Understanding Lagrange Multipliers Visually

  Рет қаралды 364,836

Serpentine Integral

Serpentine Integral

Күн бұрын

Пікірлер: 347
@scalex1882
@scalex1882 Жыл бұрын
This is one of these things where you are sitting in university, getting fed the final formula with an absolutely insane proof of the formula that makes you question reality and when you see this video it takes no more than 10 minutes to understand the entire concept. Absolutely incredible, thank you so much!
@lehninger2691
@lehninger2691 Жыл бұрын
Wait, you guys are getting an absolutely insane proof???
@ico-theredstonesurgeon4380
@ico-theredstonesurgeon4380 Жыл бұрын
Why the heck dont they teach these things visually in university?? This video is literally higher quality education for free. It makes no sense at all
@ico-theredstonesurgeon4380
@ico-theredstonesurgeon4380 Жыл бұрын
@@pyropulseIXXI that's true but I would argue that sometimes visualisations really speed up the learning process, and teachers are often not the best at drawing.
@ahmedbenmbarek9938
@ahmedbenmbarek9938 Жыл бұрын
​@@ico-theredstonesurgeon4380it is not free it is sponsored by a university. The main issue with understanding math is to have a teacher who really understands maths to begin with. Most math teachers are simple folks looking for a fat salary. Maybe themselves do not understand the concept so they simply regurgitate what another teacher did to them. Anyway all thanks to KZbin that allowed brilliant teacher to explain mathematics from simplest concepts to the most complicated ones.
@ico-theredstonesurgeon4380
@ico-theredstonesurgeon4380 Жыл бұрын
@@ahmedbenmbarek9938 it's free for us, that's what I ment. I do not have to pay to watch this. Plus i think even competent and passionate teachers would boost the overall engagement of the class if they started to use modern visualization methods more.
@GiulioDean
@GiulioDean Жыл бұрын
I'm doing a PhD in aerospace engineering and never have I seen a video so clear on this topic. chapeau!
@paganaye
@paganaye 5 ай бұрын
Chapeau = "hats off."
@josuequintero-td5gh
@josuequintero-td5gh Ай бұрын
A PhD in engineering and you're reviewing a basic calc 3 topic?
@shayansa6865
@shayansa6865 12 күн бұрын
@@josuequintero-td5gh I bet you never got anything higher than a high school diploma or a bachelor! Only a fool says that! Even a Professor at MIT might forget things constantly and have too look up for those!
@QuangNguyen-fk6md
@QuangNguyen-fk6md 10 күн бұрын
@@josuequintero-td5gh People forget things that they haven’t revisited in a while.
@rintepis9290
@rintepis9290 2 жыл бұрын
I am so impressed by how clear this video manages to explain the intuition behind the Lagrange Multipliers. The only part I had to pause and ponder is to show the gradient of f must be perpendicular to the level curve when the point is a local maximum on the boundary curve.
@euqed
@euqed 2 жыл бұрын
Same, if anyone has an intuitive explanation, please do share it !
@jozsefnemeth935
@jozsefnemeth935 2 жыл бұрын
@@euqed it was explained by the ball on the slope: a perpendicular barrier to the ball trajectory will stop the ball, hence the barrier is in the horizontal plane.
@gdvirusrf1772
@gdvirusrf1772 2 жыл бұрын
@@euqed If you imagine the parametrized curve of the boundary of f(x,y), you'll know that the maxima/minima occur at points where the derivative of the parametrized curve is equal to 0 (the single variable calculus way of solving the problem). The thing is, if the derivative is nonzero, then it must either point to the right (positive derivative) or to the left (negative derivative) on the parametrized curve. But this must also mean the gradient vector on the actual function f(x,y) itself must _also_ point to the right or left! Another way to say this is that for a point on the boundary of f(x,y), any deviation in the gradient vector away from perpendicular _must_ imply that the derivative of the parametrized curve of the boundary is nonzero at that point, and hence it _cannot_ be a max/min. So only the points where the derivative of f(x,y) is perpendicular could possibly be a max/min.
