Calculus of Variations ft. Flammable Maths

[@vcubingx]

Support me on Patreon! patreon.com/vcubingx Join my discord server! discord.gg/Kj8QUZU Just a quick clarification why we set the integrand equal to 0 by lucabla: " The key is that delta_q is an arbitrary continuous function (I don't know if the continuity of delta_q is mentioned in the video, but it should be) and a theorem from calculus states: If f: [a,b] -> IR is continuous and the integral of f*g from a to b is equal to zero for all continuous functions g: [a,b] -> IR, then f=0. Applying this with g=delta_q and f being the Lagrangian, then you get the result. " Another explanation by Bram Lentjes: " This is not trivial. It is called the "vanishing lemma" in one of my analysis books. This is true when the integrand is a continous function. The proof of this is quite nicely. Let the integrand be denoted f as a function of x. Suppose by contradiction f > 0, then there exist a point in the domain, let's say x0 such that f(x0) > 0. Since f is continous, for a given epsilon > 0, there exist a ball around x0 of radius delta > 0 such that |f(x)-f(x0)| < epsilon. Now choose epsilon like f(x0)/2 > 0, hence. 0 < integral over the ball of f(x0)/2 < integral over the ball of f(x) < integral over domain of f. A contradiction, since we assumed the integral over the domain was zero. 😀 "

[@prometheus7387]

@vcubingx I am having trouble joining the server.

[@vcubingx]

@@prometheus7387 what trouble are you having?

[@nadiyayasmeen3928]

Oh you have a discord server too? Joining

[@johnny196775]

I would like to understand this video, but I don't have the greatest hearing. You spoke slowly and clearly, but as soon as the fellow with the extremely thick accent started talking swiftly, I lost any ability to follow the explanation. I am so disappointed, I don't think I want to spend time with any of your other videos. Also, when you introduce the second speaker, the style of explanation changes and the whole feel of the production alters. It is incredibly distracting when you do that. You introduced the idea at the beginning that you were going to hand it off to him in the middle, and then when he took over, he unnecessarily introduced himself and spent about a minute off of the explanation you were in the middle of. This is the WORST thing you can do to a student - interrupt your explanation of the math by introducing a lot of information suddenly that has nothing to do with the math and then he picks up where you left off, without reestablishing what has been said already. This is a real bad production if you are interested in teaching people who don't already know the material involved.

[@joaomiguelmachado8178]

P

[@PapaFlammy69]

[@davidescobar7726]

Pappa :'v you guys make an excellent team.

[@luischinchilla-garcia4840]

Amazing work on the derivation!

[@maxwellsequation4887]

@@davidescobar7726 100 th like

[@pattiknuth4822]

This has got to be THE most enthusiastic presentation of the Euler-Langrange equations ever made.

[@DipsAndPushups]

All of his videos are like that, I used to watch them a lot.

[@BJCaasenbrood]

For those interested in mathematics, the approach discussed in the video is also called Hamilton's variational principle.

[@kennedyonyelowe6345]

I saw the solution to the end, then how does the entire process apply to a physical function like compression of concrete over time of curing in days? How does one get back to relate the whole process to the dependent variable of a situation like the one cited here? you can respond via my email at [e-mail] if you dont mind.

[@maxwellsequation4887]

Thank you

[@TheHumanHades]

How ironical since a different mechanics called Hamiltonian mechanics itself exists 😂

[@aptrock327]

I don't think that "For those interested in mathematics" is necessary to say here, we're on vcubingx's channel haha

[@klam77]

The "approach" is from the broadest first principles, almost definitional. A general integral is transformed using first Leibniz's rule, then chain rule, and integration by parts! Why call it Hamiltonian?

[@joem8251]

You're the first person I ever trusted who draws their integrals from bottom to top.

[@vcubingx]

I don't, Jens does :p

[@PapaFlammy69]

:DDD

[@jakobr_]

I’d heard the calculus of variations mentioned before in passing, but before this video I had absolutely no clue how they got it to work. I was imagining all the different ways to change a function, all the uncountably infinitely many variables (one for each x value) approaching zero you’d need to take into account, instead of just the one variable (h) we use in regular differentiation. How on earth is that manageable? A stroke of genius, that’s how. Just multiply all that variation by a single variable (s), and let that approach zero! It’s so simple! Beautiful video, thanks for sharing it. I learned a lot

[@DipsAndPushups]

Usually when I can write the solution of a math problem without looking back at the material I consider it to be understood, but this time even after writing this derivation myself I still don't understand how any of this is related to the potential and kinetic energy in physics and why did we assume that function F takes in derivative as its input. What does F represent? How did we jump from the example of minimalizing the length of a line to this F which takes in t, f and f' as its arguments?

