Lagrange Multipliers | Geometric Meaning & Full Example
HTML-код
- Опубликовано: 30 сен 2024
- 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.
****************************************************
YOUR TURN! Learning math requires more than just watching videos, so make sure you reflect, ask questions, and do lots of practice problems!
****************************************************
►Full Multivariable Calculus Playlist: • Calculus III: Multivar...
****************************************************
Other Course Playlists:
►CALCULUS I: • Calculus I (Limits, De...
► CALCULUS II: • Calculus II (Integrati...
►DISCRETE MATH: • Discrete Math (Full Co...
►LINEAR ALGEBRA: • Linear Algebra (Full C...
***************************************************
► Want to learn math effectively? Check out my "Learning Math" Series:
• 5 Tips To Make Math Pr...
►Want some cool math? Check out my "Cool Math" Series:
• Cool Math Series
****************************************************
►Follow me on Twitter: / treforbazett
*****************************************************
This video was created by Dr. Trefor Bazett. I'm an Assistant Teaching Professor at the University of Victoria.
BECOME A MEMBER:
►Join: / @drtrefor
MATH BOOKS & MERCH I LOVE:
► My Amazon Affiliate Shop: www.amazon.com...
by just seeing that graph, I immidiently understood something my professor talked about for 2 freakin hours 😂
now i understood something my teacher talked about for 1 mounths lol
Stop it man, if you're no longer hungry after eating 2 pizzas, remember to pay respect to the first one.
@@hubenbu If the first pizza was a 5" when I ordered a 13", I wouldn't pay respect to the first one
😂lol
Bruh, ikr
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!
I could say practically the same. Thank you very much.
Hahaha! Such a great comment! You are amazing!
Wow that's absolutely satisfying
unlucky
Same is true for me -- but I'm 69.
His beard is as good as his explanation
YESSSS that is an amazing beard
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.
Genius answer, you helped me a lot. Thank you.
@@sandroelful thanks! You made my day
Wow I like the way your brain works!
Been trying to understand this for a while and you totally helped me!
This is genius
Mr. Bazett, I think this version of explanation is the best one in whole RUclips, thank you very much!!!
I agree
By far
3blue1brown`s one is way better. But it is hard to beat 3blue1brown. All in all a good video.
Such a nice video. Very enthusiastic presentation. The graphics are some of the most explanatory one for Lagrangian Multipliers that I've ever seen.
@@DrTrefor 5:54 could you explain why the
gradient is always normal to the level curve ? you have any video on that ?
@@firsttnamee3883 ya he has a video called gradient vector or something like that just search in his multivariable course
@@sashamuller9743 yes. Thank you. i got that
Thank you very much. I find that geometric interpretations of math concepts often make it significantly easier for me to understand
me too
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.
@@THELORDVODKA complex analysis: *hello there*
@@mastershooter64 It really isn't hahaha
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.
Uhmm that describes pretty much all math channels on RUclips lol
@@lorentzianmanifold718 All good ones, that is....
Definitions and proofs are important, you will never understand math without them
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.
Thank you so much, this is on my entrance exam to Japanese University
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.
I honestly needed this great intuition, thank you sir for the demonstration
Sir ,i have a doubt .. why we need this we can simply solve for y in terms of x , y²=1-x²
And then put this y in our function to maximize.
F(x)=xroot(1-x²)+1
And now we can use simple optimization and get the same result .
Why we need Lagrange multiplier then ??
Sir pls reply ..
What if the contour with highest value is not tangent to the constraint curve instead passes right through it and intersects at two points?
Will the explanation be valid then?
How would you then explain Lagrange multipliers since the grad. of constraint is not perpendicular to contour.
The argument is that this is not possible. Because if it did intersect at two point, you could climb still higher until that last possible moment when it only intersected at one point.
love your passion in math and it definitely motivates me! thank you, thank you, thank you!!
a question please: why is grad(g) perpendicular to g's curve ? for me it makes more sense to have it tangent to g.