@sender1496
@sender1496 2 жыл бұрын
It follows from the definition of the gradient. At a local min/max, the slope of f is zero along the boundary curve, meaning that f doesn't change in that direction. The gradient gives you the direction and magnitude in which a function changes the most and is thus perpendicular to this. In other words, if the gradient were to have a component in the "boundary curve"-direction (ie not perpendicular), then surely it couldn't have slope zero since f would be increasing/decreasing when wandering on the boundary.
@jozsefnemeth935
@jozsefnemeth935 2 жыл бұрын
@@euqed another way to look at the problem: we search for points where a level curve of the f-surface is tangent to the constraint curve. The perpendicular to these curves belonging to the X,y plane will be the same. By definition, the gradient on the respective surfaces provides this perpendicular.
@omargaber3122
@omargaber3122 2 жыл бұрын
I can't believe I managed to understand Lagrange Multipliers after all these years!!!!!!! , how magical math is when it's understood, thank you so much
@mobilephil244
@mobilephil244 Ай бұрын
How magical it is when explained by someone who CAN be bothered to explain.
@richardvondracek496
@richardvondracek496 9 ай бұрын
I have been waiting for this video my whole life. Although I did many calculations with Lagrange multipliers in my life It never clicked in my brain the way other things did. Close to half century old and you have just completed my brain. ♥♥ Thank you so much for this. ♥♥ Damn.. this feel good. You are my new hero!!
@hatelovebowel4571
@hatelovebowel4571 2 жыл бұрын
this is fking amazing. The best explanation and Calculus should be taught with geometry, it is so clear.
@yendrian44
@yendrian44 Жыл бұрын
Holy shit when you said that lamda in this case is called the Lagrange multiplier I could literally feel the creation of new neuron connections in my brain. This video is a masterpiece
@shlokpunekar8503
@shlokpunekar8503 11 күн бұрын
It has been DAYS that i was struggling to understand the Lagrange multipliers and this visualization has literally blown my mind. You deserve more recognition and today you have gained a new subscriber.
@leonvonmoltke7923
@leonvonmoltke7923 2 жыл бұрын
I would like to say that it is not often that people explain things better than khan academy. Well done sir.
@NemoTheGlover
@NemoTheGlover 2 жыл бұрын
once you go past Cal I, khan academy content isint that great in my opinion
@agrajyadav2951
@agrajyadav2951 Жыл бұрын
@@NemoTheGlover what
@golchha_J
@golchha_J 4 ай бұрын
KA video on this topic was crap
@firstkaransingh
@firstkaransingh 2 жыл бұрын
I salute you for taking a complex concept and breaking it down to understand at a very basic level. More power to you.
@gergerger53
@gergerger53 2 жыл бұрын
That whole framing in terms of terrain, seas and what counts as the shoreline are fantastic metaphors to aid the conceptual understanding of this method. Very, very well represented, here.
@JulianHarris
@JulianHarris 9 ай бұрын
Outstanding. Just spent a whole morning trying to understand these things and the visualisations really really crystallise the relationships. Obviously this is an advanced topic and the prerequisites involve simultaneous equations, a little bit of linear algebra and partial derivatives. But once you’re in that position, I think this is possibly the best way to understand Lagrange multipliers.
@Sajad-d1q
@Sajad-d1q 2 күн бұрын
Lagrange's idea was insanely ingenious.
@dannis5165
@dannis5165 8 ай бұрын
that rolling ball analogy is so insane. i never understood a concept more clearly before.
@StarContract
@StarContract 5 ай бұрын
In my opinion, good mathematical education should strive to develop your mathematical intuition, which in turn you would be able to turn into formality. This video is literally perfect.
@jperez7893
@jperez7893 20 күн бұрын
My only regret is that my calculus teachers never taught me the concepts as well as you. Brilliant and lucid presentation
@eklhaft4531
@eklhaft4531 9 ай бұрын
I have no idea why they couldn't explain it like this at the university instead of just throwing a bunch of boring letters at us but here we are. I feel like you just removed an ulcer from my brain that's been sitting there for couple of years. Thanks.❤
@krittaprottangkittikun7740
@krittaprottangkittikun7740 2 жыл бұрын
This video is way underrated, it is very clear and nice!
