The Curl of a Vector Field: Measuring Rotation
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- Опубліковано 29 чер 2024
- This video introduces the curl operator from vector calculus, which takes a vector field (like the fluid flow of air in a room) and returns a vector field quantifying how much, and about what direction, the flow is locally rotating, or swirling, at every point. The curl is a fundamental building block in vector calculus, and is used to measure the vorticity in a fluid. A tornado is a region of high curl.
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This video was produced at the University of Washington
%%% CHAPTERS %%%
0:00 Introduction & Overview
5:22 Simple Example
9:07 Interpretation of the Curl
15:02 Intuition for Curl as Solid Body Rotation
24:29 Curl(Grad)=0 and Div(Curl)=0 - Наука та технологія
Be careful at 13:59 since the video is flipped, this is actually showing the "left hand" rule, not the right hand rule. I am actually left handed, so it is extra confusing...
The point I never noticed in previous lectures, it looks u r in the other side of mirror we r watching but not seeing ourselves. Anyway ,great job👍👍👍
"my right hand ..."😀
There's a really easy solution to the ambiguity: simply label the direction as xy or yx instead of z. The two are negatives of each other, but clearly and explicitly indicate which direction they're going rather than relying on a "right-hand rule" or "left-hand rule."
@@angeldude101 You may already know this, but this is what's done in exterior algebra. The cross product is replaced with the exterior product x ^ y. This creates what's called a bivector which lies in the plane of rotation, going in the direction of the first variable to the second, so rotating from x to y in this case. It makes much more sense than invoking a third dimension when working in 2d and having to use your hand to figure out the direction.
@@OmnipotentJC Yup! I mainly come at it from the perspective of geometric algebra, but it still applies to the exterior algebra as well.
We must protect this man at all costs
That's why he teaches from behind a buller proof glass
@@erickgomez7775 hahahahahahha
@@erickgomez7775good one
BROO best playlist I've ever laid my eyes upon
This man is what we need more in math teaching: Someone who thake the time to build the knowledge from the base just to be shure that everyone understand.❤❤❤❤
I have been recently watching your playlists. They are all great. One can see how much you are motivated to teach and your explainations seem to be very intuitive to understand.
I am actually arabic and we study math and physics in french and now am learning about gauss's divergence Theorm and mass con... All of this in english .. but i am still enjoying it
Love the "Ok, Good!" transitions. Keep up the great work prof!
Once again: brilliant video! Thank you for explaining this so clear and concise. Keep up the good work!
this is so on time, I have my Calc 3 final soon and your explanation is just perfect
This man advances my carrier. Thank you so much.
Any SB fan who knows who is the SB of probabilities?
Thanks professor for respecting our minds and talking to us at the intuition level.
Great lecture, professor! As always, you provide clean and concise explanations with examples that help the essence of the mathematical concepts to be understood. Are you intending to cover related topics in physics and engineering as well? In any case, thanks for these high quality videos!
Yes absolutely. We will get to conservation laws soon, and then to PDEs and engineering applications.
By the time Steve said he’s left handed, I finally realised that the video was flipped all the time. That’s a big reveal to me.
Anyway, I always love your videos. Great work and amusing.
He writes from the back of that window. So if not flipped we would see everything written from right to left.
Thank you for your lecture, I like it very very much!
Strongly suspected a fellow south-paw, and mirrored video. Partly the wedding ring but mainly that *no one* is that good at mirror writing! superb presentation. I did basic vector calculus at the Cavendish, but switched to other sciences before the course went on to tensors. It's all flooding back! Thanks. Chris
Thank you for your effort and time. It would be great if you could create a playlist for your videos. Your videos are incredible but finding all the videos related to vector calculus is a bit difficult.
Great video!
Thank you. amazing lecture
This guy is simply the best
Beautiful! thank you!
Never imagined soneone could create this awesome videos on calculus
Thank you!
Thanks for the amazing series!
BTW, the curl in both examples seems to omit multiplication by vector k - which is a bit confusing, because a curl is expected to be a vector rather than a scalar.
Thank you very much for all your work! Really looking forward to your continuation on this topic!
As a somewhat funny side note:
Up to this video I always thought that you, Sir Steve Brunton, were drawing and writing in a mirrored fashion. This few seconds of you seemingly waving with your right hand and saying you are left handed threw me off balance so hard!!
