Follow-up on Green's Theorem! -> kzbin.info/www/bejne/boTafHdrlLt_a9E Support me on Patreon: patreon.com/vcubingx Here's a small intuition correction: When I said "projecting the integral", that isn't particularly correct. It's more of a "stretching out" the curve so it becomes straight, without distorting its length. This is why we have the factor of sqrt(x'^2 + y'^2) in the conversion formula. (Credit to Dr. Steve Trettel for pointing this out!) MISTAKES! At 4:11, the problem should be (1,-1). I read it wrong and typed it wrong. Wow. Also in the first example the integral's function should be 3(3-2t)^2-2(6-7t). ALSO IN 8:15 I MISSED A MINUS SIGN HOLY
@md.rafiqurrahman7925 жыл бұрын
vcubingx You put the parametric equation such as 3(3-2t)^2-2(6-7t)^2 where the main function is (3x^2-2y). I think you put a square by mistake.
@vcubingx5 жыл бұрын
@@md.rafiqurrahman792 sheesh you're right
@AdamsAppleseed2 жыл бұрын
Just wanted to add onto this that if you calculate the problem at 4:11 to be to (1,1) then you should get an answer of 6 * (sqrt(29)) At least that's what I got lol
@theInternet6335 жыл бұрын
This kinda looks like 3Blue1Brown content. Not to say this isn't awsome still. That's more of a compliment than anything else but i'll be damned if you aren't an avid viewer of him.
@vcubingx5 жыл бұрын
Haha yeah I use the animation engine he created for his videos, which is why it looks so similar
@borekworek695 жыл бұрын
When I saw the first seconds of the video my first thought was "Yes! More Manim users!" I think Manim should be a standard for math-related videos. It's such a great library and it's a shame that not that many people know about it. Sorry for my poor English :)
@zakirreshi67374 жыл бұрын
Same here...and i love these types of videos
@bostangpalaguna84673 жыл бұрын
you should check out: manim.
@quitest48504 жыл бұрын
I am a simple man, I see videos helping people to understand the concept of math, I hit like
@Miguelocod2 жыл бұрын
As an electronics engineering I needed to refresh this 6 years after taking Multivariate calculus, and this was an excellent explanation. Thanks!
@steventhijs69214 жыл бұрын
There is a minor mistake at 4:50, is should just be '-2(6-7t)', not '-2(6-7t)^2'. The answer of 8 × sqrt(53) is correct though if you meant a line segment to (1, -1)
@michaeljuergen43263 жыл бұрын
I was thinking the same
@andrewshain8992 жыл бұрын
@@michaeljuergen4326 I was second guessing myself The two mistakes on one screen was really getting to me.
@DougChemist-oj4ev8 ай бұрын
How is 8 x sqrt(53) correct given the corrected '-2(6-7t)' term and the (1, -1) endpoint? Evaluating the eqn at bottom of 4:50 (with the correct term substituted) at t=1 gives 5 x sqrt(53), and for t=0 one gets -15 x sqrt(53), so isn't the correct answer -10 x sqrt(53)?
@vashon1007 ай бұрын
@@DougChemist-oj4ev Because 6 -5 = 1, not 6-7 = 1 . (3,6) to (1,1)
@DougChemist-oj4ev6 ай бұрын
Sorry I forgot to take the antiderivative when evaluating the final integral; the given answer of 8 times the square root of 53 is correct@@vashon100
@johnstone68485 жыл бұрын
your parameterization for the curve C at 4:25 is incorrect. It should be 6-5t not 6-7t
@wduandy5 жыл бұрын
How to do this parametrization?
@mrdiin.dev_4 жыл бұрын
@@wduandy n order to parametrize a line, you need to know at least one point on the line, and the direction of the line. If you know two points on the line, you can find its direction. The parametrization of a line is r(t) = u + tv, where u is a point on the line and v is a vector parallel to the line. There are lots of possible such vectors u and v. To find one such vector v, find the difference between any two points on the line. In this example, we have A(3, 6) and B(1, 1) or I guess he meant to say B(1, -1) hence his result. So using the latter, the direction of the line will be AB = B - A = (1, -1) - (3, 6) = (-2, -7). Thus us our v. The line can then be written as r(t) = A + tv or B + tv, it doesn't really matter, but he used r(t) = A + tv. So r(t) = (3, 6) + t(-2, -7) which means x = 3 - 2t and y = 6 - 7t.
