Lagrange Multipliers | Geometric Meaning & Full Example

Lagrange Multipliers solve constrained optimization problems. That is, it is a technique for finding maximum or minimum values of a function subject to some constraint, like finding the highest point on a mountain subject to the fact you can only walk along a trail. In this video we study the contour lines or level curves of a function and see geometrically why they are maximized when they are tangent to the constraint curve. That tangency condition leads to the algebraic formula that the gradient of f is equal to lambda times the gradient of g. In this video we will visualize the geometric meaning and then walk through a concrete example.
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This video was created by Dr. Trefor Bazett. I'm an Assistant Teaching Professor at the University of Victoria.
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[@BloobleBonker]

At the age of 66 after trying to understand Lagrange multipliers since the age of 18, I think I've finally got it. Conturs and gradients. Excellent graphics!

[@raulsimon2218]

I could say practically the same. Thank you very much.

[@asdfbaseqsfawefbaef7738]

Hahaha! Such a great comment! You are amazing!

[@KSM94K]

Wow that's absolutely satisfying

[@fraserpye9667]

unlucky

[@kevinbyrne4538]

Same is true for me -- but I'm 69.

[@gamingmonts9737]

by just seeing that graph, I immidiently understood something my professor talked about for 2 freakin hours 😂

[@john.z3822]

now i understood something my teacher talked about for 1 mounths lol

[@hubenbu]

Stop it man, if you're no longer hungry after eating 2 pizzas, remember to pay respect to the first one.

[@grapplerart6331]

@@hubenbu If the first pizza was a 5" when I ordered a 13", I wouldn't pay respect to the first one

[@raptor2245]

😂lol

[@themasstermwahahahah]

Bruh, ikr

[@nathanborak2172]

This is not the way I have usually thought about it but it's equivalent. The way I've usually thought about it is that you imagine walking along the constraint and observing the gradient of f as you go. If the gradient of f has any component along the constraint, it means you can keep walking along the constraint and get higher (or lower) values of f, since the directional derivative is just the component of gradf(f) along the direction you're moving. Therefor you keep walking around the constraint until you reach a point where the gradient of f is normal to the constraint, since at this point f is instantaneously not changing. To me this is more intuitive than thinking the level curve of f should be tangent to the constraint, even though the gradient of f being normal to the constraint IS the level curve being tangent to it. Different strokes I guess.

[@sandroelful]

Genius answer, you helped me a lot. Thank you.

[@nathanborak2172]

@@sandroelful thanks! You made my day

[@mackenziemurray9270]

Wow I like the way your brain works!

[@vishvakravichandran6797]

Been trying to understand this for a while and you totally helped me!

[@offtheleash6440]

This is genius

[@Lawrance36]

Mr. Bazett, I think this version of explanation is the best one in whole KZbin, thank you very much!!!

[@markbell8967]

I agree

[@Septumsempra8818]

By far

[@cauchyschwarz3295]

3blue1brown`s one is way better. But it is hard to beat 3blue1brown. All in all a good video.

[@shemsnow3711]

I think you're the first person I've ever heard explain math without either focusing too much on precise definitions and proofs that no one cares about or just expecting us to memorize formulas. Nice step by step relevant instructions. Very nice.

[@lorentzianmanifold718]

Uhmm that describes pretty much all math channels on KZbin lol

[@yousafe007]

@@lorentzianmanifold718 All good ones, that is....

[@marcourielmedinamandujano5872]

Definitions and proofs are important, you will never understand math without them

[@Aruuuq]

Such a nice video. Very enthusiastic presentation. The graphics are some of the most explanatory one for Lagrangian Multipliers that I've ever seen.

[@firsttnamee3883]

@@DrTrefor 5:54 could you explain why the gradient is always normal to the level curve ? you have any video on that ?

[@sashamuller9743]

@@firsttnamee3883 ya he has a video called gradient vector or something like that just search in his multivariable course

[@firsttnamee3883]

@@sashamuller9743 yes. Thank you. i got that

[@tedskins]

Thank you very much. I find that geometric interpretations of math concepts often make it significantly easier for me to understand

[@zahraazad8441]

me too

[@THELORDVODKA]

I think Geometric interpretation is the whole point of it. The "discoverer" of it probably thought of it in this way itself, it is as if spirit joins othervise empty shell. Even though mathematicians like to portray these stuff on less visual basis, more "universal" logical one, this is how visionaries think in my opinion.

