Vector Calculus: Understanding Curl

[@dysxleia]

I can't believe you posted only one video and killed it. This was so clarifying. I hope you continue

[@leeknowbiased4644]

I have seen dozens of videos about curl. This one is by far the best video I have ever seen.

[@octavioaguayo5366]

One of the best math videos I've ever seen. It explained the curl definition in such a simple way, I loved it

[@greibert1447]

I think that this is the best video on math I have seen since 3Blue1Brown dropped his linear algebra series. Very well done.

[@hiltonmarquessouzasantana6883]

Nice job! A nice complement could be the idea that , locally, the force field is a sum of a stretch and a pure rotational force field. The pure rotational part is the curl. This idea is known in the business as Helmhotz decomposition

[@angeldude101]

In a different business that I'm more familiar with, it's called the Geometric Product. The case of divergence and curl is just when one of the multiplicands is the 4D spacetime gradient.

[@kitzelnsiebert]

Thank you for the video. The calculation of circulation presented here is similar to what I see in most calculus textbooks, but I think it really comes to life with your animations.

[@BarkanUgurlu]

I got the whole intuition behind the curl operator now. Many thanks!

[@evertonsantosdeandradejuni3787]

More, more, more, please. Well explained, very clear, adds to the state of the art on vector calculus present on youtube. if you actually do another video I'll watch, for sure. Congrats on the views too !

[@POPO-kk6nh]

OMG Your explanation was beyond my imagination. The animation helps a lot, too. Thanks

[@alexfoo_dw]

Wonderful explanation and motivation. Thanks for this!

[@ObsessiveClarity]

First comment on this great channel. :) Please make more videos like this :)

[@kostyaalekseev6138]

I watched a lot of videos about curl, and yours are the best, thank you, it really good and simple

[@Number86Rules]

This is the best video I have ever seen

[@kikilolo6771]

This video is incredible ! Everything is perfect !

[@blakec542]

Dude hella appreciate this, I was lookin everywhere for a video on this. If you could cover the 3D and 4D versions next time that would be awesome

[@MrRyanroberson1]

In N dimensions, there are N(N-1)/2 components to the curl function. each value of the curl simply corresponds to each possible plane. Notice: 2d -> xy 3d -> xy, yx, xz 4d will have 6 components, and so on. due to the linearity of everything, the case of N+1 dimensions can be generated from N dimensions: every vector can be broken up as an element parallel to the N+1st axis, and an element perpendicular to it. The perpendicular components have a curl of the same kind as in N dimensions, and notably any curling occurring in N dimensions will have no effect on the N+1st axis. Finally, for each of the N dimensions, the N+1st axis forms a plane, which can then experience curl.

[@angeldude101]

@@MrRyanroberson1 N(N-1)/2 I suppose is one way of formulating it, but I usually see it as N choose 2.

[@MrRyanroberson1]

@@angeldude101 in the most literal sense, there are (n choose k) many unique K dimensional orthonormal subspaces in N dimensions, yes.

[@orstorzsok6708]

Aesthetic animation, quite neat! Thanks!

[@MrW3iss]

Great video, Joey, thanks. Hope you make more!

[@Nikolaos.I.P]

Very good. keep on good work!

[@ko-prometheus]

ok!! We look forward to more such lessons, such wonderful lectures !!!!! -------------------------------------------------------------------------------- Ждем больше таких уроков, таких прекрасных лекций!!!!!

[@satyampanchal-1016]

nice video, for 3D curl, i build upto it by thinking not about how stuff rotate in 3D but splitting that rotation in three components, each one can be explained by the methods of your video, and all combined i see it as bringing a small.sphere to each point in space and seeing how it spins, then connecting a vector to the spin using right hand rule. That hiw i visualize curl

[@diegopg7186]

What a wonderful explanation! I am impressed!

[@96sam26]

Amazing video!

[@ArthMaxim]

Best explanation.

[@Anik_Sine]

Would I be right to interpret curl as being similar to the net torque on a wheel with multiple forces applied to it, divided by the area?

[@patrickmichel7371]

Great work. Thank you!

[@bernardofitzpatrick5403]

Will there be more vids? This was awesome !

[@michaelfisher295]

Subscribed, very nice! More please...

[@corrompido7680]

every time you play 6:00 a mathematician dies from an anneurism

[@michaelcollins7192]

Really nice 👍🏻👍🏻, thanks!

[@Tom4this]

Great video and topic. Thx.

[@jhanolaer8286]

Hi sir ,how do you put vector in grid?

[@djentgod45]

Amazing!