@onemadscientist73054 жыл бұрын
SInce we're working with a segment here, one better way to think about this is in terms of barycenters between the two endpoints with coefficients between 0 and 1. If A and B are the endpoints, then a point P inside the segment can be written as follows : P=t*A+(1-t)*B where 0
@onemadscientist73054 жыл бұрын
Also, the parameterization in the video is definitely wrong and this actually changes the value of the integral ! For a start, , it should be ds=sqrt(29)*dt and not ds=sqrt(53)*dt...
@gd3xsilon7834 жыл бұрын
I think he made a mistake in the definition of C. He said in the video “C is the line segment from (3,6) to (-1,1)”.
@tandlose5 жыл бұрын
This is great! I like the balance you have between questions and explanation, 3blue1brown never uses questions which is a shame because it is a nice way to apply what you've learnt and get a breather from the learning. On the other hand the examples are quick and you don't waste time explaining basic calculus so it doesn't feel dragged out and uninformative like many other math videos who uses two thirds of the time for examples. Great work and I can't wait to see more!
@understandtheuniverse2199 Жыл бұрын
Bro you don't know value of basic calculus and we need to think of it every time
@Harry-ub2fv4 жыл бұрын
This encouraged me to learn Vector Calculus.
@mahatmawisesa97063 жыл бұрын
8:14 isn't that integral should be -3(7+5t)-5(33+39t) ?
@fyu19455 жыл бұрын
Really good but I think you make the same mistake 3Blue1Brown made when he was starting. Talk a little bit slower. But otherwise great video!
@Michallote2 жыл бұрын
Is it weird to watch this videos on 1.5x?
@agrajyadav29512 жыл бұрын
@@Michallote yes
@howmathematicianscreatemat9226 Жыл бұрын
The downside of being gifted means not being able to see what others cannot see.
@isaacaguirre18816 ай бұрын
I am a first year EE student studying for my vector calc exam . This is the best channel i have found this year regarding this topic. Keep it up, you're the best.
@martinsanchez-hw4fi2 жыл бұрын
If C was from (1,1) to (3,6), shouldn't the parametrization be [3-2t, 6-5t]?
@vashon1007 ай бұрын
Yes, vid is wrong.
@photon27244 жыл бұрын
indians are finally making quality content.
@krishnabharadwaj46382 жыл бұрын
Yes finally something other than jee and neet
@ritishbehera47112 жыл бұрын
Is he indian? I thought its a foreign youtube channel
@utkarshsaxena35114 жыл бұрын
Thank you for giving such details. I don't know why these are not written in books clearly, and we have to search the internet for better understanding, especially for mathematics.
@quantumleap79643 жыл бұрын
the animations really helped solidify the concept, great job.
@vcubingx3 жыл бұрын
Thanks! Glad you liked it!
@gururajmundargi79393 жыл бұрын
sir you have started to teach difficult math using animation it really helps to me.iam a engineering student from india. please dont stop uploding this kind of vedio
@PTGaonkar Жыл бұрын
ತುಂಬು ಹೃದಯದ ಧನ್ಯವಾದಗಳು... ಇದೊಂದು ಅದ್ಭುತ ಪ್ರದರ್ಶನ... ಸಂಪೂರ್ಣವಾಗಿ ಅರ್ಥವಾಯಿತು...
@alexdeac29514 жыл бұрын
such a great video! concise, intuitive and covers all the relevant information
@bostangpalaguna84673 жыл бұрын
dude... your visualization is crystal clear! thank's a lot !
@thehighground5833 жыл бұрын
THANK YOU for helping for giving me a visual intuition of where the formula for arc length comes from at the start of the video!! That was something that I had previously lacked. Also love the background music!