[@mastershooter64]

@@THELORDVODKA complex analysis: *hello there*

[@THELORDVODKA]

@@mastershooter64 It really isn't hahaha

[@ayushthada9544]

I feel whenever I need to brush up on my knowledge of calculus, I always end up on your channel. Your channel is a great learning resource. Thanks for posting these videos. Wish you were teaching Differential Geometry of Manifolds.

[@abhinavsharma3188]

Every person who has ever taken an optimization course should see this short video! It gives you so much mathematical intuition to the concept of constraints and Lagrange multipliers!

[@slurperslurpslurp2670]

Absolutely wonderful, thank you!!! I saw other explanations without showing geometry and using too many jargon that are much longer and fail to explain the simple method. Thank you!

[@Love2ne1BigBang]

Thank you so much for this perfect visual representation of the Lagrange multipliers! I was so use to doing the same calculation techniques without really understanding what they mean and you just clarified everything!

[@borannchanrathnak8256]

Thanks for your explanation, Dr. Trefor Bazett. I was trying to imagine the stuffs in my head and it didn't work until I came here to see your graph visualization. Thumb up for your great work.

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The internet is a blessing, because of people like you

[@parkjessica4444]

love your passion in math and it definitely motivates me! thank you, thank you, thank you!!

[@kasyapdharanikota8570]

best professor teaching maths ,great explanation , very thankful to you

[@marvellousngidi3765]

This is one of the "insane" videos I have ever seen on Lagrange multipliers🙌. You inspire me, keep saving the world 👏👏

[@vishank7]

Can’t thank you enough for this amazing explanation. Please keep up the good work!

[@lewiduressa]

Your videos are REALLLLLYYYY helping me understand my Calc 3 class concept, and you explain it way better than my teacher. Thank you!!

[@bentaakinyi2248]

Thanks Dr for the clear and precise explanation.It is the best so far I have seen regarding Lagrange Multipliers.Very Intuitive!!This is what we need in Mathematics ,not just formula.Thanks once more!

[@mathveeresh168]

His beard is as good as his explanation

[@fireemblem2770]

YESSSS that is an amazing beard

[@nunopires2106]

Was looking for this explanation for days and finally found it! Ty very much sir!

[@anish_cr]

the great graphical representation made it very easy to understand. thanks for the enthusiastic explanation.

[@yarenkaya7872]

I honestly needed this great intuition, thank you sir for the demonstration

[@jrt6722]

Thank you so much, this is on my entrance exam to Japanese University

[@abbieeagle5343]

thank you, you explained what was going on greatly through the diagrams, really helped me out

[@adrianrivero6104]

That explanation about the tangent gradients was very clear and helped me a lot. Thanks

[@briandwi2504]

A joy to listen to your explanations. Lovely bit of maths!

[@GoutamDAS-ls1wb]

Fantastic use of computer graphics to explain concepts. Lots of hard work. Thank you so much!

[@Eric-xh9ee]

I usually don't "like" videos but this is an excellent video, so I gave you a thumbs up! Thank you, Professor!

[@155mushfiqurrahman5]

Your explanation is really excellent ever i see on multi variable calculus....may Allah increase your knowledge more

[@jamyllecarter3016]

Beautiful visualizations. Thank you!

[@RespectThePouch1028]

My textbook had the same explanation, but your visuals and simultaneously lucid explanation finally helped me start to get it. Thank you!

[@DrTrefor]

Glad it helped!

[@johannesvanm.3467]

Time and again you are so incredibly helpful, Dr. Bazett.

[@oliviab6415]

Dr. Bazett! Amazing work! Very concise and clear, graphics were incredibly helpful! My Calculus 3 teacher recommended this video on our lesson and I feel so enlightened. Thank you for your contribution, and keep up the great work.

