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Curl, Circulation, and Green's Theorem // Vector Calculus
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- Опубліковано 7 сер 2024
- his video is all about Green's Theorem, or at least the first of two Green's Theorem sometimes called the curl, circulation, or tangential form. Consider a smooth, simple, closed curve that encloses a region in the 2D plane, together with a Vector Field. One thing we could do is compute the circulation along that curve, which would be a large-scale or global property. Separately, at any point in the enclosed region we could compute the circulation density or curl at that point, which is a small-scale or local property. The power of Green's Theorem is that it relates these two concepts. The circulation or line integral along the curve (i.e. which only depends thus on the boundary of the region) is equal to the double integral over the entire region of the circulation density. Amazing!
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The reason behind this channel being underrated is..that most of the students just study for marks not for the concepts (schools has made them like this) .....there are very few people who look for intuition of the concept...and sir u are a blessing for us...love from india🇮🇳
Same here,yeh banda sahi hai
👍
I agree. If I dont understand or AT LEAST have an idea what is happening behind all of these math formulas, I feel like I am not really learning nor understanding the topics
I study for marks but my dry memorization sucks. Therefore I am here learning the fundemantals lol.
That why I like Indians. Hong Kong people always memorizing stuffs but don't want to understand it
For me a physics student, this channel I just found is a goldmine...
You can´t imagine how poorly math is lectured by our professors.
Thank you and all the other great youtube channels like 3blue1brown!!
My undergrad was in physics so I always like when physics students find me:D
you're a wizard man, these videos are so clear I find myself knowing the next sentence sometimes before you even said it. Thank you.
Great lecture again - I used to think Green's theorem was difficult - you just made it easy!
Undoubtedly the best educational math channel on UA-cam.
I finally understand the intuition behind all of formulas in my calc lectures, makes it a million times more interesting (and MUCH easier to remember)!
Thanks for the amazing content!
Thank you so much!
The moment when you mentioned the relationship between Integration as the area calculation and yet determining something which is just confined to the boundary kinda made me pause the video and think for a few minutes! A hell of an insight there.
That's because Stokes theorem generalizes FT of Calculus. The "closed curve" in that case is an interval of the real line
Best explanation of all UA-cam videos on circulation in a very small area. congrats. After this video, line integral concept is much easier. You articulate well and presentation sequence is very logical and understandable.
You are the greatest teacher of all time with the amazing graphical representations and concepts!!!
#ULTRA_LEGEND_OF_MATHEMATICS♥️♥️😊😊
These are the best math videos on the internet. Very good for studying for math exams. I'd be happy though if there was a good stochastics lecture for undergrad.
Eloquent and really great intuitive professor thank u!
brilliant analogy to the fundamental theorem of calculus in the end. thank you 😊
The last misconception mentioned in the video was totally my confusion! Thx for solving this problem, and now Im really clear whats green theorem is talking about! Great video!
The best ever channel to learn vector calculus....
You have passion for what you do.
This is why I like your approach- visual and intellectual
Well well well, i was waiting for this video. Thanks Dr.
Hope you enjoyed it!
It makes sense how the middle circulation impacts the outer. Compare it to water moving in a circle, if you begin stirring in the opposite direction inside the circle, if would affect the inner flow. Question for myself: The left part of the equation is the circulation around the edge, while the right is the circulation in the middle (as well as on the edge). Why are they the same? Must be because it's not circulation in the middle, but circulation density, which is how much it circulates in a given area. Times it by the size of that area and you only get the circulation. The definition of circulation is "The amount of force that pushes along a closed boundary or path". It's the total 'push' you get when going along a path, such as a circle. So by computing all the small spinning propellers inside an area, you can find the force that's exerted at only the edge of that area. I assume the same way you could change the area, and through knowing the circulation density, you could predict the force needed to go through that line. Thus, is you know the circulation density anywhere, you can calculate the force needed to transverse any simple and closed path.
your lectures are just awesome.
