Good luck!

Oh, OK, I was just listening and not watching. Yes, I’m right on one point (your slides show the dimension difference as being between n and n+1, not n+1 and n-1).

Yes, sorry the text is correct, I misspeak. I should say N+1 and N, nice catch!

@@casualscience Overall, an excellent video! (I think my other point is just empty semantics: in some sense, the lower-dimensional optimization problem is "unconstrained" once the constraint is set, but relative to some hypothetical higher-dimensional problem, it is "constrained"...it seems like both capture the same mathematical meaning).

Thank you! 💯 Will be making more videos, just had a super hectic last year, hoping it will calm down in the next few months

Thank you, that's incredibly kind :)

That doesn't bode well for the quality of my video 😨

Thanks for letting me know Hannah!

Hi this is covered from 7:20-9:40. But it's because you need G to have a fixed value (normally 0), if grad(G) is nonzero, you are moving in a direction where G changes, e.g. It goes from 0 to not 0. Since we only want to explore the space where G is 0, we have to move only in the directions perpendicular to grad(G), i.e. Along directions where G is not changing

Hey thanks my dude!

Might be useful to note that it is possible for the quadratic term in your Taylor expansion to also be zero, in which case we'd want to look at the cubic term. If we want this point to be a stable equilibrium, we'd require this term to be zero also, so we'd want to look at the quartic term. In this example, the small-amplitude solution would not resemble a harmonic oscillator. Incidentally, we know such "phi^4" terms do get considered as perturbations in QFT, although they're usually still starting from a harmonic theory like the Klein Gordon equation.

Manim

Thanks for the comment Trần, really makes the work feel worthwhile to know I'm helping people learn!!

Thank you man! Hope to get it out soon!

haha, well it's the gradient field of circles centered at the origin, so I feel like it existed before Clipping.

Hi, sorry I'm not sure I understand the question, are you asking to maximize the volume of some shape given that it will fit within another shape? Because I don't believe that problem has a simple solution. Also, I would take a look at math.stackexchange.com, that's a good place to post these kinds of questions.

@@casualscience yes any arbitrary volume inside other volume tells weather it could fit or being maximized

Hi Franco, I give a short proof in chapter: 9:23 - Types of Extrema. I also linked a short history in the description that might give some context: abel.math.harvard.edu/~knill/teaching/summer2014/exhibits/lagrange/genesis_lagrangemultpliers.pdf But I'll say it's pretty hard to know exactly what the old mathematicians were thinking when they came up with ideas; the culture around mathematics in the past was much more closed off. I will say, however, that Lagrange was an absolute master of finding ways to solve math problems by introducing a functions whose derivatives give the solution. He was considered the best mathematician of his time, holding the chair of the prussian academy after Euler. The lagrange multiplier is only one example of these "Lagrange functions". Another famous example his Lagrangian from classical mechanics: en.wikipedia.org/wiki/Lagrangian_mechanics Unfortunately that's the edge of my knowledge on the history/development. Lagrange's work is all in French, and I remember having difficulty finding English translations in grad school. I agree there is still a small leap there that doesn't flow naturally, perhaps that is simply Lagrange's brilliance... or perhaps a better historian will come along and have some more to say on the subject. Thanks for the comment!

@@casualscience Thank you! I asked because I tried to find a proof integrating, but I am not really sure if that's ok. I kinda hoped there was a smarter way to prove it. Also I'm not sure if I fully understand the paper, but thanks anyway!

Hahaha, I know that feeling man, have to pick your battles! Thank you for the kind words!!

Thank you very much Ali, I really appreciate that!

This would be similar to the situation at 1:44, here you are optimizing inside of a disk (in two dimensions that disk has volume). You would need to do a free optimization, then manually check which solutions are within the unit circle. You might also have a maximum on the boundary, so you'd want to also include any solutions that you get from using a Lagrange multiplier with g(x,y) = x²+y² -1

You might want to check out the Karush-Kuhn-Tucker conditions, which generalize Lagrange multipliers to inequality constraints: en.m.wikipedia.org/wiki/Karush%E2%80%93Kuhn%E2%80%93Tucker_conditions

My plane of existence is actually x5=42069

@@casualscience seriously, what’s the meaning of the amplitude lambda in extra dim? Its thickness or rigidity? Eg?

​@@hosz5499 In Lagrangian mechanics in physics, it has the interpretation as a type of force multiplier for the constraint. physics.stackexchange.com/questions/47651/how-are-constraint-forces-represented-in-lagrangian-mechanics In general, I think all you can say is that at the zeros it's the ratio of gradients.

@@casualscience i think it means the largest eigenvalue, if f is a quadratic of (x,y) locally (near zero). So local eigenvalues such that f and g gradients align or rescaled contours coincides

See kzbin.info/www/bejne/bKC9hWpoYtOhr6s

Thanks a lot, thats very nice of you to say!

Thank you max!

Exactly! 3blue1brown is possibly the least educative math channel. Why show pretty graphics without showing the math used to make them?

Hi, N-1 is correct there. You have one equation which relates a single variable to the N-1 other ones, so once you fix those N-1 numbers you immediately know the Nth, giving you N-1 DOFs. I do have a mistake at 2:48 though, I should say "choose the N-1 parameters", not "choose one of the ..." That might be the source of your confusion. thanks for the comment

Thank you so much! I appreciate the kind words, and that you took the time to comment; it means a lot to me!

Hey thanks so much, it really makes me motivated to make more videos knowing that I'm helping students. I too was once a physics undergrad!