Stokes' Theorem and Green's Theorem
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- čas přidán 15. 06. 2024
- Stokes' theorem is an extremely powerful result in mathematical physics. It allows us to quantify how much a vector field is circulating or rotating, based on the integral of the curl.
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This video was produced at the University of Washington
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0:00 Stoke's Theorem Overview
6:59 Green's Theorem
12:22 Geometric Explanation
16:30 Examples
18:45 Green's Theorem to Compute Land Areas - Věda a technologie
This channel is a gold mine. Thank you!
It is *quite* the gem.
100/ true
I am a gold digger
I wish I had such a professor as you, sir, during my vector analysis course. Thank you for the amazing work, now I feel like I can fill in some gaps ;)
Me too, I hate that shitty profs are even allowed to teach. Just let them do research. Excellent profs like Dr. Steve Brunton do far more for STEM in general; Mathematics in particular, than all the pushing of STEM in the media. I majored in Applied Mathematics and Biology, and I noted how many students dropped out of the harder more challenging classes in part due to profs not giving a damn whether the students grasped the material or not.
TLDR; great videos, love your stuff please keep doing what you're doing.
Everything in this video is perfect, except the "thank you" at the end. It was such an amazing lecture, and I found myself saying "What?! No, thank YOU!"
This is easily my new favorite channel. Content is interesting and very well explained. Thanks!
Thank God for this man who knows how to make the complex simple. The textbook fr makes this way more confusing than it has to be.
This series has been great. Steve, is such an effective teacher.
This guy has a great knack of connecting a lot of seemingly unrelated concepts
Great video professor Brunton. Your teaching skills are immaculate
Great content. Your lessons are grounded with intuition and context. I usually end up teaching myself the math courses I've taken, because the intuition component is missing. Thank you!
...I have the sense that you've been waiting a long time to use the Get Stoked! pun :D Well, it got me to engage, so take your upvote ;)
I just bought a copy of your book the other week, and I'm looking forward to diving into it along with your lectures here. I haven't gotten a chance to watch this video yet, I've been meaning to say thank you for posting every video that you do. I'm a software engineer that really didn't fit in academia, and had learned a good chunk of the prerequisites for all of your material when teaching myself the math needed for physically based rendering. I started to casually branch into machine learning to satiate my curiosity and was surprised how much of the math I'd learned was transferrable. Your videos have really been elucidating, and have helped me deepen my understanding greatly. You're a great educator, and I really thankful that you've decided to share your knowledge so openly. The format works really well for me, and having the visual aids while we're able to see your face really helps me stay engaged.
All to say: Thank you, and please keep the content coming!
Lol, believe it or not, I just came up with that the day before I released it. Thanks for buying the book -- I hope you enjoy it!!
Thank you soo much for such an insightful conceptual explanation! Highly appreciated the lecture!
Please never stop making videos!
Never stop never stopping
Awesome lesson… when I was studying Analysis II all these theorems were so hard to understand, they were taken “as is”, and seemed so mysterious.
Great series, Dr. Brunton. Thank you!
Awesome, the professor is able to explain complex concepts into simple way.thanks
Amazingly described... Thank you
Thank you so much! I couldn't have wished for a better explanation!
This ability to tour complex ideas into easy ones is awesome ..... and also, this is the best way to learn
Stoke's Theorem (And Green's too) were the kind of theorem I just accepted as a bizarre true of the Universe when I first met them in the undergraduate course. Nowadays I understand them a little bit better. Watching prof. Button's class reminded me I still find both theorems incredible and some how magic, almost a supernatural mathematical property of the universe. The fact integrals of both sides of the equation are equal stills amuses me.
Professors in our college just write the formula without explaining much. Everything is marks oriented.
Lots of that comes from students, though. Many push back when expected to understand principles, unfortunately. Probably heritage of the school system
I am doing my computer engineering degree and your playlist not only help me understand vectorial derivative, but also make want to make my own vectorial field processing program on my server. Keep going ❤❤
Thank you math bro. This is what I needed in my life. Mostly Green's theorem, I should have paid attention to that a bit more in EM.
Never disapointed... amazing explanation, as alwaya
his teaching is just wow💛
Cool stuff! Great explanation! 😂 I'm watching this along with your series on complex variables!
Thank you, Doc. B!
This is a great explanation! I like how just in the way of explaining and demonstrating with the grid on the kidney bean shape, one can draw similarities between Gauss' Divergence Theorem and Green's Theorem just changing operator. It is also a visual representation that does not require mnemonics at all!
I have as well a question:
Is there a way to compute surface areas in 3d using Stokes Theorem? Which field should we apply?