[@leonid9680]

A less known way to derive the Euler Lagrange equations is the way Euler did it originally: He took a discrete version of the functional (a sum of functions in n variables and discrete difference quotients representing slopes), then differentiated this discrete version of the functional with respect to n variables and took the limit as n (number of variables) goes to infinity, so the array of n variables converge to a function, the sum converges to the integral and the difference quotients converge to the derivative of the function. At the time though, Euler did not prove rigorously the convergence but it turned out to be correct. That way you can also visually see the terms in the euler lagrange equations and where they come from. Euler's argument is quite intuitive. The way it is usually derived (due to Lagrange) is more efficient from a computational point of view.

[@erebology]

Thanks! Im gonna look into Euler's argument....

[@comrade_marshal]

Thanks mate. Can you please tell me about the source?

[@8685mike]

Where is this other proof!!??

[@shuoyang9536]

Is there a source I can look up

[@pubgplayer1720]

Since I'm a physics boi; I recognised it as the Lagrange Equation with different variable names! I always associate this with a mechanical system.

[@vcubingx]

Yep!

[@YashMRSawant]

I would like to tell for one of the tricky parts in the proof, at 19:00. To make integral to zero, \delta q can be made a dirac delta with centre at point x which is b/w x_1 and x_2 (which simply means perturbing f only at point x) and therefore right-hand side of Euler-Lagrange equation evaluated at x is zero. This random point therefore can be chosen in any of points b/w x_1 and x_2 which gives us Euler-Lagrange equation.

[@GundamnWing]

This is the first time I've understood the derivation, thanks so much!

[@devtech1015]

Your videos help my through 3rd year mathematics and field theory. Very intuitive.

[@vcubingx]

Great to hear!

[@matron9936]

Wow!! Collab with papa?! Amazing!!!

[@jacoboribilik3253]

I'm new to this channel and this video is fantastic. Great, great piece of work.

[@vcubingx]

Thank you so much!

[@caio868]

Both are amazing, but I can really see the difference between the extremely didact approach of v3x that really is able to foresee any kind of doubt someone can have when following the lecture, basically making the learning process as smooth as possible. Flammable also does it - and he is really really good (maybe a bit nervous in this video), but for some reason, it makes me think much more to grasp any point of the lecture. But thank you, it was amazing! This is also used in graduate level Economics (Macroeconomics), for example, to find the optimal savings decisions of a household in the economy, btw!

[@klam77]

.... And Hotelling rule

[@ianprado1488]

12:55 flammable looks like he is questioning if too much weird came out for this collaboration

[@jimklm3560]

In 19:02 instead of saying that the function in the integral must be equal to zero, we could better say that the expression [θq(L)-d(θq'(L))/dt] must be equal to zero so that the integral of δq*[θq(L)-d(θq'(L))/dt] be equal to zero for EVERY noise-function δq ?

[@nafrost2787]

18:50 but the definite integral can equal 0 when the function is odd and the boundaries are the negative of each other.

[@duncanhw]

I thought the same, see pinned comment

[@maye5307]

Papa's becoming international tho. Daaaamn ;)

[@vcubingx]

International :o

[@RohanDasariMinho]

thanks for the very nice video and it’s super informative!

[@vcubingx]

Glad it was helpful!

[@paperclips1306]

Excellent. Midnight now. Great work.

[@nadiyayasmeen3928]

Ah calculus of variations. One of my favourite topics

[@trigon7015]

Wow, manim looks so good but I can’t code... if only

[@vcubingx]

You can do it!

[@hoodedR]

@@vcubingx It is fun but I don't have anything to animate 😂. Guess I learnt it for no reason.

[@nidhyaneducation7123]

@@vcubingx Please guide me on how to master manim?

[@NovaWarrior77]

@@hoodedR Make more science videos! The world isn't gonna run out of the need for your unique perspective!