But yeah, now that it's said, it's pretty obvious... Ups :P
Nice! I always have fun with the handedness :)
Professor, Thanks a lot for this lectures. They are really awesome.
Btw, no need to worry because u r a left hander and u need to use the right hand rules. You can verify the direction of the cross product or so without leaving your pen/ marker and use right hand to verify the direction and keep on with your calculations using left hand 😅.
Thank you so much!
At 12:59, curl of (-yi+xj) = 2 should come with basis vector k.
Thanks!
Great lecture! Actually the material isn't new for me. I'm radioelectronic engineer. I studied vector analysis and PDE in the university 10 years ago. And also I used all this maths stuff in electrodynamics applied to radar antenna arrays design. But your explanation is the best! I use your lectures to refresh my knowledge.
Thank you Professor
Thank You, sir !!
Dear Prof Steve,
thanks for the video,
I am interested to know the technology and techniques you are using in presenting and preparing your amazing videos
I would really appreciate your response
Thank you for also using the word "zed" for us Canadian viewers.
thank you so much sir
Steve, thanks again for what and and your team do! I started a machine learning course and for my birthday my friends presented me 2 books (they asked which I want): Deep Learning by Goodfellow, Bengio and Courville and guess what - your Data-Driven Science and Engineering! Both are published in color in Russia, which was a surprise. So far I like your book very much!
That is so nice to hear!!
Love it
Damn Underrated i just hit a gem mine.
@steve Brunton why do you use cross products instead of wedge products when calculating curl?
In case it doesn't go without saying, I think these videos are great. I believe there is a generalization of the curl operator using wedge products of differentials. Would you be willing to do a video covering that?
Thanks, glad you like them! That is a great point, and I'd love to do a video on the wedge operator at some point in the future. Might take a while before I'm back in the studio, but I'll put it on the list!
Does this mean Steve is left handed?
Yes, one step up from here is Differential Forms and Exterior Calculus. Would be great to see it covered by professor Brunton!
If you are curious, there is an excellent series on this by Keenan Crane (application in geometry processing, but very relevant and approachable!)
everything is exactly what I want to learn. Perfect lecture. But could you try to mitigate the scratching sound. That is a little distracting.
Thanks for posting these videos! Are there recommend books that take a similar approach to explaining PDEs?
I really like Vector Calculus by Marsden and Tromba.
@@Eigensteve Great thanks for the response. Going to Amazon now 😊
The MIT OCW, 18.02 Multivariable Calculus, course is really great. The instructor, prof. Denis Auroux is excellent. Vector Calculus is the second part of the course. The course is complete with video lectures and class notes. They use the Vector Calculus part of Thomas' Calculus, I found it easy to understand.
@@laszer100 Thanks for sharing. I know the topic of this video is vector calc but the plan is to tie it in with PDEs and I am curious if any authors or classes take this approach. I am a bit more studied on vector calc but want to take a deeper dive into PDE land
Love from India 😊
I wish my prof actually put in the effort and explained this
13:59
This is where the technology used to record this lecture failed : the in-camera mirroring used to record from the other side of the transparent board conveyed the direct opposite of the the right-hand rule intended by the instructor: The viewer using their right hand on the rotating vector field correctly have their thumb pointing out of the the screen ( toward them) while seeing the instructors thumb pointing INTO the screen.
Good point. I saw the same thing when I processed. I'll make a pinned comment pointing this out.
@@Eigensteve BTW, what's the name/brand of that video system you use to record through the transparent board ?
Thank You
Hello, 23:15， So r is radius, v is velocity, but what is the curl of v in physics?
No longer a student... been a minute since I've done any vector calculus and differential equations. There has to be a connection between divergence/curl and eigenvalue solutions to ODEs... right? (sadddle, sink, source, complex solutions, etc)
Absolutely there is! I will try to explore the connections more in this series on vector calculus and PDEs (should be some other videos in the playlist)
You keep referring to omega cross r as if it were a curl operation through out the second part of this video, it is just a simple cross product.
Agreed. The differential operator “curl” is not yet involved when doing v = omega x r
Yes, you are absolutely right, and this is a good point. This is not the curl, simply a cross product. It is supposed to be giving some "intuition" for how the curl is measuring the solid body rotation of the fluid. Probably could have said that better in the video.