@prathmeshpatel56574 жыл бұрын
Dude this is great. It made everything so much clearer. Thank you for making this
@sidimohamedbenelmalih71332 жыл бұрын
This is the first time when I started to understand the integrals, and it was great, like a schematization to what's Waiting for me, and be prepared to it
@erickperez59304 жыл бұрын
Your videos are great! Please keep the great job you have been doing so far.
@vcubingx4 жыл бұрын
Thank you so much!
@myonlynick4 жыл бұрын
4:24 it would help us a lot from confusion if you take the extra step and correcting the point (1,1). It should say (1,-1). Maybe embed an annotation in the current video?
@gibranayala39032 жыл бұрын
underrated content very good
@visbellum3 жыл бұрын
It's my pleasure to watch it!! Thank's for doing it!!!
@vcubingx3 жыл бұрын
Thanks!
@morgengabe14 жыл бұрын
The instrumentals in this video are something else lol. Stylishly done!
@mariogalindoq Жыл бұрын
There are a mistake: the line segment from (3,6) to (1,1) is not C=(3-2t, 6-7t) 0
@thevegg32755 жыл бұрын
Thanks! What is the precise method of parameterizing C? Looks easy to figure out for a straight line but a wavy curve?...can't be done without a specific method. Can you define that method in detail please?
@vcubingx5 жыл бұрын
You can't always parametrize the curve by hand. For example if I draw any squiggle on a paper, you can't always parameterize it. But, when solving problems by hand you would probably get curves that can be easily parameterized, like circles, ellipses, lines, quadratics, etc. You can just Google "parametric curves" and that should answer your question if you're curious on how to parameterize those curves. Thanks for watching!
@andrewshain8992 жыл бұрын
kzbin.info/www/bejne/iqjToH94lt5-j68 This video here does a great job of explaining starting at 1:32.
@EngineerNick4 жыл бұрын
Really good video thankyou so much :) I am confused at 6:26 - I think the red and yellow vector arrows should be in the same direction? I think the red arrow in the circle is facing the wrong way; for the dot product to be negative there must be an obtuse angle between the vectors? Still the animation is amazing well done :)
@LagAttacktoSlay2 жыл бұрын
Finally, people are recognizing 3b1b's Pythonlib lmao. Good video!
@Linaiz4 жыл бұрын
Thank you for all your helpful videos. I think you're doing an amazing work. If I could suggest one thing, once your channel expands more, you could hire a professional voice-over guy. I think that would facilitate understanding, because visuals are already top notch, and I only ever get lost in the audio explanation. Best of luck with your channel!
@oo69554 жыл бұрын
4:53 I read comments, but I have one question. Parametrization for curve C is f(x(t),y(t))=3(3-2t)^2-2(6-7t), but you integrate 3(3-2t)^2-2(6-7t)^2. First one is 8, but second one is -23/3. First one is right? Or second one is right?
@vcubingx4 жыл бұрын
Yeah I made a mistake my bad
@oo69554 жыл бұрын
@@vcubingx Oh, Thank you for answering my question
@vikramnagarjuna35495 жыл бұрын
Hi, I am from India, that was great video sir... 👏👏👏👍
@Iris-zr1rf4 ай бұрын
I like to think about the line integral as being the total elevation graph as you take a hike (Curve C) through a mountainous region (f(xy))
@kalaiselvanrajasekaran83524 жыл бұрын
One suggestion pal,if you could have used a normal curvy surface like a parabolic roof top kind and explain the line integral concept then it would be made easily more comprehensive than now.But it is a good explanation pal.Impressive one.
@lok73965 жыл бұрын
Nicely made.
@zprogress40044 жыл бұрын
3:52 the line integral is supposed to be the area of the orange colored region right? the arrow of the integral is pointing on the projected shadow on the xz-plane?
@saiprasadmohanty83394 жыл бұрын
Even i have the same doubt so can you please help me incase if you have got your doubt cleared. Thanks
@predatoryanimal63972 жыл бұрын
the lo-fi track is a nice touch
@wgwandawg3 жыл бұрын
This is so well explained. I also liked the way you played the animation again so that we could get a clearer picture of whats going on. Thank you!