@SerpentineIntegral
@SerpentineIntegral 2 жыл бұрын
@joseph ramos Hey, hello! I still make new videos, but not on this channel anymore. I put all my new stuff on a new channel called Morphocular. You can find it here: kzbin.info/door/u7Zwf4X_OQ-TEnou0zdyRA
@shapooopiefour7173
@shapooopiefour7173 Ай бұрын
Thank you! Did these last year and forgot what they did. I haven’t needed to use these yet, but it’s frustrating to forget something that you learned.
@qwerasdliop2810
@qwerasdliop2810 2 жыл бұрын
Absolutely incredible! Can't believe something so simple yet incredible was fit into such a simple set of equations, just under the surface!
@derrick20
@derrick20 2 жыл бұрын
A neat way to conceptualize this idea is to think of the constraint function as a filter of sorts, since we know every point along the constraint curve has a gradient perpendicular to the curve (this can also be understood in the sense that everything is a local extremum, since they are all equal, so the direction of max increase shouldn’t be biased to either side similar to the ball analogy in the video). So, when setting the gradients of the two functions equal, we just filter only the extreme in the objective function
@CG119Animator
@CG119Animator 6 ай бұрын
That explanation was stellar! You broke down a tough concept without frying anyone's brain cells.
@christostsaggaras1821
@christostsaggaras1821 6 ай бұрын
Read the Wikipedia article and then came back here. While there it was fairly understandable, here the explanation was absolutely brilliant.
@gohanmineiro
@gohanmineiro 2 жыл бұрын
Simple, clear, and concise explanation. Kudos.
@flatmajor6802
@flatmajor6802 9 ай бұрын
This presentation of L.M is much easier than the presentation that the level curve of the max of f is tangent to the level curve of g. Completely bypasses the need to show why they would be tangent at all. Ty🔥
@NicolasMartinezAngulo
@NicolasMartinezAngulo Жыл бұрын
Could not have explained it any better. Probably top 3 math videos I've ever seen.
@Hinchey613
@Hinchey613 5 ай бұрын
The animation at 2:50 was incredible, definitley ignited a light bulb moment in my head.
@laodrofotic7713
@laodrofotic7713 Жыл бұрын
This is a good video, congratulations on helping millions around the globe with this.
@plekkchand
@plekkchand 2 жыл бұрын
Wonderful, direct, lucid, free of affected cuteness and cosmic background music. Thank you!
@egeecagan
@egeecagan 5 ай бұрын
best explanation ever without killing some of my brain cells
@verracaelum5258
@verracaelum5258 5 ай бұрын
agam bu tarz animasyonlarla anlatan başka bildiğin kanallar var mı bu adamın az videosu varmış böyle
@boutainabenhmida6071
@boutainabenhmida6071 2 жыл бұрын
never seen a visual explanation better than this
@klevisimeri607
@klevisimeri607 2 жыл бұрын
This video is more valuable than gold!
@rhke6789
@rhke6789 Жыл бұрын
Best explanation of Lagrange multipliers on KZbin. Congrats and thank you
@zhuleung2938
@zhuleung2938 2 жыл бұрын
excellent work. you've just made me understand what confuse me throughout my whole collage life.
@ktgiahieu1
@ktgiahieu1 11 ай бұрын
Thank you very much for such impressive video. The concept used to be so blurry to me, yet it is as clear as bright day now!
@alperyldrm4788
@alperyldrm4788 Жыл бұрын
That is wonderful how you visualize and construct the idea step by step! Grateful!
@canowow11
@canowow11 2 жыл бұрын
really good video on a difficult math problem, but visually you made it easy
@franciscorivas4036
@franciscorivas4036 Жыл бұрын
Best explanation I've found so far about lagrange multipliers. Thank you.
@محمداحمد-ز4ل6ط
@محمداحمد-ز4ل6ط 2 жыл бұрын
every teacher should teach like this! very excellent illustration
@KYosco
@KYosco 11 ай бұрын
That makes it extremely intuitive! I don't think one can explain it any better than that.