[@DrTrefor]

Thank you!! Can I ask what school you are at? Always love when I get a teacher recommendation:)

[@todianmishtaku6249]

@@DrTrefor First off, let me express my deep gratitude about this great explanation. Many tutorials seem to skip important moments in explaining the geometric intuition behind the main equation of scalar multipliers. Still I have something unclear. When we say that the gradients of f and g have the same direction, it seems to imagine them lying in the same plane; and, this plane is the same where the contour line of f lies in. Isn't it like this? If so, it is obvious that two vectors that lie in the same plane and are perpendicular to the same straight line, then they are parallel one to another. BUT, the gradient of f in fact does not lie on the plane defined by contour line; it is a vector in 3D space (which, for instance, points to the top of hill). How do we know that the gradient of g at the tangent point is parallel with the gradient of f? Or when you say that the contour line of f and the one of g are tangent, do you have in mind a common tangent line or a common tangent plane? You show that real gradients in 3D are "projected" into 2D. In other tutorials, people see just the ones in 2D and then the analysis that gradients are collinear is quite easy because the analysis about parallelism seems to be based on 2D, but as I mentioned above my concern is related to the fact that the real gradients we put in the equation are in 3D; hence showing their parallelism remains a bit unclear. In other words, when we say that the gradient of f is perpendicular to f, that can be true even if the gradient does not lie on the plane defined by the contour line. Could you please shed more light on these "paradoxes" (which may be only my paradoxes) ? Could you please draw both gradients (for the f and g) on the graph of the left side? Could you please pick up two or more g functions?

[@xanderx8289]

man. you rock! finally, someone who actually TEACHES! not reads a precooked textbook rigid abracadabra.

[@padynz9869]

Thank you very much for really explanatory videos.Could you please provide videos with examples on Fixed step, Optimal step and Conjugate gradient algorithms?

[@mackenziemurray9270]

Thanks for this video! My calc teacher assigns us your videos to watch and we love your graphics!

[@Harry-ub2fv]

Most beautiful explanation on Lagrange Multipliers.

[@chethanar8274]

Such a wonderful explanation. You are the ones who prove that math is interesting. Thank you so much.

[@DrTrefor]

You're very welcome!

[@Festus2022]

This is the best explanation of the Lagrange multiplier I could find online. Thanks. Nice graphics!

[@DrTrefor]

Glad it helped!

[@sushantgarudkar211]

You are at maximum respect for me with constraint that I got entire concept so easily. Thanks a lot for the video and efforts you along with maybe your team if you have it put in to create such content along with matching and timed visuals! Superb Explanation!

[@corbinaquino]

Thanks for making these videos! I’m trying to self teach myself and these videos have really nice visual explanations

[@DrTrefor]

Glad you like them!

[@albertesayas13]

you amazed me, thanks

[@ankitparashar6370]

The 3D visualization helped a lot. One of the best explainations on internet.

[@johnholme783]

An explicit lecture on Lagrange multipliers! Thank you!

[@alfcnz]

This video is simply awesome! I understand now I can push further on mixing human + math animation videos. I understand you're using a green screen with a static blackboard photo on which you draw math and graphs. At the beginning I was thinking you were going to actually write on the board with a chalk. I didn't notice it was artificial.

[@Cheeriot]

I love your teaching style, keep it up!

[@everelement1092]

Man, this video deserves more views and likes. I definitely need these 3D graph to understand it.

[@DrTrefor]

Glad the graphs helped!

[@emerald_eyes]

God bless you, I've been trying to understand this for hours. You explained it so elegantly.

[@nitika9769]

you keep me motivated to do what i am doing, by showing how beautiful math is , im so grateful for having people like u

[@brianwilson3656]

I second all these comments. Wonderful example and wonderful enthusiasm! Thank you

[@Z4yx]

Thanks for the graphics, i understand better now.

[@catchphraseish]

Brilliant explanation. The visual aids help make it more intuitive. Thank you for this!

[@DrTrefor]

Glad they helped!

[@mdmijanurrahman9695]

I wish I had watched your video 4 months ago. This would make my life a whole lot easier. Anyways, watching now will also help me in my exam. You are amazing. I wish you all the best.

[@DrTrefor]

Good luck on your exam!

[@nathanzbrazil]

This is *extremely* well-explained!

[@orangehat1924]

Explained Beautifully, bravo!

[@conintava514]

diagrams helped significantly! thanks for this

[@StratosFair]

You definitely have a gift for teaching, thank you for sharing it with the world

[@user-ex6xc5ox3k]

Damn, this is exactly what I was looking for. Wonderful explanation!

[@kartikshrivastava582]

Very well explained with graphics...!! Thanks for making these videos

[@VietnamSteven]

Just wow! Thank you sir!

[@timothypulliam2177]

Excellent video, perfectly explained. Thank you

[@edwarddi3833]

man, with the help of your video, simply save 50% of my study time for struggling in the textbook.