@Mirabel1514 жыл бұрын
Thank you so much for this video! it helped it to better understand line integrals for my finals!
@crimsonarmor.5 жыл бұрын
Excellent explanation and awesome animations! Subscribed :)
@jagsproduction3 жыл бұрын
Dope animation, it helps in understanding better
@sheepphic2 жыл бұрын
You can make parentheses scale to the size of their contents in LaTeX by using \left( and ight) instead of plain ( and ); this might work better for things like your derivatives at 2:07
@filomenamendes53492 жыл бұрын
Incredibly helpful! Thank you so much
@126sivgucsivanshgupta24 жыл бұрын
I really like the song choice in the background ... And offcourse the mathematics
@SuperCuteAnimeWaifu4 жыл бұрын
Thank you for the visualizer
@ignassablinskas9175 Жыл бұрын
Hello, you wrongly parametrized the curve C at 4:25 , it should be < 3-2t , 6-5t >
@sonu76423 жыл бұрын
Finally i understood the concept..........thankyou 😎😎😎😎
@edissonstevencastanomesa4841 Жыл бұрын
Nice video! Just a comment, for better notation, it should be used a "dt" for last shown equation at minute 8.11.
@DevPatel-nf5mp2 жыл бұрын
Great work 🔥🔥🔥
@arrabax5 жыл бұрын
great work!
@DiegoMathemagician5 жыл бұрын
0:28 just pointing out this little mistake i did noticed f(x)=x^2+y^2; it should be f(x.y)=x^2+y^2. Thanks for doing such a great video, by the way.
@vcubingx5 жыл бұрын
Thanks for pointing that out!
@mohammedkhan49904 жыл бұрын
Excellent presentations. Your videos helped me do much. Thank you.
@vcubingx4 жыл бұрын
Thanks for watching
@ingridp82334 жыл бұрын
such a great explanation, thank you!
@NoNTr1v1aL5 жыл бұрын
AMAZING! KEEP IT UP!
@gby98613 жыл бұрын
Thank you for the help! Cemented my understanding!
@vcubingx3 жыл бұрын
Glad it helped!
@speedychicken8313 жыл бұрын
this is betufial
@dordavid30322 жыл бұрын
4:25 the parametrization is incorrect! should be C=(3-2t,6-5t) the correct answer is sqrt of 1044
@mrdiin.dev_4 жыл бұрын
Chillhop music is my favourite and made me love this video more! What software did you use to animate though?
@vcubingx4 жыл бұрын
I used manim: github.com/3b1b/manim. Glad you enjoyed!
@bharathsundar23785 жыл бұрын
great work mate...
@szilike_104 жыл бұрын
3:57 how could the projected area be tha same as the area under the graph? I mean if we'd stretch that curve to a line, the area of it would be much bigger. Isn't it like saying that given a line y = mx, on the OXY plane, and a segment a to b on the X axis, the distance between (a, y(a)) and (b, y(b)) is the same as d(a, b)? The pythagorean theorem says the exact opposite. I know it is not the same, I reduced the number of dimensions, but I'm still curious if I'm not noticing something. However great video, keep up the work.
@szilike_104 жыл бұрын
Also, if I have a sheet of paper its shadow can be either a line or the shape of the paper depending how I keep it in the air. The area of the paper (the line integral) won't change however the area of the shadow (projection area) will.
@probablyapproximatelyok81464 жыл бұрын
I don’t think the area of the projection is the same as the area under the curve. I’m very much new to these topics from multivariable calculus but my best guess as to why it is being projected onto the OXZ plane is b/c the integral is being taken ‘with respect to’ x (with the dx at the end) so it is only multiplying the height of each rectangle by the x component of its respective ‘step’, and when the integral has the ds at the end it is equal to the area under the curve b/c it multiplies the rectangles by the actual magnitude of their respective ‘step’ instead of a projected version of it. Maybe if it were dy instead of a dx it would be projected onto the OZY plane. Hope this helps! (Sorry if it doesn’t or if I misunderstood your question)
@bugi66064 жыл бұрын
Thank you for that video! Do more such videos !!!