This is absolutely the best explanation of Green's Theorem
I have to say, this was absolutely amazing!!! That last connection to FTC at the end was so beautiful I could've cried; that connection between activity at the boundary and inside the boundary seemed a bit less abstract than before. One question: Since this is a double integral with a function of x and y inside the integrand, does that mean that we are technically doing a volume integral? Or, even if we are, would we really be interpreting that number that we get as a volume? Thank you!
Thanks....your playlist s helping me developed much needed intuition.
These videos are pure gold. The derivations and intuition are top notch
Thanks so much!
Best explanation ever. Thank you so much 🙏
Fantastic, thank you.
I remember watching this for the first time during my calc 3 time. I hadn't seen a more perfect and easy to understand explanation than this. Being able to visualize calculus makes it so much more fun. Coming back and rewatching it now makes it all nostalgic. Thank you Dr Bazett!
Great job sir.
Sir thank you so much for this amazing video💖
You're a legend. A math god.
Thank you sir for again making maths interesting for me❤
Love from India 🇮🇳
This explanation was amazing, thank you!
Glad it was helpful!
that was really helpful. Thanks a lot ❤
Excellent💯💯
THIS VIDEO IS GOATED!
Thank you ❤
Here is what I'm looking for. Thanks 🙏
I fell good with vector calculus thank you
Masterful.
Sir u are my life saver. love from india
Thank you very much :)
Informative and interesting class.
Glad it was helpful!
Why this channel has that small number of views? It's great.
thank you so much!
You are a legend
I think subtitles should be added to the video, even though the subtitles are automatically generated
Awessooommeee!!!
Best Math Teacher Ever
Thankyou
Without a doubt you’re making me enjoy my Calc 3 course, even though I’ve been having a bit of a rough time.
I'm sorry to hear that but happy I could help:)
@@DrTrefor I made it through! Thanks for your videos!!!
Thank you math man
You reignited my love for math
I’m so glad!
I missed so much of this my first time through calculus
I'm currently doing my PhD and deal with Stokes' theorem a lot. Particularly using partial integration and product rule on Stokes' theorem to regularize certain singular integrals in Boundary Element Method. Would love for some discussion and exchanges with you :)
Thanks, that was a nice video lecture-but the unintuitive scenario you describe at the end begs me to try and 'disprove' it by way of a counterexample. When I can't disprove it, I'll be satisfied. To the whiteboard!
but in green circulation theorm- when we integrate sum of all curls on dA... (curlF). k̂ dA....then dot product of two perpendicular vector should be 0 i.e (curlF). k̂ should be =0??
I have a doubt here. So to get rectangular, we cut the shaded area into rectangles (which is require a lot or i can say infinetely cut). So we can't ignore the narrow boundary, can we?
Does Green's theorem imply that dQ/dx = dP/dy, because of Cauchy theorem on closed and analytic curves?
Fantastic Video! Content is top notch. Audio does seem to be clipping a bit, if you can try set your gain on your mic down just a touch. Your voice is too enthusiastic your mic can't handle it :D
hero
Sir, where the second integral came from in Green theorem? As the circulation density has no integral.
The way I think of this is that the circulation is the sum of (i.e. integral) all the circulation densities at all the points.
Very helpful video. Could you make a video on normal and tangent versions of Green's Theorem, with pictures as usual? Thank you.
Yup, check the rest of the vector calc playlist
I will, thank you.
4:57 how does the single sum change to a double sum ?? any clarity on that please ? it wasnt covered in the video. so you have delta x and delta y but just one sum for i. dont we need a j as well to make it into a double integral ?
Firstly, it was a single sum because it is only sum of curves. Then it is a double sum because it was sum of areas and areas is consist of 2 variables
Why is the value of curl not the negative of the difference, what is the negative curl value defined, something like negative substance?
