Understanding Lagrange Multipliers Visually
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- เผยแพร่เมื่อ 22 ส.ค. 2021
- When you first learn about Lagrange Multipliers, it may feel like magic: how does setting two gradients equal to each other with a constant multiple have anything to do with finding maxima and minima? Here's a visual explanation.
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This video was funded by Texas A&M University as part of the Enhancing Online Courses grant.
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The animations in this video were mostly made with a homemade Python library called "Morpho". You can find the project here:
github.com/morpho-matters/mor...
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!
Wait, you guys are getting an absolutely insane proof???
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
You should start reading the textbook and doing the proof yourself. This stuff in the video is basically just straight from the textbook. As for visualizations, you should be visualizing this stuff in your head.
If your 'learning method' is to just sit in lecture and let a professor program you, you won't ever learn anything, which is why you'll be confused all the time until someone basically does the learning for you (like this video).
@@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.
@@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 TH-cam that allowed brilliant teacher to explain mathematics from simplest concepts to the most complicated ones.
I'm doing a PhD in aerospace engineering and never have I seen a video so clear on this topic. chapeau!
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.
Same, if anyone has an intuitive explanation, please do share it !
@@shouligatv 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.
@@shouligatv 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.
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.
@@shouligatv 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.
I would like to say that it is not often that people explain things better than khan academy. Well done sir.
once you go past Cal I, khan academy content isint that great in my opinion
@@NemoTheGlover what
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
this is fking amazing. The best explanation and Calculus should be taught with geometry, it is so clear.
That explanation was stellar! You broke down a tough concept without frying anyone's brain cells.
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.
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!!
I salute you for taking a complex concept and breaking it down to understand at a very basic level.
More power to you.
best explanation ever without killing some of my brain cells
agam bu tarz animasyonlarla anlatan başka bildiğin kanallar var mı bu adamın az videosu varmış böyle
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.
This video is way underrated, it is very clear and nice!
@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: th-cam.com/channels/u7Zwf4X_OQ-TEnou0zdyRA.html
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.❤
that rolling ball analogy is so insane. i never understood a concept more clearly before.
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.
The concept is quite simple
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
Simple, clear, and concise explanation. Kudos.
Wonderful, direct, lucid, free of affected cuteness and cosmic background music. Thank you!
Absolutely incredible! Can't believe something so simple yet incredible was fit into such a simple set of equations, just under the surface!
Read the Wikipedia article and then came back here. While there it was fairly understandable, here the explanation was absolutely brilliant.
I'm sure this video did absolutely nothing for your ability to actually solve problems. That is, you didn't learn anything; you only felt like you did
That is wonderful how you visualize and construct the idea step by step! Grateful!
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🔥
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!
Best explanation I've found so far about lagrange multipliers. Thank you.
never seen a visual explanation better than this
Best explanation of Lagrange multipliers on TH-cam. Congrats and thank you
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.
This is the best explanation on this topic that I've seen, after seeking them out for years. I really wish math would be taught more like this, where the intuition comes first, and then you see how it is just notated in equations (that will then have some conceptual meaning.) _Very_ nicely done!
This just blew my mind. This is what I was looking for. Great work.
This video is more valuable than gold!
Excellent work I ever met ! Tanks a lot ,deer professor!!!
Very clearly explained, this clarified a lot for me thank you so much
This is exactly the intuition I had trying to understand Lagrange Multipliers!
That was the best explanation I’ve ever seen in multivariable calculus, definitely subscribing
That makes it extremely intuitive! I don't think one can explain it any better than that.
excellent work. you've just made me understand what confuse me throughout my whole collage life.
Unbelievably super simplified explanation 👏
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
Could not have explained it any better. Probably top 3 math videos I've ever seen.
every teacher should teach like this! very excellent illustration
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.
This is beautiful! I wanted something to help me explain Lagrange Multipliers better as a tutor and this was brilliant. Thanks
Excellent and clear explanation. Thanks very much!
Interesting presentation! Love the graphics! 😊
Solid content 👍🏾Thanks for spending the time to create and share 🤙🏾
Very nice for developing intuition re Lagrange multipliers.
This video was executed perfectly. Great job.
Wow! Thank you for this video. Visuals GO A LONG WAY my brother. Cheers and you have a new subscriber :)
Brilliant explanation and visuals!
really good video on a difficult math problem, but visually you made it easy
best video for understanding lagangian multipliers - now i understood it :-)
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.
This is a good video, congratulations on helping millions around the globe with this.
Fantastic explanation. Thanks!
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
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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
This is THE way to explain things. Thanks!
Amazing explanation !
Pure gold
Outstanding visuals. Thanks a lot!
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...
The visuals are soooo well done
Brilliant graphics and explanation.
Wow, that is really well and clearly explained.
thank you for your time and persistence
A bit repetitive in the explanation, but finally a good explanation of this concept. Thanks a lot!
Absolutely amazing video! Subscribed.
Amazing explanation and graphics!
very very useful and amazing explanation.Thank you so very much.
Excellently explained.keep it up sir 💪
Thank you for the clear explanation!
This video made my brain tingle, thank you very much!
Awesome just awesome because of the perfect visualisation
Brilliant presentation!
Such a great visualisation
Incredible explanation this helped me so much
Thank you so much for making this video.
Superb explanation, subscribed
Thank you. I love this explanation.
Amazing graphics really help to understand Lagrange Multipliers. My middle name must be "Lambda" because I don't contribute to the solution :)
insane video. cant express how much this helped me
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?
Absolutely insane. Thank you so much.
Very good explanation thank you
amazing explanation
The best video by far on the topic!!!
Hats off, man, really good one. Thank you very much.
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 !
Fantastic explanation 💗
beautifully done
Excellent explanation 🙂
Clearly explain! thank you so much
Fantastic video. Thank you very much
This is just awesome. Really thanks
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!
thank you so much for your extraordinary video! this helps me a lot!
Thank you. I was struggling with this.
What an impressive explanation, Thank you!
Really nice video, Thanks!
Excellent, many thanks to you .
Amazing video!