[@gudmundurjonsson4357]

great video

Dude, awesome video! Loved it!

[@SurinderKumar-os5il]

I understood this more Thanks for guidance

[@SuperMaDBrothers]

Right when I thought the world didn’t need ANOTHER curl video, you proved me wrong :P I already derived all this stuff out of curiosity a while ago but good job! The only thing is, you didn’t say why dg/dx and df/dy can just be added, without adding, say, d(f+g)/(x+y) or some intermediate direction. You can give a BS reason that x and y are independent, but I want to hear a real answer :P

[@albertyeung5787]

excellent

[@FahimaAkter-v5h]

So underated!!! ❤️‍🩹

[@sorry4all]

Well done!

[@blancaroca8786]

Big R seems like a radius of circle going to zero really It is the area of the circle. Couldn’t we use A or S or dA ?

[@Uriakatos]

Love it. Thanks!

[@plasticpeepsperformances]

Nice and clear! Maybe in a future video (like the one about 3D curl) you could give an intuition of why certain pairs of directions should be negative (i.e. what makes df/dy nagative but not dg/dx, and why the df/dz term in " i*stuff - j*(dh/dx - df/dz) + k*stuff " has to be double negatified). It's always something I've just accepted. All I can figure out is that it somehow corresponds to how many times you've switched a pair. For example, xyz is positive (i, d/dy, h), but yxz is negative (j, d/dx, h).

[@angeldude101]

"All I can figure out is that it somehow corresponds to how many times you've switched a pair." That's pretty much exactly what it is. I personally don't like thinking about it in terms of signs at all, especially since they won't work the moment there is more than one plane the field can be rotating in. The two directions of rotation in 2D are x->y and y->x. It's not too hard to see that rotating in one direction is equivalent to rotating backwards in the other direction, so x->y = -(y->x). In practice, most representations of curl just pretend that the value is a scalar in 2D and instead represent the direction with a sign. Which direction is positive and which is negative is completely arbitrary, but convention usually sets x->y as positive and y->x as negative. df/dy is x->y and dg/dx is y->x. In order to add them into a single rotation, you need to flip one so that they have the same unit: -dg/dx in x->y or -df/dy in y->x. In 3D, x->y and x->z are along completely different planes and therefor can't be converted into each other with just a sign flip. x->y->z on the other hand is equivalent to z->y->x, while also opposite to x->z->y. In the 3D volume case, I tend to think of a helix that you can rotate all you like, but can't be mirrored without changing the sign, though that shouldn't be relevant for curl, which only cares about the 2D plane.

[@plasticpeepsperformances]

Ooh, nice point about the helix. Also, maybe the curl-ing itself only cares about the 2D plane, but you still need an axis. So I think your imaginary helix is still justified. I suppose the sign-switching = rotation-switching trick might not be all that helpful for explaining 3D curl. Although maybe the idea can be pushed a little further. If you travel through a helix that obeys the right-hand rule, with your thumb pointing from the origin towards [1,1,1], then the closest axis to you will follow a pattern of being ...x, y, z, x, y, z... If your thumb points towards [-1,1,1] instead, keeping with the right-hand rule, the pattern becomes ...z, y, x, z, y, x... It looks to me that the other 6 octants (I learnt a new word) all work the same: reverse a sign, and you reverse the cycle (which is equivalent to flipping a pair when you have a 3 element cycle, of course). Admittedly, that feels like a slightly contrived pattern, and I'm not sure it helps much for intuiting why 3d curl works. Ultimately, I think it's more interesting to think about how pair-switching = sign-switching when calculating each term of the determinant in any dimension. [I haven't thoroughly checked, but] I think the rule can work as a replacement of the cheeky, recursive definition of the n-dimensional determinant. Although I'm not sure there's an easy intuition for why it's true above 3D.

[@angeldude101]

​@@plasticpeepsperformances "but you still need an axis." Says who? The whole point in 2D is that there is no axis. To try to find such an axis would be to invent an entire extra dimension. In 4D it gets even worse since there are infinitely many perpendicular vectors. How would you ever choose just one? The helix that I visualize is just for an oriented volume to show that it can be rotated into equivalent volumes, but can't be mirrored without changing it. The volume actually acts more like a scalar rather than a vector in some ways. In 3D, the curl is still usually just an oriented plane. In fact, the oriented volume is so much like a scalar that it's usually generated through the scalar triple product, whose use of a dot product means that it's usually addressed as the divergence rather than the curl.