[@edwarddi3833]

Hi, Trefor, could you please explain in your 3D graph about the difference between the graph of f(x,y)=x²+y² and f(x,y,z)=x²+y²+z² ?

[@irenepadre3352]

you are just the best math prof out there!!

[@khyatitrivedi3360]

Thank You So much... your videos are very helpful. The visuals make it very easy to understand Mathematics

[@JFASACM]

A natural professor! Thank you, sir!

[@DrTrefor]

Thank you kindly!

[@bobbymcdingdong]

First off Trefor : I want to say that you are beautiful and I love you! Second Off (ly) : Your series on Multivariable Calculus is a superb compliment to Denis Auroux's (also superb) MIT course on Multivariable Calculus. Your graphical representations of the problems are so much better than what was available in 2007. Many thanks

[@AJ-et3vf]

Awesome video! Thank you!

[@tommygokes]

This helped so much with my homework! Thank you! My professor in college is awful at teaching, but you're amazing at it

[@shwephyusinmoe5068]

Thank you for your explanation,Sir!😊

[@denali9455]

my midterms tmr, thank u so much dude, made this intuitive

[@muhahaha153]

i already passed my analysis 2 exam but i never understood what i was doing when using langrange multipliers, i just learnt how to use it. Now i finally understand what i have been doing all the time thx

[@RajivSambasivan]

Wow! what a fantastic explanation of lagrangian multipliers. The best I have seen. Amazing.

[@DrTrefor]

Thank you!!

[@TheGraftal]

Thank you for this amazing video!

[@SK-ww5zf]

Great video -- Thanks a million!

[@sureshjakhar8740]

Your Videos are simply awesome Dr. Trefor. Please keep on making such videos.

[@DrTrefor]

Thank you, I will!

[@sinasoltan.m4859]

Thank you so much.one of your 10 minutes videos is better than 10 years of studying at university😀

[@plainfilth1126]

amazing..really. So helpful.Thank you!

[@Tweeteketje]

Extremely well explained!!

[@joaquinacevedo9332]

Excellent explanation!

[@SwabhimanTripathy4331]

Graphs help to visualise far better than asking to do self - imagination that may be error-riddened. I hope this video reaches to all those who truly want to learn this concept.

[@continnum_radhe-radhe]

Thank you so much sir... 🔥 before seen this video.. this topic looks so complex but now it is easy

[@christophersmith8515]

Very helpful, thank you!

[@ryanguo13]

Thank you so much with the high equality graphs😊

[@Junker_1]

Superb explanation. I really love that you also show it visually. It helps me a lot.

[@DrTrefor]

Glad it helped!

[@holyknighthodrick3614]

Nice video helped a bunch, thanks.

[@PiOu812]

Fantastic job!

[@umedina98]

Amazing explanation!

[@levinsonherman6777]

this is so great sir ! thankyou !

[@yeast4529]

Great explanation as always, thanks

[@kelecsenyizoltan274]

Very good! Thank you!

[@eduardoramsesballesteros-r2241]

excelente explicación, gracias por compartir

[@khalidmuntasir9230]

Thank you very much for your excellent lecture.

[@freedomofmusic2112]

for the algorithm! love your videos Dr. Bazett!

[@tommasotrognacara8832]

Simple and effective explanation, thanks!!!

[@DrTrefor]

Glad it helped

[@DaveFer]

Thanks for doing this video. It's great. I'm only sad that I discovered it only after I've already understood it. I've been looking at these Lagrange explanations for years and yours is approximately the best. I love your graphics and your explanation. (Maybe the audio could be a bit better.) If anyone is curious as to a real-life application of this type of solution, think about a consumer that has a budget constraint of, say, $12,000 / year to spend on either food (f) or clothing (c) and the utility U they derive from that food and clothing is given by U = f * c + 1. Their (budget) constraint is the line given by $12000 = (Pf)(f) + (Pc)(c). This method helps us find the combination of food and clothing that maximizes the consumers utility given their budget constraint. Yay! :)

[@ArtutMTMartins]

Great explanation, thank you

[@HermanToMath]

I finally know the whole story...... thanks a lot!

[@DrTrefor]

Glad it helped!

[@themasstermwahahahah]

Jesus Christ, just seeing the two gradient vectors makes it immediately obvious why this works! I have been staring at equations when all I needed was teo pictures This is amazing!

[@daggerdeadshot1694]

Love from india sir keep on the good work ...education learning wisdom unites people