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- เผยแพร่เมื่อ 14 ต.ค. 2024
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This video aims to introduce green's theorem, which relates a line integral with a double integral.
Line Integrals: • The Line Integral, A V...
Divergence and Curl by 3b1b: • Divergence and curl: ...
For the example at 5:07, the equation of the vector field has the x equation and y equation flipped. It should be F=(-yx + x^3, 6y-9x).
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Divergence Theorem: th-cam.com/video/UOG3mOhv5Xo/w-d-xo.html
CORRECTIONS
For the example at 5:07, the equation of the vector field has the x equation and y equation flipped. It should be F=
Saw your pinned comment here after I could not quite believe your Curl(F) result, and got -y+2x^2-6 trying to calculate it myself... There is another little inconsistency you have here, at least in nomenclature: The result of Curl(F) is a vector field itself. On the left you have a dot product of F and dr, which becomes a scalar. On the right, you need to make dA a vector (i.e. represented as a normal vector on the area element), and then use a dot product with the curl vector to make the right side a scalar also. In 2D you only have a z-component as the result of the curl operation (i.e. "as if" it were 3D, with z component of F being zero). That z-only result of the curl, dotted with the dA vector then becomes a simple product of numbers, essentially...
@@PeterBaumgart1a You're right. In fact, I realized this mistake after rendering the video, but I decided to not fix it as it would take another hour or so to render everything else with the dot product. I left it in because it does not hinder the understanding of the topic, but I appreciate you pointing this out!
@@vcubingx So how about a video about the Green's Function now, maybe even showing how it's connected to Green's Theorem? That'd be quite awesome, if you could pull it off! Do you have ideas how to make that leap and into the corresponding graphics?
@@PeterBaumgart1a I'm actually not very familiar with Greens Function and it's applications, gonna read up on it first before making a video. I appreciate the suggestion though!
Do you have a stutter or are poor at reading?
Keep it up dude. The more videos like this, the less people will struggle with math and the more people will learn to enjoy it.
Thank you very much!
And hence, there would be more scientists, and hence, more developement in science, and hence, more enlightenment, and hence, closer to the nature
...
Wow, that's escalating
@@maxwellsequation4887huh
Excellent video! For a field F=, my understanding is that the 2D curl = dQ/dx - dp/dy . For this field, 2Dcurl = 3x^2 - y - 6
:)
👍💕
Can't like this enough
same here man,,, idk how he got the last one correct
... (curl of a vector is always a vector)
exactly
I've taken a class on advanced engineering mathematics that was heavily focused on vector spaces and green's theorem etc. a year ago. I passed the class. But now I'm looking at this video and wondering, how the hell did I actually pass I do not know any of this. So I will watch your videos to actually learn the subjects this time Thanks a lot for the quality videos !
I REALLY hope I can take an engineering mathematics course sometime in my college career. Looks like amazing stuff.
That is Stoke's theorem though. Stokes and Green are the same in 2D anyway.
Regarding 2:48 adding the line integral around a closed curve; a planimeter is a mechanical instrument that is used to trace around the perimeter of a closed curve. What a planimter does is to calculate the area inside of the closed curve based on the principle of Green's Theorem.
Elementary: I love math! It’s easy
College: I want to die
high school: omg it's sooo dumb i want to learn advanced stuff this is too easy
doing a physics major: yaayy advanced math so interesting and amazing
But not really. Elementary school math is very dry and boring. Even if you find it easy you just straight up can't find it interesting.
College is where you should start to appreciate math, as you now have context to work with - applications that give background to the material. Before college is rote memorization with very little intuition
Hey dude, amazing channel, keep coming with these!
Just a constructive criticism, if you don't mind. When you explained the cutting the section in two parts, it'd be cool to go in a bit more detail on why it works (I'm guessing it's because the line integral of a small square approaches its curl as it gets smaller, and the fact that since the line integral cancels inside, the curl also cancels, leaving only the “outside curl”, so, the line integral... but I don't really know, just a guess), as, at least for me, it was not clear why the curl of the small pieces should approximate its line integral (it's understandable that the sum of their line integrals would do it, since they cancel, but it only makes sense for the curl to approximate as well if the curl approximates the line integral for those small pieces)
But other than that, great video man, hard to find such intuitive and understandable explanations out there, people like you make math much easier and fun, keep it up!