[@NovaWarrior77]

Here is a complete tutorial series: kzbin.info/aero/PL2B6OzTsMUrwo4hA3BBfS7ZR34K361Z8F And here is an as of yet incomplete tutorial series by THESE VERY TWO GUYS: kzbin.info/www/bejne/gJfKpZ9jdrxohcU&t Can the world get any better?

[@gergodenes6360]

So we don't need to take "The shortest distance between two points in Euclidian space is a line" as an axiom anymore? FINALLY, AFTER ALL THESE YEARS, ONE AXIOM LESS!

[@shambosaha9727]

It was never an axiom

[@gergodenes6360]

@@shambosaha9727 Please give me a sufficient geometric proof then good sir.

[@sravan_krishnan]

@@gergodenes6360 You're not quite getting the meaning of an axiom. Good sir above is right because even though it is true that the proof is never taught to students it doesn't mean that it was axiomatized. An axiom is just a 'starting point' that holds true in general and results are derived from it. The statement that the shortest distance between points in Eucledian space is a straight line is not something that is assumed to derive results from it. It is a statement that has already been proved using *even more elementary* axioms.

[@tomkerruish2982]

@@gergodenes6360 In geometry, only line segments have length. Using the Triangle Inequality Theorem, one can show that a single line segment has a shorter length than any (finite!) sequence of connecting line segments. Lengths of any other curves can only be defined in calculus. (If the length of a circular arc could be measured in geometry, the circle could be squared. Line segments and circular arcs are the only curves which even exist in geometry.)

[@angelmendez-rivera351]

Tom Kerruish Actually, the impossibility of squaring a circle has nothing to do with whether the length of a curve exists in geometry or not. It factually does not. The circle cannot be squared because the ratio between the area of a circle of radius r and the area of a square with sides ar, where a is an algebraic constant of proportionality, said ratio is a transcendental number and not an algebraic number, and such a transcendental number is transcendental if and only if π is transcendental, which it was proven to be. The transcendental properties of π have nothing to do with the length of curves.

[@nunetoyamato]

Thanks a lot for this presentation which helped me understand such important principle. Not that it changes the final result, but just for the sake of being thorough, I think the derivative result should actually be f´´(x) / ( (1 + f´(x)^2)^3/2 ). IIt's easier to get it right when you use the "MATH INPUT" feature.

[@A_Random_Ghost]

Thanks. I did it by hand and was so confused.

[@turel528]

Finally I understood variational calculus. Thank you so much!!!

[@kebman]

Within programming we call those callback functions, meaning a function that uses another function as input, and then makes a callback to it. They can also be used in recursive functions, or recursive programming, meaning a function that calls itself.

[@erebology]

I gotta give props to callback functions though, because they go beyond plain maths into algorithms. Callbacks allow asynchronous calculations and race conditions, which express ideas beyond math functions alone. They are different and both have value... so to speak.

[@DipsAndPushups]

I thought callback functions only refer to event handlers, user input and things of that nature. I didn't know that callback function is any function which goes as an argument to another function.

[@klam77]

You're saying callback functions are like functional?

[@andrewzerwekh6855]

"... let it die in México somewhere getting eaten by hyenas..." Jajajaja the hyenas are all over, man!

[@andrcarb]

Great stuff my G!!!!!! Keep it up

[@vfurt6c]

Amazing Amazing!🤩😍💯

[@haydnenthusiast]

Papa: For example, take a curve, for example, but make it a little bigger, for example...

[@comrade_marshal]

Finally, we get a proper definition of displacement

[@staristo2355]

Subbed to both. Immediate like. Keep rocking. Wish many more subscribers to vcubingx!

[@RockHardWoodDaddy]

[@simplySwag7567]

why is there no channel like this and 3B1B for High school math and physics :( things are just soo much clearer in these videos...

[@comrade_marshal]

Coz real fun of science and academic pain begin from college

[@That_One_Guy...]

Why don't you just go to Khan Academy for those stuff

[@ilsennodipoi]

The exposition was to me very clear and lucid until I started to wonder about how the hyenas got to Mexico.

[@mario__265]

tysm for this my control design professor was rly bad at his job

[@XanderGouws]

Love it

[@vcubingx]

Your video was fantastic too ☺️

[@XanderGouws]

@@vcubingx Thank you!