@Matheus-yx6qb3 жыл бұрын
There's an error in 7:15. The red vector inside the circle was supposed to be the (vec r) derivative, but it seems that the red vector was (vec r) itself
@BoZhaoengineering4 жыл бұрын
It is a better interpretation than textbook. But I would suggest go to the text book first after getting the concept of line integral and back to the animation video. The idea of line integral in a vector field becomes much more clear! Thank you for your nice video and interpreting
@vcubingx4 жыл бұрын
I agree! I think I've improved in my explanations over time, might revisit this topic in the future!
@BoZhaoengineering4 жыл бұрын
vcubingx cheers. Today’s learning resource and landscape has changed to the better than before. Digital tutorial video becomes indispensable as a learning companion. Before the rise of Khan academy , it seemed nobody predict that while learning ‘stem’, the help could be made from enthusiasts by digital technologies.
@DOPEBEATZBOYS4 жыл бұрын
the first example was wrong wasn't it? the parameterization seems off
@DOPEBEATZBOYS4 жыл бұрын
and in the animation, you had a square for the y and it wasn't in the problem, not sure if you used that for the final answer
@vcubingx4 жыл бұрын
Yeah both the examples have problems. I addressed all of them in a pinned comment. Thanks for pointing them out!
@DOPEBEATZBOYS4 жыл бұрын
@@vcubingx oof sorry, i should have looked for that first
@rayyan21d2 жыл бұрын
Thank you so much, this was really helpful!
@zakirreshi67374 жыл бұрын
Thanks for these beautiful videos..
@abhinaamrawat16803 жыл бұрын
Just have one word to say "exquisite"
@conzpicuouz3 жыл бұрын
Awesome video. Thank you very much for your help.
@vcubingx3 жыл бұрын
Glad it helped!
@jindeakshaykumar28635 жыл бұрын
Mind blowing
@vcubingx5 жыл бұрын
Thanks
@AlePreludioFinal5 жыл бұрын
Hey! A great video. Thank you
@vcubingx5 жыл бұрын
Thanks! You're welcome!
@veerstar52 жыл бұрын
Wow!
@ayooluwakolawole Жыл бұрын
Great vid, really helped me build intuition! Only critique is some of the coordinates of the curves didn't match the graphs you put
@xsli28764 жыл бұрын
Excellent video and excellent the plotting web sites provided in the description!!! Thank you very much!!! I am a little puzzled at 6:25: my understanding is that the orange arrow inside the circle at the bottom indicates the traveling direction, which is the tangent direction of the curve, which is the yellow arrow direction on the curve. But at this moment, the yellow arrow is pointing toward between clock 12 o'clock and 1 o'clock direction, while the orange arrow within the circle is pointing toward 7 o'clock direction. I am puzzled. Besides that, I follow the video for the most part. It is fantastic video with nice explanation, same as others made by this man. Thank you! God bless!!!
@xsli28764 жыл бұрын
I also understand that at this moment(6:25), the yellow arrow and the vector at that spot basically is in opposite direction, therefore their dot product is negative. The yellow value below the ruler near the bottom is indicating the dot product is negative at this moment. However, within the circle, the orange arrow and the green arrow is basically in the same direction. If so, their dot product should be positive.
@pastorofmuppets76545 жыл бұрын
Nicely done! Can you suggest some good Manim tutorials?
@muhammadyaseen89235 жыл бұрын
Yep. Moar! manim tutorials.
@vcubingx5 жыл бұрын
Try Todd Zimmerman's. But the thing that works best for me was going through Grant's projects.
@muhammadyaseen89235 жыл бұрын
@@vcubingx thanks for suggesting, can you link? I'm not sure which "Todd Zimmerman" you are referring to.
I think your parametric representation of the line should be . Am I wrong ?