@davidebic
@davidebic 2 жыл бұрын
This is exactly the intuition I had trying to understand Lagrange Multipliers!
@manueelrubik
@manueelrubik 5 ай бұрын
this video is low key the best math lesson even made, congrat s
@BitsNBytes_
@BitsNBytes_ Ай бұрын
This video is awesome, the clear explanation and the intuition behind the Lagrange Multipliers is great, thank you!
@elyjamesuzu
@elyjamesuzu 2 жыл бұрын
this channel is highly underrated...
@PB-sk9jn
@PB-sk9jn 2 жыл бұрын
haha.. when I was final year undergrad we had a prof of theoretical physics teach us lagrange multipliers, who inexplicably said he couldn't explain it and had never found a good explanation. I figured this out and explained it to him and to my classmates. So bloody obvious I thought...
@harshal8956
@harshal8956 Жыл бұрын
This just blew my mind. This is what I was looking for. Great work.
@mase4256
@mase4256 6 ай бұрын
That was the best explanation I’ve ever seen in multivariable calculus, definitely subscribing
@yolo2709
@yolo2709 9 ай бұрын
The day I understood this with one of my friend was the day it stopped being a weird cooking recipe and Lagrange multipliers finally started to make sense!
@camel2666
@camel2666 5 ай бұрын
single-handedly saving my vector calc grade!
@breitbandfunker4332
@breitbandfunker4332 2 жыл бұрын
best video for understanding lagangian multipliers - now i understood it :-)
@autumnreed2079
@autumnreed2079 11 ай бұрын
This is beautiful! I wanted something to help me explain Lagrange Multipliers better as a tutor and this was brilliant. Thanks
@BarryKort
@BarryKort 5 ай бұрын
In order to actually find the extremum of a function subject to constraints, it's typically necessary to determine the actual values of the Lagrange multipliers. One of the better behaved algorithms is to replace the scalar Lagrange multiplier by a convex curve which can be adjusted by means of an iterative solution process. This method, known as the Generalized Lagrange Multiplier Method is mathematically related to another important branch of mathematics called Duality Theory. Such Primal-Dual Methods were explored by myself and Professor Dimitri Bertsekas in the early 1970s, when we were both at Stanford University. The resultant algorithm is spelled out in one of Dimitri's textbooks on the subject of Optimization Methods.
@sandeepmandrawadkar9133
@sandeepmandrawadkar9133 11 ай бұрын
Unbelievably super simplified explanation 👏
@user-dz9eb7fu2f
@user-dz9eb7fu2f 2 жыл бұрын
Very clearly explained, this clarified a lot for me thank you so much
@marcods6546
@marcods6546 Жыл бұрын
A bit repetitive in the explanation, but finally a good explanation of this concept. Thanks a lot!
@bravepsa850
@bravepsa850 22 сағат бұрын
Thank you for sharing this explanation. I could easily understand the concept.
@hereigoagain5050
@hereigoagain5050 Жыл бұрын
Amazing graphics really help to understand Lagrange Multipliers. My middle name must be "Lambda" because I don't contribute to the solution :)
@gossipGirlMegan
@gossipGirlMegan 2 жыл бұрын
Excellent work I ever met ! Tanks a lot ,deer professor!!!
@Amprichu
@Amprichu 2 жыл бұрын
YOU ONLY HAVE 1.5K SUBS???????? THIS VIDEO WAS SO HELPFUL WHAT
@sepehr__byt
@sepehr__byt 2 ай бұрын
The video was phenomenal and truly amazing; thank you for providing such valuable content!
@dufrain79
@dufrain79 2 жыл бұрын
A very good informative video for beginners in optimisation. Very good entry level for understanding Lagrange Multipliers. Such a beautiful use of the Morpho library under Python.
@lh2738
@lh2738 Жыл бұрын
Thanks a lot for such a well explained and drawn video, it really helps a lot to understand the subject. This channel is pure gold.
