Understanding Lagrange Multipliers Visually

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...

[@scalex1882]

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]

Wait, you guys are getting an absolutely insane proof???

[@ico-theredstonesurgeon4380]

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

[@pyropulseIXXI]

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).

[@ico-theredstonesurgeon4380]

@@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]

​@@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.

[@GiulioDean]

I'm doing a PhD in aerospace engineering and never have I seen a video so clear on this topic. chapeau!

[@rintepis9290]

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.

[@shouligatv]

Same, if anyone has an intuitive explanation, please do share it !

[@jozsefnemeth935]

@@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.

[@gdvirusrf1772]

@@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.

[@sender1496]

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]

@@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.

[@CG119Animator]

That explanation was stellar! You broke down a tough concept without frying anyone's brain cells.

[@leonvonmoltke7923]

I would like to say that it is not often that people explain things better than khan academy. Well done sir.

[@NemoTheGlover]

once you go past Cal I, khan academy content isint that great in my opinion

[@agrajyadav2951]

@@NemoTheGlover what

[@hatelovebowel4571]

this is fking amazing. The best explanation and Calculus should be taught with geometry, it is so clear.

[@richardvondracek496]

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!!

[@omargaber3122]

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

[@firstkaransingh]

I salute you for taking a complex concept and breaking it down to understand at a very basic level. More power to you.

[@krittaprottangkittikun7740]

This video is way underrated, it is very clear and nice!

[@SerpentineIntegral]

@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

[@Murphyalex]

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.

[@plekkchand]

Wonderful, direct, lucid, free of affected cuteness and cosmic background music. Thank you!

[@arthurnascimento6325]

Simple, clear, and concise explanation. Kudos.

[@qwerasdliop2810]

Absolutely incredible! Can't believe something so simple yet incredible was fit into such a simple set of equations, just under the surface!

[@JulianHarris]

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.

[@pyropulseIXXI]

The concept is quite simple

[@derrick20]

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

[@alperyldrm4788]

That is wonderful how you visualize and construct the idea step by step! Grateful!

[@boutainabenhmida6071]

never seen a visual explanation better than this

[@klevisimeri607]

This video is more valuable than gold!

[@harshal8956]

This just blew my mind. This is what I was looking for. Great work.

[@ktgiahieu1]

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!

[@lh2738]

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.

[@franciscorivas4036]

Best explanation I've found so far about lagrange multipliers. Thank you.

[@rhke6789]

Best explanation of Lagrange multipliers on KZbin. Congrats and thank you

[@gossipGirlMegan]

Excellent work I ever met ! Tanks a lot ,deer professor!!!

[@davidebic]

This is exactly the intuition I had trying to understand Lagrange Multipliers!

[@user-dz9eb7fu2f]

Very clearly explained, this clarified a lot for me thank you so much

[@jasonspencer7267]

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!

[@dannis5165]

that rolling ball analogy is so insane. i never understood a concept more clearly before.

[@KYosco]

That makes it extremely intuitive! I don't think one can explain it any better than that.

[@zhuleung2938]

excellent work. you've just made me understand what confuse me throughout my whole collage life.

[@user-ky5ve8ss3x]

every teacher should teach like this! very excellent illustration

[@VectorSpace33]

This video was executed perfectly. Great job.

[@meirgold]

Excellent and clear explanation. Thanks very much!

[@zacharydavis4398]

Solid content 👍🏾Thanks for spending the time to create and share 🤙🏾

[@mase4256]

That was the best explanation I’ve ever seen in multivariable calculus, definitely subscribing

[@sandeepmandrawadkar9133]

Unbelievably super simplified explanation 👏

[@flatmajor6802]

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🔥

[@anthonytafoya3451]

Wow! Thank you for this video. Visuals GO A LONG WAY my brother. Cheers and you have a new subscriber :)

[@NicolasMartinezAngulo]

Could not have explained it any better. Probably top 3 math videos I've ever seen.