Just in case anyone is interested, he has a newer video on the divergence theorem, that follows the same ideia, and there, he nails it.
This channel's gonna be big, just wait and see 👀
Got it. I read this comment as I was writing my script for the next video, so I'm gonna make sure everything is detailed and clear. Thanks for watching and commenting!
@@luiz00estilo same I've facing problem it's mind boggling that approximation part! He should have explained in detail but thanks for your explaination man!
I guess there is a mistake.. when I calc curl from the example mentioned (5:39) I get: -y+3(x^2)-6 !?
I think you're right
yeah dude, i calculated it myself and also got the same answer as you. Initially I thought I had made a mistake so I went to this link: www.geogebra.org/m/jWfTBWWT and set : F1=6y - 9x, F2=-y x + x³ and F3=0 and it spat out the same result. I do think there's a mistake in the video.
Yeah there's a mistake. The equation of the vector field is supposed to be F=. My bad. I mentioned the mistake in a pinned comment.
Yeah I saw that too. I was so confused until I saw you had commented about it
Off! I thought that i was the only one to get this solution 😅 thanks!
Very nice visualization. Your explanation "clicked" for me at 2:38 :) Finally a video where they actually explain what it is rather than just apply it. You just earned yourself a subscriber.
Thank you!
Hope u will keep making such videos. I believe that learning math visually is much better for applied mathematicians, engineers, and physicists because those math practitioners needs to know what that math does, not what is its definition!
Wow. This along with the video about the divergence theorem are pure gold! Thanks!
That is where the mobius vector is time placement {p~n}3.14`1
Thank you, I had never considered that the line integral of a surface is the integral of the curl of an infinite amount of areas approximating the surface.
I don't remember who made this analogy of Green's theorem, but it stuck with me: Imagine laying out the whole of the Sunday New York Times on the floor of a gymnasium. By reading every word on the edges, you know the content of the whole newspaper. Not a perfect analogy, but it conveys the profound nature of this theorem.
I’m glad it works for you, but it’s so wrong!!
@@ironsideeve2955Absolutely savage, but you are right
I would add that a physical way of thinking about this theorem, is that if you know the in and outflows at the edge of your area, the flow and curliness on the inside is basically already determined.
I finally found my new favorite channel
Try 3Blue1Brown as well; he uses the same art tools, similar style, and I think had a lot more history doing this.
Just found this channel randomly. I’ve been trying to learn different math visualization software like manim because this looks amazing. I absolutely love this video and your explanations!
Thank you!
Your visual proof/demonstration of Green's theorem was cool, but I don't think that knowing "curl measures how much the vector field rotates about a point" is enough to conclude the result at 4:05 for each small square. Also, shouldn't |r| on the right side of the equation be delta A so that when we take the sum of all the pieces, we get a double integral with respect to dA? Since curl(F) and the distance of the square from the origin |r| are both finite, the right side would blow up to infinity when we sum all the infinitessimal pieces together.
Yeah to me the explanation didn't make much sense
I don’t understand this part either.
These videos are incredible; keep up the amazing work!
I like this. It's like Vice is giving me a calculus lecture, wonderful!
Thank you. I can't fully understand this before your video!
Universities need you as a lecturer!
The visual approach is so helpful! Keep it up!
Such a great video!! came here from Reddit. Keep it up, please!
Videos like this remind me to visualize like Michael Faraday and crunch analysis like James Maxwell.
can you please explain why you specifically took a teardrop shape and not a circle with centre origin for the line integral?
Awesome video! Can't wait for the next!
Thank you!
