MOON matches the revolution; AS TIME is NECESSARILY possible/potential AND actual ON/IN BALANCE; AS WHAT IS E=MC2 is taken directly from F=ma. By Frank Martin DiMeglio

Einstein never nearly understood gravity or time.

yupp, It's that bad....

Blaxploitation movies all start like that

@@crehenge2386 lmao

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

Thank you.

Im undergrad and i like it

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

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

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

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

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 :-(

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

@@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!

@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!).

@@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.

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

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

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

Haha, exellent :D

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

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

@@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

@@frankdimeglio8216 ✨🔥👏👏👁️👄👁️

Good comment. I also agree

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

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

Thanks!

I'm glad. Cheers.

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?

Thank you.

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.

I'm glad that someone else recognized the music.

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.

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

High school? Covariant derivative in high school?

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

@@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....

@@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.

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.

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

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.

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

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.

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.

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.

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

I do that in this video: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-TZHxJ77ovzE.html&ab_channel=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.

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.

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.

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.

Thank you!

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.

Oh, and the drawings were done in PowerPoint directly.

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

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

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

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

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.

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:

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:

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.

He can't handle criticism.

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

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.

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.

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.

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.

Okay, thanks again.

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?

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.

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.

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?

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.

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

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.

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

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