Thanks a lot for this... I used to be apathetic towards calculus in my ug course. This makes it so interesting.
I loved the trick for computing the area!
I just that lecture on Stokes and Green's theorem yesterday
Very well explained
This is understandable thanks professor
Thank you professor Steve
this is the best content on the internet
Great lecture! I find interesting is that both Stokes theorem and Gauss theorem have their own versions of the green theorem that is one dimension lower than them
Beautiful lecture. I am a UW student in mechanical engineering, and this is awesome. However, I was slightly confused where the F = [ -y, x] come from in the example
Thanks. I mean for all of us could be every interesting to hear about fractal derivative and practices using fractal derivative) Thanks.
Amazing better than books and papers i hope to be like you in one day.
very must crystal clear explation
I like your videos a lot. May I know what kind of tools/software do you use to make your video?
just realized you were writing backwards the whole time... amazing
Loved the thumbnail😁
This series is fantastic. Since you mentioned potential flow for the next video, I would love to hear a bit about the Helmholtz decomposition and how its potentials relate to the flow of vector fields. Polthier and Preuß (2003; Identifying Vector Field Singularities Using a
Discrete Hodge Decomposition) suggested a method for identifying singularities in flow fields by finding the local extrema in the Helmholtz decomposition potentials. However, I don't understand why these potentials are more informative than the local extrema in the divergence and curl of the same field?
Thanks, and great questions. Hopefully the next couple of videos start to address these.
I'm very much looking forward to those! Especially the last question is a mystery to me. 🤔
This guy really know how to teach respect❤
Thanks!
Old video so I doubt I'll get a response, but I don't understand the justification for 20:39 when the vector field [-y x] is introduced. it feels arbitrary to make the equation work, but I am really interested in this relationship between the rotation of a closed curve and the area it encloses!!
Although the notation gets heavier, I find Stokes Theorem, and differential forms at large much easier to contemplate when I take the initiative to control the limit implicit in the definition of an integral. Hopefully someone groks this and it strengthens their understanding! Stokes Theorem is close to my heart, as is Fourier’s.
Thanks for your intuition on this -- always helpful to see things from a variety of perspectives!
I thought stokes and gasses theorem reduces to the same thing in differential forms
@@Martin-iw1ll yea p much. That's what makes differential forms op.
I love this series! Just a question about the graphical intuition for the curls cancelling - it doesn't seem obvious to me that the curls in the neighbouring cells will be equal and hence cancel. I have one idea to answer my own question. These areas are infinitesimal and hence so near to each other that neighbouring curls are the same because they are essentially "in the same place". Is this correct? If not, please would you explain!
i think you can see the arrows canceling each other out. in the middle of the surface its obvious.
then the arrows on the upper side of the surface get canceled by the arrows on the lower side of the surface
i think.
Steve: you wrote:
A=(1/2)*integral(x dy - y dx), and that is true.
But it is also true:
A = integral(x dy) = -integral(y dx), so basically you are computing the area two times and then dividing by 2.
Nice video, congratulations.
Thanks Prof. Brunton. May I ask you why are the last two videos not visible?
Gracias.
just a little complement, if u wonder why outer product can be calculated by determinant, you can check a concept called wedge product.
Excellent presentation! Can you connect with a mechanical device called a "Planimeter"? It measures area by walking around the perimeter. Still useful even today!
Will you also do line/surface integrals?
Why the vector field is [-y x] for the area calculation of the irregular surface ? Love your lectures ❤
I am the first viewer of this lecture.
Thank you
It's cool to think about how this shows that there is a "balance" between weather in the northern and southern hemispheres
What happens when The Fluid has curls in different direction because then they add up for example when there ist a positiv curl and left from it one with negative curl. They would add up in the middle and not cancel each other out so that greens or strokes equation are no longer useable.
The mental image that Stoke's theorem conjures for me is imagining if one day, all the wind on Earth blew from the west to the east, at least along the equator. That motivates thinking of the surface integral half as a "net" or "average" curl over the surface, and in this example the "net" curl of the northern hemisphere has to sync up with all the wind on the equator blowing in the same direction.
This mental image is intuitive for me in understanding the physical consequences and reasoning behind Stokes' theorem (assuming I understand it correctly hehe)
Since the equator integral represents the total “swirl” or curl in the northern hemisphere, does Stoke’s theorem mean that the total curl in the northern hemisphere always equals the total curl in the southern hemisphere? … and if a contour is confined to the border of a hurricane, then that integral says the total curl in the storm equals the total curl in the rest of the world?
is there any reason as to why the vector field F for computing acres of land is ? thanks
Sir I understood about curl of the vector but why is it dotted with da???
