Understanding Lagrange Multipliers Visually 

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!

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.

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

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

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.

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!!

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!

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

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

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

Simple, clear, and concise explanation. Kudos.

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

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

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.

never seen a visual explanation better than this

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

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

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

This video is more valuable than gold!

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

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!

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

Best explanation of Lagrange multipliers on RU-vid. Congrats and thank you

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

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

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

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

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

best explanation ever without killing some of my brain cells

every teacher should teach like this! very excellent illustration

The video was phenomenal and truly amazing; thank you for providing such valuable content!

The animation at 2:50 was incredible, definitley ignited a light bulb moment in my head.

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

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

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

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.

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

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

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

this video is low key the best math lesson even made, congrat s

Excellent and clear explanation. Thanks very much!

Unbelievably super simplified explanation 👏

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 !

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

single-handedly saving my vector calc grade!

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.

would solving like (lim m->k {d/dz (f/(g-m))}) = 0 work?

This video was executed perfectly. Great job.

Wow, that is really well and clearly explained.

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

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.

Absolutely amazing video! Subscribed.

Hey, nice video, could you tell what animation tool you use for the animations here?

The visuals are soooo well done

This is THE way to explain things. Thanks!

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

Very nice for developing intuition re Lagrange multipliers.

this channel is highly underrated...

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?

Thanks! I had to watch a few times, but it makes sense now

Incredible explanation this helped me so much

thank you for your time and persistence

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

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

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!

Thank you. I love this explanation.

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?

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

Thank you so much for making this video.

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!

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...

Can someone please explain how the gradient of F became parallel to the gradient of G? I know they share the same tangent plane so the gradients must be parallel but i was not able to understand the logic that was said in the video.

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.

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.

Awesome just awesome because of the perfect visualisation

insane video. cant express how much this helped me

The best video by far on the topic!!!

Excellently explained.keep it up sir 💪

Outstanding visuals. Thanks a lot!

Interesting presentation! Love the graphics! 😊

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

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..

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

Brilliant graphics and explanation.

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

Your explanation is another level. You link every step with a question, which is an excellent way for people to follow well

Absolutely insane. Thank you so much.

Does this only work for points of the surface of f(x,y) constrained by the equation g(x,k)=k? In other words, if you were looking for a maximum using this method, would it give you: 1) a point on the boundary or 2) the point in the center of f(x,y)? I suspect 1), and you would get 2) if the constraint was g(x,k)

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

Amazing explanation ! Pure gold

Fantastic explanation. Thanks!

Very good explanation thank you

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.

Thank you. I was struggling with this.

Thank you for the clear explanation!