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:

[scalex]

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!

[GiulioDean]

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

[rintepis]

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.

[leonvonmoltke]

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

[omargaber]

I can't believe I managed to understand Lagrange Multipliers after all these
, how magical math is when it's understood, thank you so much

[hatelovebowel]

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

[richardvondracek]

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!!

[firstkaransingh]

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

[yendrian]

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

[gergerger]

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.

[krittaprottangkittikun]

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

[StarContract]

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.

[CGAnimator]

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

[derrick]

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

[boutainabenhmida]

never seen a visual explanation better than this

[laodrofotic]

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

[klevisimeri]

This video is more valuable than gold!

[gohanmineiro]

Simple, clear, and concise explanation. Kudos.

[qwerasdliop]

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.