Hello sir I have a doubt .... I understood that circulation density of a vector field and that we can split up a curve into multiple curves(which are rectangles in this case) but at the boundary of the curve we can never truly overlap the curve using rectangles......in standard integration I understood that error shrinks to zero but in this case we are calculating the line integral so I dont understand how the error here shrinks to zero.....
Ps I am just a highschool student so plz explain it in detail.....I have seen your multivariable and vector calculus playlist but I have severe doubt in understanding green theorem
Thank you very much sir
Where's the "like" button Prof?
👌🔥🔥🔥
Green thought of all this up in the early 19th century. wow.
How u wrote double integral...is that any way I can feel that how how this come in picture
In the 2D case, curl is a scalar, that is positive when CCW and negative when CW.
Since the curl is a scalar, imagine it as the height of a hill above a reference level we call zero elevation. The volume of this hill, tells us the total line integral around the vector field, enclosing that region, according to Green's theorem. Volume above the reference level we call positive, and volume below the reference level, we call negative.
By convention of a right-handed coordinate system, we consider CCW rotation to be positive curl, and CW rotation to be negative curl.
🙌🙌
Hello Dr. Bazett, Thank you for your thorough and easy to follow explanation!
I still can't quite understand why the circulation density of a uniform rotation vector field is non-zero and the circulation density of the "whirlpool effect" is 0. The latter field is F=(-y/(x^2+y^2))i+(x/(x^2+y^2))j. Do you have any thoughts about this peculiar field? Thank you again for your videos!
Imagine a leaf floating in such a field. It would just go around in a big circle with the flow, but that doesn’t mean the lead itself is spinning.
@@DrTrefor Thank you, that helps a lot!
🙏❤
🙏🙏🙏
Sir could you do a video on why curl of velocity field is twice of angular velocity
Start with a generalized rigid body, spinning at a rate of ω, centered at the origin, spinning CCW around the z-axis
At any given point on the body, its linear velocity is given by:
v =
Take the curl of this vector field.
curl v = d/dx (ω*x) - d/dx (-ω*y)
Carry out derivatives:
d/dx (ω*x) = +ω
d/dx (-ω*y) = -ω
Thus:
curl v = ω - (-ω)
curl v = +2*ω
@@carultch thankyou very much
Audio is better!!! Ears are not mad :)
fyi, there is still room for improvement, still feels a little top heavy... but ONLY if the volume is too high. It's PERFECT at 40% on my laptop :))
haha nice! I'm hoping the vids coming out in about a week or two will be best. Mic/Camera/Room insulation/Post-Processing all finally on point. People will still complain I"m sure:D
Sir my foundation in algebra is weak but iam good at Calculus what should I do for improvement
practice practice practice. When you find something you are weak at, take note of the specific challenge. Then master it so you never struggle with that specific thing again. Math is often a lot of rather small details all put together so master all those details.
Thnks sir but iam weak in algebra due to word problems
@@AshishSingh-753
So you're saying that if you got the equations without the words, you're fine?
Write a "Givens:" and a "Find:" try to convert it into NOT a word problem, and then you don't have the word problem issue. Try easy word problems until they get hard as well and you're good.
@@briendamathhatter816 Thnks man I think I have imposter of not doing word problems well I try my best to put your suggestion into math
Sir! I want to watch the next video in this playlist but it says join the channel although i have subscribed your channel.
It will be coming out this week. The (paid) membership grants early access to videos, but I'm releasing them at a rate of three a week.
@@DrTrefor ok sir.
I swear if I pass multivariable calculus I will give this guy a free burger
Lol u got this!
Did you pass the exam?
Dr Strange of Mathematics
Anyone of IIITD here?
Definitely had a few commenters mention this
@@DrTrefor We have weekly quizzes and your topics match with them. In fac this topic will be asked in tomorrow's quiz :P.
Btw these are gr8 videos. Thank you a lot :D
Pog
R u converted Muslim ?