Suppose we have a curve given by g(x,y)=0. Here x and y are implicitly function of t. Then if we take d/dt of both sides we get by chain rule (grad ) dot =0. So the is the tangent and the grad is the normal as the dot product is zero.
How can we say that the level curves r not touching each other.......I mean. While projecting the countour curves there can a chance of overlapping on one another.....???...please solve my doubt ....,!,!...superb xplination
Sir , facing some problem in curve tracing.....
I think only you solve this...
Please sir , make videos on curve tracing....
best professor teaching maths ,great explanation , very thankful to you
It was like being in a powerful sermon.
Brilliant Dr. Trevor, thanks a lot for your excellent explanation.
The internet is a blessing, because of people like you
5:54 could anyone explain why the
gradient is normal to the level curve ?
Thank you so much sir... 🔥
before seen this video.. this topic looks so complex but now it is easy
If I could, I would give this vid 10^99 likes. So we’ll explained!
Thank you! You taught it better than my MIT multivariable calculus professor lol
Which professor teaches you 18.02?
plls12 Are you an MIT student
Matthias Takele Nope, I applied to the transfer program so hopefully I’ll get in. I ask because I am watching 18.02 on OCW by Denis Auroux and so far he’s been phenomenal
@@pllagunosWell Larry Guth teaches 18.02 but because of the coronavirus, he is no longer able to go back on campus to record lectures. So for that reason, everyone in the class is going to use OCW to also watch Denis Auroux
Hi thanks for the explanation. What always confused me was the notion of the gradient of the constraint g(x). What I understand is the gradient is equivalent to the normal vector of the level curve g(x)=0 and can of course visualise this. And visualise why the normal of this and the gradient of the surface f(x) are scalar multiples. What I can't understand though is the gradient of g(x) when g(x)=0 is just a curve. How can a curve have a gradient. Or how can its corresponding surface g(x) (which doesn't exist) have a gradient. What exactly does del g(x) stand for. If you understand what I'm saying.
Yes, it started with g(x)=0 a curve and moved to using the gradient of g without first discussing that g(x) is a surface with a gradient.
There a lot of echo that not seem to be good, but the explanation is great ,make sure to solve that....🙂
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!
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!
I keep seeing halfplanes and convex functions everywhere!!!!
Just wow! Thank you sir!
You explain better than my professor at Cambridge XD
I finally know the whole story...... thanks a lot!
Glad it helped!
I never understood Lagrange multipliers. I was taught to hate calculus, which is really sad. Calculus is a fascinating subject.
SOOOO many people have that experience, and it really is sad because calculus can be so intuitive and understandable when done right
9:05 Not the focus of the video, but a slightly neater approach is to square the first and second equations and sum them up resulting in x² + y² = 4λ²(x² + y²). By the third equation, it follows immediately that λ² = 1/4. Then, proceed the same as before.
Man, this video deserves more views and likes. I definitely need these 3D graph to understand it.
Glad the graphs helped!
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.
Your explanation is really excellent ever i see on multi variable calculus....may Allah increase your knowledge more
Excellent way to explain this!
how can i visualize this concept in one dimension? currently, i have the problem of maximize the variance (of n variables) through lagrange multipliers (PCA). The function f (variance (points projected on a vector n-dimensional w)) depends only of arbitrary unitary vector w (which maximize the variance).
for the algorithm! love your videos Dr. Bazett!
Thank you for your explanation,Sir!😊
Your videos are REALLLLLYYYY helping me understand my Calc 3 class concept, and you explain it way better than my teacher. Thank you!!
Thanks for the graphics, i understand better now.
Thank you for this amazing video!
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!
Pretty nitpicky, but there is a small error at 9:20 where Trefor says y^2 instead of lambda^2, though the onscreen text is correct...
I'm calling this a "speak-o" :D
dammn just discovered your videos, you are a hero.
