Green's Theorem, explained visually 

Support me on Patreon! patreon.com/vcubingx Join my discord server! discord.gg/Kj8QUZU Divergence Theorem: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-UOG3mOhv5Xo.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=

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.

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

That is Stoke's theorem though. Stokes and Green are the same in 2D anyway.

Elementary: I love math! It’s easy College: I want to die

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 guess there is a mistake.. when I calc curl from the example mentioned (5:39) I get: -y+3(x^2)-6 !?

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.

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.

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!

Wow. This along with the video about the divergence theorem are pure gold! Thanks!

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!

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.

I finally found my new favorite channel

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.

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.

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.

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!

These videos are incredible; keep up the amazing work!

That is where the mobius vector is time placement {p~n}3.14`1

I like this. It's like Vice is giving me a calculus lecture, wonderful!

Awesome video! Can't wait for the next!

The visual approach is so helpful! Keep it up!

Videos like this remind me to visualize like Michael Faraday and crunch analysis like James Maxwell.

Thank you. I can't fully understand this before your video! Universities need you as a lecturer!

Such a great video!! came here from Reddit. Keep it up, please!

Thats stoke's theorem

I love mathematics even it is sometimes hard for me to understand and visualize.

A very nice theorem used often in Electrodynamics, will see it at home after downloading

can you please explain why you specifically took a teardrop shape and not a circle with centre origin for the line integral?

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

why is the calculated curl of 5:41 is x-9? when i calculated it,the result is 3x^2 -y -6

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

5:37 who else noticed that he miscalculated the curl and it should be 3x^2-y-6?

Fantastic explanation, thank you

I did not understand why the curl is a good approximation of the line integral of very small pieces.

Very well done. 💚💚💚 That video cleared all my misconceptions 😍😍😍.

Divergence of vector field is -x-9

Thank you for the video, good stuff here

very helpful

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.

Awesome video! Thank you!

todo muy claro, muchas gracias profesor.

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 :)

Please add a pop for the calculation correction at 5:40. The error worsens an otherwise great video.

It's good, keep it up my man.

1:47 the Suggestion is very good.

you're the goat, keep this shit up, this helped so much

I got the intuition, really good bro

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.

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

Isn’t del X F a vector output? Where’d he get x-9 from

I think it is stokes theorem

will you do a video like this for stokes' theorem?

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

are you sure that you calculate the curl of F correctly??? i think there's a mistake

Good effort.. sir. Thank you so much..

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.

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

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?

greens theorem becomes so simple after watching this video, damn

I have look for your correction but still got difference curl that - 9 + x , is that true?

Just me or is there a mistake in the computation of curl(F)? isn't it supposed to be 3x^2-y-6?

There might be something wrong with the curl calculated at 5:40 .

Is implicit differentiation valid in the condition for Greens theorem?

Can u make physical interpretation for residues and singularity in complex analysis

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.

I'm a bit confused about 2 dimension cross products in the example at the end. Why is it giving us a scalar?

Bing Gpt-4 Approves this.

Thank you.

Thx dude

Awesome!

thank you sir....

I can do the traslantion to spanish... ! Great Video and chanel...! Saludos desde México...

3:23 to summarize

Thanks dude

what library are you using for the animations

How you make the letters pop up one by one

Can you upload the Manim codes??

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

I have learn from Wikipedia that this is a stoke's theorem(curl theorem)

Can u provide a proof for this?

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.

Yes

Great!

Awesome

Loved this video! It helped me a lot to understand vector calculus!

Wow, nicely explained. Comparable to 3blue1brown.

A Christmas themed ad in July?

Either wrong name for the theorem or shoddy notation on the double integral, but good video otherwise

title of music, please?

1:00

WHEN NEW VIDEO

Please share code on github

But I think it's called stroke theorem

It's green-reimann

Voice crack at 3:45

The computation of the curl is wrong!!

I can't hear you even with sound turned up all the way.