@harrymorris5319
@harrymorris5319 Жыл бұрын
4:07 for Lagrange multipliers to work - need to have the constraint expressed as some expression involving x and y set equal to a constant x^2 + y^2 = 4 6:57 8:33 10:30 11:20 The max or min of a function f(x,y) which has a constraint g(x,y) = k must occur where ∆f (gradient of f) is parallel to ∆g (gradient of g) . If two vectors are parallel one is a scalar multiple of another. So ∆f = λ ∆g and λ the scalar multiple is called the Lagrange multiplier How to solve 12:13
@brianyeh2695
@brianyeh2695 11 ай бұрын
Your explanation is another level. You link every step with a question, which is an excellent way for people to follow well
@federicoferraro7080
@federicoferraro7080 Жыл бұрын
Even yhough I knew the answer, this helped to visualise the concepts and even helped me make links with other concepts (fluid mechanics). So thanks a lot !
@Words-.
@Words-. Жыл бұрын
The visuals are soooo well done
@mehdiardavan
@mehdiardavan Жыл бұрын
Fantastic video. Well visualized and explained. I was just wondering what you used to make the graphical effects while showing LaTeX formula rotate in 3D?
@joaogoncalves-tz2uj
@joaogoncalves-tz2uj 4 ай бұрын
this is the best video I've seen on this topic and it still doesn't clear all the doubts about it. Why does it when the tangent line at the 3d curve of g is parallel to the plane xy we can say the gradient at f is perpendicular to the "level curve" of g? Also, given that there are infinite lines perpendicular to a given line, how does it guarantees grad f // grad g?
@anthonytafoya3451
@anthonytafoya3451 2 жыл бұрын
Wow! Thank you for this video. Visuals GO A LONG WAY my brother. Cheers and you have a new subscriber :)
@LucaSalemi
@LucaSalemi 2 жыл бұрын
Brilliant explanation and visuals!
@Mathematics_and_physics
@Mathematics_and_physics Жыл бұрын
It is worth noting that g(x,y)=k defines some differentiable manifold , and the gradient vector is expanded in terms of the basis of the orthogonal complement to the tangent space of the manifold.
@전호진-f1g
@전호진-f1g Ай бұрын
Holy this was awesome Thank you for great visualization
@kaytea2983
@kaytea2983 9 ай бұрын
Very nice for developing intuition re Lagrange multipliers.
@VectorSpace33
@VectorSpace33 7 ай бұрын
This video was executed perfectly. Great job.
@readjordan2257
@readjordan2257 2 жыл бұрын
I really enjoy this channel. I love the presentation and explanations. I watch a lot of math channels, but this one is (for me) just as good as any of them.
@meirgold
@meirgold 2 жыл бұрын
Excellent and clear explanation. Thanks very much!
@zacharydavis4398
@zacharydavis4398 Жыл бұрын
Solid content 👍🏾Thanks for spending the time to create and share 🤙🏾
@user-wr4yl7tx3w
@user-wr4yl7tx3w 2 жыл бұрын
Wow, that is really well and clearly explained.
@dorol6375
@dorol6375 5 ай бұрын
Idea for finding the extrema on a boundary: use that boundary's parametric equation and plug it into the function which will result in a 1d function. From that it's as trivial to fund the extrema as it would be on a 1d function!
@NoNTr1v1aL
@NoNTr1v1aL 2 жыл бұрын
Absolutely amazing video! Subscribed.
@ronaldjorgensen6839
@ronaldjorgensen6839 2 жыл бұрын
thank you for your time and persistence
@ThePiMan0903
@ThePiMan0903 2 жыл бұрын
Nice video Serpentine Integral!
@agaz1985
@agaz1985 8 ай бұрын
This is THE way to explain things. Thanks!
@ΠάνοςΚΜ
@ΠάνοςΚΜ Жыл бұрын
Great vid.Only thing that sucked was english is not my mother tongue so i had to search the termology every once in a while in my language and that set my trail of thought back a couple of times
@odysseus9672
@odysseus9672 2 жыл бұрын
From the point of view of finding the minimization, lambda tells you nothing. If you're working with a Lagrangian, though, then the Lagrange multiplier tells you the force needed to maintain the constraint.