[@eklhaft4531]

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.❤

[@autumnreed2079]

This is beautiful! I wanted something to help me explain Lagrange Multipliers better as a tutor and this was brilliant. Thanks

[@canowow11]

really good video on a difficult math problem, but visually you made it easy

[@LucaSalemi]

Brilliant explanation and visuals!

[@user-wr4yl7tx3w]

Wow, that is really well and clearly explained.

[@curtpiazza1688]

Interesting presentation! Love the graphics! 😊

[@chamnil8666]

very very useful and amazing explanation.Thank you so very much.

[@laodrofotic7713]

This is a good video, congratulations on helping millions around the globe with this.

[@breitbandfunker4332]

best video for understanding lagangian multipliers - now i understood it :-)

[@agaz1985]

This is THE way to explain things. Thanks!

[@jackyyeh8763]

Fantastic explanation. Thanks!

[@jmajumder15]

Amazing explanation ! Pure gold

[@NoNTr1v1aL]

Absolutely amazing video! Subscribed.

[@Speak4Yourself2]

Outstanding visuals. Thanks a lot!

[@Words-.]

The visuals are soooo well done

[@christostsaggaras1821]

Read the Wikipedia article and then came back here. While there it was fairly understandable, here the explanation was absolutely brilliant.

[@pyropulseIXXI]

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

[@kaytea2983]

Very nice for developing intuition re Lagrange multipliers.

[@englemanart]

Brilliant presentation!

[@trippymccube8735]

This video made my brain tingle, thank you very much!

[@cadedulaney1522]

Incredible explanation this helped me so much

[@user-qs3ih3ll5f]

Thank you. I love this explanation.

[@atirmahmood7058]

Awesome just awesome because of the perfect visualisation

[@vladimirkolovrat2846]

Brilliant graphics and explanation.

[@adwaitkesharwani3569]

Thank you for the clear explanation!

[@shankhasinha1444]

Thank you so much for making this video.

[@paulgerlach2625]

insane video. cant express how much this helped me

[@suhasisroy7240]

Such a great visualisation

[@ronaldjorgensen6839]

thank you for your time and persistence

[@samfriedman5031]

Amazing explanation and graphics!

[@kensonmalupande2424]

Excellently explained.keep it up sir 💪

[@ilong4rennes]

thank you so much for your extraordinary video! this helps me a lot!

[@yosef7947]

The best video by far on the topic!!!

[@yendrian44]

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

[@dufrain79]

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.

[@jesusfuentes7589]

Hats off, man, really good one. Thank you very much.

[@Yeahagreed]

Absolutely insane. Thank you so much.

[@federicoferraro7080]

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 !

[@mahxylim7983]

Clearly explain! thank you so much

[@tranngoctuan4197]

Fantastic video. Thank you very much

[@CaRmEn899]

This is just awesome. Really thanks

[@readjordan2257]

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.

[@ascanius398]

Thank you. I was struggling with this.

[@EpicGamer-ux1tu]

Really nice video, Thanks!

[@lucialee1232]

Very good explanation thank you

[@matteomansi7499]

Superb explanation, subscribed

[@harrymorris5319]

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

[@sidhpandit5239]

beautifully done

[@marcods6546]

A bit repetitive in the explanation, but finally a good explanation of this concept. Thanks a lot!

[@gourbiswas7176]

Excellent, many thanks to you .

[@yaronyahav656]

this is so so so good. thank you.

[@jhivernuits]

amazing explanation

[@sam08090]

Fantastic explanation 💗

[@mehdiardavan]

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?

[@colins.9367]

You are a life saver, thank you!

[@phy6geniuxYTcreations]

This video is amazing! Thank you for your dedication.

[@firstkaransingh]

Excellent explanation 🙂

[@Amprichu]

YOU ONLY HAVE 1.5K SUBS???????? THIS VIDEO WAS SO HELPFUL WHAT

[@anirudhthatipelli8765]

Thanks a lot, this was brilliant!

[@hereigoagain5050]

Amazing graphics really help to understand Lagrange Multipliers. My middle name must be "Lambda" because I don't contribute to the solution :)

[@formlessspace3560]

Amazing video!