I understand the constraint is the equation of a circle, where 𝑔(𝑥,𝑦)=0 and increases in the direction outside the circle. I don't understand how a local max/min would be found "inside" the circle with the way this is being used. The min/max is only found on the circumference of the circle in the following picture. Whereas, if the equation for the circle were modified so all points inside and on circumference were g(x,y) = c, then min/max would be found inside as well.
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
Nice video helped a bunch, thanks.
man. you rock! finally, someone who actually TEACHES! not reads a precooked textbook rigid abracadabra.
the great graphical representation made it very easy to understand. thanks for the enthusiastic explanation.
Could you visualize the case with 2 cóntraints function?
Great video -- Thanks a million!
Ahhh, thanks man, superb video
This is one of the "insane" videos I have ever seen on Lagrange multipliers🙌. You inspire me, keep saving the world 👏👏
you amazed me, thanks
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!
Awesome video! Thank you!
Very helpful, thank you!
A joy to listen to your explanations. Lovely bit of maths!
this video definitely deserves nobel prize
Thanks for the explanation sir, really nice explanation,
But can I know why should the both curves touch on tangent?
If we take some other functions than xy+1 can't there be a maximum point which cut through this plane?
Wow that visualization is amazing
Such a wonderful explanation. You are the ones who prove that math is interesting. Thank you so much.
You're very welcome!
Great explanation. Is there a way to figure out which lambda correspond to max or min without plugging back in? Say we are dealing with multidimensional with several variables, what is the general approach?
Most beautiful explanation on Lagrange Multipliers.
Can’t thank you enough for this amazing explanation. Please keep up the good work!
Damn, this is exactly what I was looking for. Wonderful explanation!
That explanation about the tangent gradients was very clear and helped me a lot. Thanks
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.
Good luck on your exam!
this is so great sir ! thankyou !
This was a great help for the last question for my last math assignment for the year - phew! I can finally go to sleep...
This channel is gold 💙💙💙
My textbook had the same explanation, but your visuals and simultaneously lucid explanation finally helped me start to get it. Thank you!
Glad it helped!
Thanks for this video! My calc teacher assigns us your videos to watch and we love your graphics!
Beautiful visualizations. Thank you!
you save my world Dr Trefor!
I second all these comments. Wonderful example and wonderful enthusiasm! Thank you
I have a question about plotting. Can you show me how you made that surface graph at 1:50? Thank you! I am doing something and it'd be nice to make a graph like that.
I did it in MATLAB
@@DrTrefor Thank you, Professor Bazett. I have been using python, but I'll add MATLAB to this too :)
great video, great explanation, deserves much more views! but the sound quality is just soso, you forgot your clip on mic!
Great video! Nice and easy
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
Very good! Thank you!
man, with the help of your video, simply save 50% of my study time for struggling in the textbook.
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 ?
You deserve a noble prize 🤝
There should be one as math is the queen of science🤔
So on economics, the utility function tends to be logarithmic, while the constraint is the price if two goods, which is typically linear.
Brilliant explanation. Would you be able to make a video of KKT conditions please?
thanks a lot for making this video, really helped alot! just a small suggestion: maybe carrying a small mic will make the sound quality better
Fantastic use of computer graphics to explain concepts. Lots of hard work. Thank you so much!
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.
Love from india sir keep on the good work ...education learning wisdom unites people
Thanks for making these videos! I’m trying to self teach myself and these videos have really nice visual explanations
Glad you like them!
I usually don't "like" videos but this is an excellent video, so I gave you a thumbs up!
Thank you, Professor!
I didn't understand in actually what's the meaning of gradient??
Can you please help me to understand?
I have a whole video on the gradient vector in the multivariable playlist, check it out:)
@@DrTrefor okay!! Sir
Thanks
No chalk good use of color. Dosent block view of text or diagrams. Text appears in blocks not appearing from disracting handwriting. Best method of lecturing,similar to onebrown2 blue. Chalk is obsolete.?
Brillinatly explained.
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.
Thank you!! Can I ask what school you are at? Always love when I get a teacher recommendation:)
@@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?