@gaboqv
@gaboqv 2 жыл бұрын
It actually also tells you how a little change in the constraint could make this max much higher or lower, in economics this is important as optima with very high sensititivity could mean that having the correct measurements of constraints is paramount.
@PacoCotero1221
@PacoCotero1221 2 жыл бұрын
Its also, in microeconomics, the marginal effect of budget variations in utility + budget constraint problems in some instances
@arjunpukale3310
@arjunpukale3310 11 күн бұрын
So clearly explained. Thank you ❤
@curtpiazza1688
@curtpiazza1688 10 ай бұрын
Interesting presentation! Love the graphics! 😊
@vladimirkolovrat2846
@vladimirkolovrat2846 2 жыл бұрын
Brilliant graphics and explanation.
@cadedulaney1522
@cadedulaney1522 2 жыл бұрын
Incredible explanation this helped me so much
@Witwas-m5k
@Witwas-m5k Жыл бұрын
Good video. Lambda in economics context have meaning call "SHADOW PRICE"
@JanPBtest
@JanPBtest 2 жыл бұрын
10:27 Wouldn't a better way to say it be that the local max point (on the red curve) is tangent to the level curve of _f(x,y)_ through that point? This way the perpendicularity of the gradient of _f_ would be obvious, given the previous explanation of the perpendicularity of the gradient of _g._
@vkessel
@vkessel 2 жыл бұрын
At 10:39 I could imagine a counterexample by deforming the surface. Realized the deformation would result in partial derivatives that don't exist because they depend on the direction of the limit. Mentioning in case someone else runs into that line of thought.
@IamyetHere
@IamyetHere 2 жыл бұрын
A nice explanation all in all. Very powerful visualisation of the topic, making it clear. Anyway, when you say "flat" I don't understand what you mean.
@ebenenspinne4713
@ebenenspinne4713 Жыл бұрын
Awesome video. There is only one thing I find misleading here: at 7:10 you show the gradient vectors as vectors in the literal direction of steepest ascend, implying them to be 3-dimensional vectors. In my opinion this is misleading and gives a wrong intuition for the gradient that I myself had for a long time. Remember how the gradient is defined. Then it follows clearly that the gradient vectors are 2-dimensional vectors for a function like f or g which only has two inputs. It helps to visualize the graph of f or g in one's head as a plane with colours indicating the magnitude of the output and the gradient vectors pointing in the direction of "steepest ascend" of the temperature/colour. Then it follows clearly that the gradient vectors are 2-dimensional, perpendicular to the level curve and all in one plane passing through the level curve (as shown correctly at 9:55). *This should not change* just because we change how we visualize the graph/magnitude of the output of the function. 10:43 has the same issue. With this small correction/improvement, this video is very good!
@피타코라스
@피타코라스 11 ай бұрын
yes you're right i also think that is wrong! in 3 variable, gradient vector should be normal to the tangent plain.
@CG119Animator
@CG119Animator Ай бұрын
I believe this is crucial, especially for beginners learning optimization, as it can cause significant confusion. If they mistakenly think that the gradient vector of 𝑔(𝑥,𝑦) is three-dimensional, it will lead to trouble in learning the concept. Specifically, around 11:55, when the equation ∇𝑓 = 𝜆∇𝑔 is introduced. One vector is in three dimensions while the other is in two dimensions, and this mismatch cannot simply be resolved with a single 𝜆.
@kensonmalupande2424
@kensonmalupande2424 2 жыл бұрын
Excellently explained.keep it up sir 💪
@chamnil8666
@chamnil8666 2 жыл бұрын
very very useful and amazing explanation.Thank you so very much.
@nathanryan12
@nathanryan12 5 ай бұрын
Thanks! I had to watch a few times, but it makes sense now
@NCPROF.
@NCPROF. Жыл бұрын
What an impressive explanation, Thank you!
@프로틴요플레
@프로틴요플레 2 жыл бұрын
The first thing I come up with when considering Lagrange Multipliers is that it is a pure hella substitutions if the number of constraints are less than the number of dimensions..
@atirmahmood7058
@atirmahmood7058 Жыл бұрын
Awesome just awesome because of the perfect visualisation
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