A very nice theorem used often in Electrodynamics, will see it at home after downloading
The transition at 2:50 doesn't really make sense to me. Are those rectangles what we are splitting up our curve into? Also, how does a line integral correspond to rotation of a vector field (as is said shortly after 2:50)? I'll check out the articles and your video on line integrals to try and dig deeper though :)
I am amazed by ur so cleared concepts. Ur are great dude
I love mathematics even it is sometimes hard for me to understand and visualize.
It would be nice if you said at around 4:00 that in adding the microscopic curls together, the edges of the interior bits cancel each other out and only the macroscopic edge of the curve stays, or something like that.
th-cam.com/video/Hy9PMkJ0DME/w-d-xo.html
why is the calculated curl of 5:41 is x-9?
when i calculated it,the result is 3x^2 -y -6
im getting the same. I was wondering if Im the only one who got what you’re getting
@@TheReemkareem Ive confirmed this too. I think he made a mistake in the video. he calculated the divergence, not the curl LMAO.
This is so beautiful and elegant. I don't understand anything, but I had a lot of fun trying so I'd say it's a win
Haha, I'm here to answer questions/clear doubts/redirect you to places to learn (also in description!) if you need it
I suppose vector analysis is more interesting if you are studying physics. For me, as a first year maths student, it was just something that we did and we never really developed any deeper intuition for it. (It should be noted that we covered the subject through video lectures during the corona pandemic, so that could have something to do with it). Many proofs and definitions were dodgy and avoided important details. At least that was my experience of the vector analysis part of my multivariable analysis introductory course.
Here's an explanation I came to with regards to Green's theorem.
When you take a line integral with respect to ds, of a vector field F, you're finding the integral of F•ds.
Since dot product gives you how much the field F moves with the direction you go around the curve, this makes sense that if all the components of F were along the closed curve, you'd 'spin' the curve a lot. This is the net circulation.
Curl's magnitude gives you how much a thing spins. More curl magnitude = more spin. The negative and positive is the direction in relation to right hand rule, so really it's just a mathematical agreement everyone makes on how to orient spinning things.
Taking the integral of curl(F)dA is adding up the curl of every tiny area of your simple region.
Green's theorem means that adding up all the curls of your tiny areas in your bigger region is the same as finding how much a vector field will spin the outside of your big simple region.
It's like measuring how much a plank in water is spinning by figuring out how much each point on it spins and adding it up vs looking at how much the outline is being spun
I screenshots your answer
I understand the idea, but I have some problems with the physical meaning or importance, for instance, the circulation in a fluid flow I can see that the curl v measures the rotation around each point, but what if I want to apply the theorem backwards (i.e double integral into line integral), what are we measuring if we sum up all the tangential components of the velocity around the contour.
Great video btw
I found your videos extremely helpful as having a visual representation makes everything much more intuitive. Is it possible hat you can make a video about Stokes Theorem Pleeeeeaaaasss :)
1:47 the Suggestion is very good.
Very well done. 💚💚💚 That video cleared all my misconceptions 😍😍😍.
greens theorem becomes so simple after watching this video, damn
Please add a pop for the calculation correction at 5:40. The error worsens an otherwise great video.
I wish I could, but TH-cam has long removed that feature
Awesome video! Thank you!
will you do a video like this for stokes' theorem?
Maybe a small footnote, but it's extremely similar to this! Just in three dimensions :)
I think that the product of the gradient and the vectorial function is wrong... Because you have to do the partial derivates of x in the second term of the vectorial function minus the partial derivate of the first term.
Thank you for the video, good stuff here
Fantastic explanation, thank you
Thank you very much :)
what library are you using for the animations
Thats stoke's theorem
I did not understand why the curl is a good approximation of the line integral of very small pieces.
The line integral of a small piece measures the circulation of that piece. Curl also measures circulation, but for a point. As the piece becomes smaller and smaller, the line integral over that piece becomes closer and closer to the curl, since the smaller piece converges towards a point.
How you make the letters pop up one by one
There might be something wrong with the curl calculated at 5:40 .