@@Eigensteve Thank you. I love your videos.
@@Eigensteve Excellent video and video series nonetheless! Looking forward to the next installments.
At ~18:40 you say to find the velocity of the point you take the curl of the w vector with the position vector r. Do you mean to say you take the cross product of vectors w and r, or am I missing something?
Good catch -- I meant cross product, not curl.
@@Eigensteve thank you, and excellent content!
Do you write in mirror images or is this a technologically enhanced board? My grandaughter is left handed and writes "Inside out"🙃
12:58 -- why did we get constant (2) instead of vector?
EXCELLENT! 😂
What is the interpretation of the "curl(v)" being 2 times the angular rate "w"? Why is there a "2" in this egality?
12:49, when the teacher asks me for the curl of a swastika lol
What about quaternions? I don't understand why we use half-angle (or full range 4pi instead of 2pi).
5 reasons I can currently think of, in roughly decreasing order of convincingness (to me):
1) 180° ≠ 180° if you rotate around different axes. The only time the rotations converge is when they reason 360°, so the value can't become a pure scalar before then
2) The angle isn't from 0 to 720°, but from -360° to 360°, encoding not just where the rotation ends up, but also the direction it took to get there. In a sense, rotation matrices encode _orientations,_ while quaternions encode _rotations._
3) The rotation can be seen as 2 reflections. When reflecting across two intersecting planes, the end result is a rotation around their intersection by twice the angle between the planes. The half angle used in the exponential is the angle between these two planes, not the angle being rotated by.
4) The coefficient in the exponential is not the angle of rotation, but the area of the sector of the unit circle covered by the rotation, which is always half of the angle enclosing that sector.
5) The Dirac Belt Trick. Turns out rotating 360° even in physical circumstances doesn't always bring you to how you started and it takes a second 360° rotation to return to normal.
In the future I will feed chatGPT 10 Steve's videos and ask it to teach me to dance salsa. Maybe then I could finally learn.
12:57 shouldn't it be a vector: i+j, since the curl of a vector is a vector?
Thank you very much for your Video lectures! One question though, maybe also some other "student" could help me, at 20:30, if I do the cross product, I calculate w*r like (0 0 wk) * (xi yj zk) and get (-wkyj +wkxi 0) but on the video it should be (-wkyi wkxj 0) how is this result achieved, what am i doing wrong?
Same question
If you are writing vectors as tuples of components, you must not implement the basis vectors in addition. Both are different means to express a vector.
In the discussion of the asteroid, either you kept swapping "curl" with "cross product" or I'm missing something.
The point he's making there is that the curl gives you information about how much the field is rotating around the axis defined by the direction of curl(f). And if curl(f) is a constant value, it correspond to the rotation of a solid body (a body that does not change shape as it is rotating). The cross product that he introduces only serves to compute the velocity field of a rotating body, but I find that the exemple helps building an intuition as to why the curl operator gives information about rotation.
This is a good point; I'm really talking about cross product here, but as an analogy for rotation in the fluid. The comment below says this very well.
@@pierrot-baptistelemee-joli820 Thank you -- couldn't have said it better!
How is it possible that there is a link between the determinant and the curl?
To calculate a determinant is purely automatic.
A curl has a physical meaning.
But what is this mysterious analogy betwwen the curl and the determinant?
Stupid question, how do we get v=-wyi+wxj?
Is position a vector?
It can be specified as a vector from some "origin"
Why do you write with your right hand if you’re left-handed?
Great work, just great work. Fix yer pens...
divergence theorem
It would be a cool alternate reality if as well as having left-handed people you literally have left and right handed life, like it had split early on before all the amino acids were locked in end up with two races of intelligent life but based on mutually incompatible biology but end up convergently evolved to look and behave much the same anyway aww
I've wondered if you were really good at writing backwards, but this video is confirmation that the video is flipped haha
India🇮🇳
Your rotation is extremely right wing..
Definitetely not a good explanation.. those fields should be understood only in terms of the geometric characteristics from which those arise, without any introduce of del operator. This kind of explanation, usuallly presented in most situation, still overlooks their deepest meaning, usefullness, and unbeilvable beautifullness…😢