@briancabral2552 Жыл бұрын
You can add Line Integral Convolution to your visuals ;-)
@matron99364 жыл бұрын
Nice explained thank you.
@vcubingx4 жыл бұрын
You're welcome!
@sollinw5 жыл бұрын
u saved the education :)
@ProfeJulianMacias Жыл бұрын
Great Msth problem
@aalexren3 жыл бұрын
Sorry, but on 4:34 you made a mistake. Parametrisation supposed to be , not the 6-7t. Maybe I didn't get the point, then could you please fix my issue. UPD. OK, got the pinned comment
@professoreisenoxid98822 жыл бұрын
Why are F(r(t)l and r'(t) at 7:16 aligned? Shouldn't r'(t) be orientied in the other direction because the displayed dot product is mit negative-it would be with a negative orientation. Therefore the (yellow) vector r'(t) in the vectorfield doesn't show the direction as the red one. Somehow I am confused 😅
@harshans77127 ай бұрын
thanks a lot for this explanation but I have a doubt, how did u get the last question's limits as 0 to 1?
@orfeaspapaioannou27555 жыл бұрын
amazing!
@sergiolucas382 жыл бұрын
great video, thanks for sharing :)
@fruxo20002 жыл бұрын
When the animation starts playing at 7:16, isn't the red arrow pointing both in the wrong direction and with the wrong magnitude? Shouldn't the red arrow, which is meant to be r'(t), be pointing in the same direction as the yellow arrow in the graph, which is also meant to be r'(t). The yellow arrow appears correct. And, since the dot product of two vectors that are more than 90d apart is negative, it would only make sense if the red arrow was on the opposite side of the green arrow, until after C crosses origo, when the green arrow should swap sides (as it correctly does) but the red arrow should remain pointing up-right (which it currently only does for the latter half of the animation). Am I mistaken?
@robertmegee90522 жыл бұрын
Ok, I'm stuck at about 2:10 into the video. You reduce your delta-x and delta-y to be derivitives but where did the delta-t come from? No explanation was given other than you can multiply and devide by the equivalent of 1. Please tell what it actually means. Where is it on the coordinate axis?
@nagesh18054 жыл бұрын
Awesome!. I'm wondering if you made these videos using manim(3b1b) library.
@abrahamlingkonkhan93403 жыл бұрын
Thank you 🔥
@sudharakafernando43913 жыл бұрын
Thank you very much...It helped a lot
@joelsagflaatholmberg39225 жыл бұрын
*Subscribed*
@karlaserit45233 жыл бұрын
Hello. How to parameterize a curve into a vector? Thank you for your answer ^_^
@Michallote2 жыл бұрын
I have the same question when dealing with more elaborate curves. But to parametrize a line is actually pretty simple. I'll try to be as clear as i can: First main idea is to use a variable t which conviniently for us goes from 0 to 1 So let's say we wanted to get from point a to b which are fixed using t as the parameter. Our parametric function will be f(t) So when t=0 the f(t=0) = a t=1, f(t=1) = b so you can think a bit about it and you can intuitively get to: f(t) = a * (1-t) + b * t So that those conditions stay true. The beautiful thing about it is that a and b can be anything, could be scalar valued, 2D vectors: f(t)= (ax,ay) * (t -1) + (bx,by) * t 3D vectors. The parametrization still linearly gets you from one to the next. More formally it is called linear interpolation look it up on Wikipedia it's awesome.
@mohamedmouh3949 Жыл бұрын
thank you so much
@ashishratnawat27115 жыл бұрын
Excellent Video! Just one thing, around 6:20 it should be "dot product indicate how aligned two vectors are" right ?
@guitsgarcia54465 жыл бұрын
i think he meant 'how parallel two vectors are' instead of perpendicular
@omjoglekar36772 жыл бұрын
3:53 why did we squash it onto X-Z plane ?
@iliasp42754 жыл бұрын
thank you
@akuo07082 ай бұрын
3blue1brown would be proud
@mnada72 Жыл бұрын
3:55 I didn't get what is the projection pointed to by the arrow.?