You represent the force vectors in field with vertor of F. You need to describe about calculation for average dimensional force in specific area as you integral over X and Y. In average magnetic field will have radius dimensions BUT you don't need to do calculation for all possible axis excepted you need to do. Guess what? One source steady will provide average magnetic field or you need to stick place it together with other magnets. 🧲
Don't forgot about tourge.
Divergence of vector field is -x-9
It's good, keep it up my man.
Good effort.. sir.
Thank you so much..
Is implicit differentiation valid in the condition for Greens theorem?
Isn’t del X F a vector output? Where’d he get x-9 from
🤣🤣🤣
its the divergence
todo muy claro, muchas gracias profesor.
I got the intuition, really good bro
5:37 who else noticed that he miscalculated the curl and it should be 3x^2-y-6?
You're right sorry about that - check the pinned comment for the correction
But what makes u conclude that a positive rotation will get u a positive line integral aswell in the first hand? Ah and is D the border of R?
Can u make physical interpretation for residues and singularity in complex analysis
I'm a bit confused about 2 dimension cross products in the example at the end. Why is it giving us a scalar?
When we take the curl of a function that has only two outputs (say 3 with [x, y, 0]), we get a vector that looks like [0, 0, Z] (try the cross product yourself!). So, what we do is just consider it to be one dimensional. When we move to three dimensions i.e. stokes theorem, you see that del cross F is dotted with dS
I have look for your correction but still got difference curl that - 9 + x , is that true?
very helpful
A Christmas themed ad in July?
you're the goat, keep this shit up, this helped so much
Question...We know the curl will be proportional to the line integral over a very small region but how do we know they are equal? You didn't prove that fact
Well think about it. The line integral measures the rotation around a curve. The curl measures the rotation of a point. As the region gets smaller and smaller, the curve approaches a point, which is why the line integral and divergence are equal.
@@vcubingx Yes, as the region gets smaller and smaller, the curve approaches a point. But how you did you know the Curl F does the same thing like the line integral.
This is the best video that explain it in the internet so far. It will be better if you explain 4:07 deeper.
Sorry if I have bad english.
a good example on why cool animations don't always compensate the lack of in depth explanation. At least Khan Academy's and Dr Trefor's video explain how to arrive at the formulas instead of just spitting them out
Check out "Vector Calculus" ~ Marsden & Tromba
www.macmillanlearning.com/college/us/product/Vector-Calculus/p/1429215089
We used the Second Edition when I took this course from Tony Tromba at UC Santa Cruz in the early 80s; Chapter 7 "Vector Analysis" has a section on "Applications to Physics and Differential Equations" which gives a detailed presentation on constructing Green Functions as solutions to boundary-value problems.
The current 6th Edition has a different layout.
Just me or is there a mistake in the computation of curl(F)? isn't it supposed to be 3x^2-y-6?
i agree
I have learn from Wikipedia that this is a stoke's theorem(curl theorem)
Loved this video! It helped me a lot to understand vector calculus!
Thanks!
I think it is stokes theorem
Wow, nicely explained. Comparable to 3blue1brown.
Thank you!
Can you upload the Manim codes??
github.com/3b1b/manim
They are already avaiable at GitHub. Grant himself has shared it.
Thank you.
I can do the traslantion to spanish... ! Great Video and chanel...! Saludos desde México...
Go ahead! I'll approve it. Thank you so much :)
Bing Gpt-4 Approves this.
Thanks dude
title of music, please?
thank you sir....
Awesome!
Thx dude
Yes
are you sure that you calculate the curl of F correctly??? i think there's a mistake
Yeah, there's a small mistake
3:23 to summarize
Can u provide a proof for this?
Great!
Awesome
It's green-reimann
Voice crack at 3:45
But I think it's called stroke theorem
The computation of the curl is wrong!!
Either wrong name for the theorem or shoddy notation on the double integral, but good video otherwise
Please share code on github
1:00
WHEN NEW VIDEO
you handwaved the part of why the curl is equal to the line integral which is actually the only important thing in the whole video then spent the rest of the time trying to explain a limit concept. what, man, what.