[@sebastiangudino9377]

Im astonished at how good your explainations were, and how bad that other dude's were omg Like: "Ah, yes, to derive this equation we are going go define the equation in terms of q-hat, and then use Leibniz rule, if you dont know that go watch a tutorial" My sweet brother, why did you think i came here looking for???

[@MathIguess]

Very interesting topic and very well discussed! :D

[@vcubingx]

Thank you!

[@matron9936]

Hi Werner

[@MathIguess]

@@matron9936 hello!

[@unoriginalusernameno999]

Principle of least action says "pssst, I'm here"...

[@xy9439]

12:55 😂😂😂

[@DragonKidPlaysMC]

Adàlia Ramon lol this is why I love papa flammy! Haha

[@fedem8229]

But we don't have hyenas here in Mexico xd

[@_seventh_son]

@@fedem8229 we got narcos

[@benjaminbrady2385]

Papa Flammy in the thumbnail. He do be lookin fresh doe. Also, a video on the Lagrangian. That's pretty cool

[@MiMaiMix]

awesome teamwork!

[@pappaflammyboi5799]

What is the shortest distance in Minkowski space?

[@maxthexpfarmer3957]

Could you treat the various functions f that go into I as countably-infinite-dimensional vectors by using the terms of a Fourier series of f as the components of the vector v, and then find the minimum of I(v)? Would that be possible? Is it an issue in physics that there are functions that do not have Fourier series? It seems like every function in physics is infinitely smooth.

[@ammarmahdi1422]

At 10:45, he says that q and q-hat are functions of t, so integrating those with respects of q will just result in some constants. Can someone elaborate more on this? I'm confused as to why exactly this would be the case (sorry, my calculus is really weak).

[@PapaFlammy69]

We integrate wrt. t . A simple example would be to evaluate int_0^1 x dx. Gives you x^2/2 from 0 to 1, makes 1/2 which is constant! :) Hope that helped ^^

[@AA-gl1dr]

Wow this just made so many things click. Thank you

[@mohannadislaieh3009]

Glorious ❤

[@ryans7536]

Good video!!

[@vcubingx]

Glad you enjoyed it

[@duncanhw]

Mistake: When you typed it into wolfram, two of the d's are seen as a variable and are cancelled top and bottom. Doing the equation without those d's gives the same as with, which shows something is wrong (if it were parsed as you wanted it to, not taking the derivative would be destructive). Doing it correctly you find - f'' / (1+f'²)^1.5 = 0, so f'' = 0. You now get (almost) the same answer. (Bar the fact that the formula in the video doesn't allow for f'=0, giving 0/0, but the correct formula does, and hence y=c is a valid solution.)

[@MrAnTiTaLeNt]

Do you happen to have the steps writted down? I have tried doing it by hand but instead arrived at [f''(t)] / [(1 + f'(t)^2)^(3/2)] = 0, giving the same answer as in the video.

[@duncanhw]

@@MrAnTiTaLeNt I did it again and got your solution, so I probably made an error. Updated the comment; the video still has a mistake. This solution is ever so slightly different from that in the video, because this solution allows for f'=0 (video's solution gives 0/0)

[@MrAnTiTaLeNt]

@@duncanhw Thanks for the quick responce. I would probably spend another hour or two backchecking it tomorrow. Anyway you are right regarding the mistake in the video. I entered the same formula as the author into WolframAlpha to check my calculations and the very first step WA takes in simplifying the formula is eliminating "d" as if it was a variable.

[@amiraamira-qp6bm]

hi i have question in the end vidéo in the exemple " comment tu as dirive d(df/y')/dx ?? merci

[@eric3813]

Hey vcubingx, i really Love pur Videos, they Are so awesome! I learned a Lot from Them And i am very gratefull :) ( i am 15 And really Love the Type of Content you do :D )

[@vcubingx]

Thanks!

[@dildobaggins2759]

I want to go back to uni to do a math degree in couple years ime 25 i just got told i have rheumatoid in both my hands i dont know if it will happen...

[@bebarshossny5148]

never give up

[@dildobaggins2759]

@@bebarshossny5148 Thats life bro you have to do best with what you got..

[@icarpio]

There is one thing I have never quite understood about the calculus of variations. What purpose does the "degree of variation" term "S" serve here. If we want to define a varied path why couldnt we simply add the functional term "delta q" defined as a functional which is the difference between the original path and the varied path? Does that make sense?