[@plasticpeepsperformances]

@@angeldude101 Fair enough, I'll tweak what I said. With rotation in 3D, you end up with an axis (or multiple I suppose). And with 2D rotation, since it's maths, you are allowed to assert the existence of a 3rd dimension as long as it doesn't interfere with anything else (I see no inherent reason why it should). But ok, you got me, maybe rotation is better thought of fundamentally as a trade between 2 (or more?) dimensions. How would you explain it? I'd be curious to know. (You are free to point me to some resources if you don't want to explain) And I can see your helix point better now that you mention the scalar triple product. I can't say much more, unfortunately, since I'm getting to the limits of my knowledge. (Thanks for responding, btw)

[@angeldude101]

@@plasticpeepsperformances I recommend looking up Geometric/Clifford Algebra. It includes an operation called the wedge/outer product that acts like the cross product in 3D, but scales to any dimension and gives an object called a bivector which acts as an oriented plane segment. x ^ y = the unit bivector in the xy plane. Since 2D only has one plane available, there's only one unit bivector. 3D has 3 orthogonal unit bivectors and each one has a correspondence to a unique orthogonal unit vector. In 4D, the number of orthogonal unit bivectors reaches 6, which is more than the number of possible vectors since there are 6 ways to combine each of the 4 basis vectors. The sign behavior also becomes rather intuitive since the bivector v ^ u is the one that rotates v to u. u ^ v instead rotates from u to v in the opposite direction. In 3D, there's actually no consensus for which basis bivectors to us, but it's doesn't actually cause a problem since the units are explicit in specifying whether you're working the the right-handed zx plane or the lexicographic xz plane. Since the ordering is explicit in the basis, it only takes a glance to tell if you're going to have to flip the sign. Regarding curl and divergence specifically, Geometric Algebra included the geometric product, which acts as a combination of the inner and outer products. As such, you can do both operations in a single step. Since the magnetic field is formed through the cross product, it can be substituted with its corresponding bivector form (which actually transforms better under reflection than "pseudovectors") and then added to the electric field without the two interfering. Similarly the time derivative can be grouped with the spatial gradient and charge can be grouped with the current (along with some unit conversions) yielding the Geometric Algebra formulation of Maxwell's Equations: ∇F = J. I'm sorry, did I say Maxwell's Equations? I meant Maxwell's _Equation._ This geometric product has an inverse by the way, so you can totally divide by the values shown. And it works in arbitrary dimensions too, though the magnetic field doesn't exist in only 1 dimension (since it requires a plane), so electrostatics are kind of boring there.

[@poisonketchup8098]

holy shit, every explanation of curl was so handwavy and it was just the most annoying thing until i found this video. i subscribed cuz i really hope to see more videos like this.

[@NexusEight]

You didn't mention the number of sets nor reps. Wait, this aint about arm curls?? Great Job on the explanation, btw.

[@gibbogle]

Why do you use R for area, when conventionally the radius is referred to by R?

[@blackimp4987]

I saw just one video simply showing how to directly calculate the rotation of the field as sum of the 3 main planes rotation without involving the concept of circuitation density which is not the same concept thou it formally equals curl

[@a.osethkin55]

Thank you!!!

[@068LAICEPS]

This componente by component way of thinking is probably the way Maxwell derived his formulas. And then the vector calculus was invented and the formulas shrinked into the compact ones that look beautiful, but are not intuitive for the inexperienced eyes.

[@realcygnus]

👍

[@BlackEyedGhost0]

Curl and cross products always annoy me because wedge products do the same thing, but better.

[@TheAzwxecrv]

My field of profession is Physics, so excuse me if I'm wrong - i think there is a serious mistake in ur interpretation. Imagine a scalar f (where there is no necessity for a force "vector f" to even exist) is associated with every point of a curve c.Then line integral of f over c gives the sum of those values of f (and notice f has different values at every point of c). So, curl has no relation to any work, but "just" gives a "value" (or call it "index" if u prefer) for the "power" of that rotation "for the purpose of comparing similar rotations".

[@KaliFissure]

The use of the wave function to describe all physics rather than using the full range of Maxwellian functions (inflow/outflow particularly) has hobbled physics imho.

[@rat_king-]

Its worse thats only planar normalised 3D Curl.

[@engelsteinberg593]

What happens with you?. If you really want to make good mathematics you have to forgot about the stupid cross prodct and use the wedge product.

[@angeldude101]

Oh hi fellow cross product hater! It's really hard to find a vector orthogonal to a plane within 2D isn't it, and it's even harder to find a unique one in 4D or higher. Having the direction made explicit with xy or yx instead of just a sign is also nice.

[@chadliampearcy]

Another cross product hater here. Either use geometric products or wedge products