Pl tell me
I'm studying mechanical engineering, in what course will i learn and use this formulas ?
@ 1:49 This is a 3D-vector, not a 2D-vector as stated here.
Legend ❤️
I don't get how to find out the area of the irregular shape property by walking around it and counting the # of steps on y axis and x axis. Can anybody help me? Thank you very much.
amazing
Thanks!
I have a question. Green theorem is still valid even that the shape has those spikes like in the last example? That does not makes issues due to the nondifferentiability?
late on commenting so i doubt you’ll need this but it’s because it’s considered “peace wise” smooth
I have to say that the way you draw the different curls of the boxes at 13:00 is kind of confusing. The way it's drawn suggests that the vector field changes direction at the cell boundaries. This is of course not the case in a continuous vector field. I had the same issue with the video on Gauss' divergence theorem.
I think a better way to make the concept intuitive would be to say that the vector field has a certain direction at the cell boundary, and explain how this would contribute to curl of opposing sign in the two cells.
Interesting point... I'll think about how to make a nice visualization of this.
Yeah I also felt the same...If all the curls calculated have positive direction as shown in the video,then it would all add up and no cancellation takes place...Why I think the cancellation takes place is that positive curl in one cell forces to have negative curl in the adjacent cell...I also felt the same for divergence theorem(I commented there also)
@@ahammedafzal7797 Well, it's good to realise that curl/vorticity is itself a continuous vector field. That means that its component in the direction normal to S can be positive over a finite portion of S. And since the division of S into cells is arbitrary, it is possible that adjacent cells have a positive value of the integral of vorticity over their area.
The value of this integral over S is equal to the sum of the integrals over the individual cells. In this sense you are right to say that the terms add up.
By applying Stokes' theorem you transform the area integrals over the cells to line integrals over the cell boundaries. The cancellation applies to the line integrals over the interfaces between cells, so that only the line integral over the boundary of S remains.
With this in mind, the arrows drawn in the video do not represent the vector field itself, but in some sense the values of the line integrals over each segment of cell boundary.
Can you do the K.epselon model
Awesome
Can we just say Stokes theorem is in 3D but Green theorem is in 2D? They are indeed the same equation. In addition, what they really do is to connect the surface to the line integral for us to solve problems.
great great
Can't you just find the area of a hypocycloid by subtracting off a circle who's radius is the same as half the side length of the related square?
damn, i have started thinking about electron flow or magnetic field like wind, that can swirl or curl.
Is he left handed?
What is the blob?
When I started watching this video I had a small headache. When I finished I had a bigger one. I guess it just seems like your are throwing away a great deal of information about the dynamics of the field by only measuring its impact on the border. When there is a hurricane approaching land, I don't really care how much its impacting the wind down in the equator; I want to know if I'm still going to have a roof on my house. Will hope further examples further explain when these aggregates are more valuable than the details of the field itself.
Are you writing backwards? Or is the video flipped once you finished recording?
The video is flipped, he is using a lightboard. Since he is left handed(you can check his older videos), it looks like he is writing normally to us.
the planimeter is an old mechanical device that is said to measure area using green's theorem. it seems earth is flat enough for green's theorem to work, haha.
Magnificent from india
what blows my mind is the fact that he has to mirror everything he writes
how do you make this video? You are opposite to the plane I'm perceiving. Are you writing everything backwards. I wonder if non-calc3 matriculates would even notice this.
is there a mirror involve?
oh, yeah in your editing software, of course
😊 you said a bean
Inner criteria equaling 1 to an outer of 0
Life was considered strange
Considering to its Masters hand not the slave of the lip
👍🏼
Am I the only one who thinks the volume is far too low?
Sorry about that... trying to fix this in the studio, but keep forgetting to bump it up in post...
Is he writing from the back or I'm i seeing things
Should admit that his hands are magic
Derivative is the opposite of the boundary 😵😵😵
Where are you from?
@@innfdtfjord3340 Russia
You learned Stoke's Theorem in high school?
Awesome 😂
im about to get stroke ._.
Second viewer
Your videos help me explore ideas related to knowledge space. Thanks for refreshing my calculus. czcams.com/video/0veCf_03yhM/video.html
Meh, his lecture ambiance is not my cup of tea… it seems this one is of a series with foundational concepts introduced earlier, for engineering students maybe
Horse shit, a length is the equal to a surface area. Why do you omit the units of the integrals?
This is understandable thanks professor
This is understandable thanks professor
Glad it helped!