[@shivamsinghaswal4995]

Ah lovely..❤️❤️ That's so beautiful [Math]

[@mazyarta]

Great performance

[@reynanhenry612]

And what does L=T-V mean (kinetic - potential). What is going to be minimized there?

[@vcubingx]

en.wikipedia.org/wiki/Lagrangian_mechanics It's an application of EL Equation

[@BJCaasenbrood]

Suppose you would choose the Lagrangian of the form: L = T+V (i.e., the total energy of the system). If you use the variational principle, and you work it out neatly, you'll find the equality dL/dt = 0. This is quiet intuitive since the total energy of the system is conserved for an isolated system. However, by stating the Lagrangian as L=T-V, you are specifically targeting on minimizing the energy exchange between kinetic en potential energy. And thus it inherently leas to the dynamics of your system, i.e., the Euler-Lagrange equation of motion. Hope this clear for you.

[@pitreason]

So there are a lot of mistakes with this derivation of Euler-Lagrange equations but the most disturbing one I guess is here 18:45 Why the function under the integral must be 0 if the integral itself is zero? The simple counter example is integral from -1 to 1 of x: it is zero but x is not the zero function

[@haroldbn6816]

Yeah but you want to guarantee that the integral yields 0 regardless of the limits of integration, thus, the only condition that can verify that is that the integrand goes to 0.

[@NotTzeentch]

I thought the solved equation should be: Y"/((1+Y"^2)^(3/2))=0 I just want to know if I made a mistake or not

[@vcubingx]

That's the same thing as the one in the video! sqrt(1 + f'(x)^2) * f'(x)^2 = (1+f'(x))^(3/2)

[@xiaozhu7219]

@@vcubingx are you sure this lhs=rhs?

[@vfurt6c]

This is correct! The input in the walfram alpha is not quite right! Regardless so, it is another amazing video! Thank you both so much!

[@sadied0g]

So why does this “finding the function that minimizes distance” translate pretty directly into finding the equations of motion for a system? Is it just that the form of the EL equations terms lend themselves to be easily interpreted as Kinetic and Potential energies of generalized coordinates? Or is there some more intuitive connection to physical principles like, how systems always tend toward states of minimum energy or something?

[@angelmendez-rivera351]

Dylan Benton You are thinking of this incorrectly. Minimizing a functional is a purely mathematical concept which, in itself, bares no meaning in the physical world. The Euler-Lagrange equations are used in physics, but they are used in every mathematical discipline of study as well. They are used in economics and any other application that uses calculus in some form. The reason the Euler-Lagrange equations become relevant in classical mechanics is because those equations are a necessary but not sufficient condition that describe the minimum of a linear functional, as stated in this video. And as it happens, the mechanical state of a system must be the minimum of some functional due to the principle of least action, also often known as Fermat's principle.

[@allaincumming6313]

The nature takes the cheapest ways to do things (in terms of energy), given certain start conditions. That's the Principle of least action, and that's why ELE and equations of motion are directly related.

[@davidzheng4772]

Can you make a video about Algebra 1? I need help solving equations.

[@kylehee]

kzbin.info/www/bejne/eKm2laZ_g5mBjLs

[@vcubingx]

@@kylehee hmmmm who is that

[@angelmendez-rivera351]

It should be noted that the Euler-Lagrange equations are only part of the answer to the question of minimization. It is necessary that a function satisfies those equations in order to minimize the functional, but not sufficient, much in the same logical conclusion that satisfying the equation f'(x) = 0 is a necessity to minimize the function, but it is not sufficient.

[@vcubingx]

Good point! I'll add a few details in the description.

[@DipsAndPushups]

He added another condition, s=0. Isn't that enough to take care of the issue? I never studied functionals, so I am just asking.

[@angelmendez-rivera351]

@@DipsAndPushups No, that is not sufficient either, as that is actually just equivalent to the Euler-Lagrange equations.

[@rayvianasampaio379]

Very good and very simple explanation ; but now how would be the following case : if we have in 3 dimensions , 2 arbitrary curves as 2 fixed curves in this space ; what would be the differential equation of a sheet that would make this sheet a minimum surface that connects these 2 arbitrary , but fixed curves in the 3 dimensional space ??? Greetings from Brazil . Ray Viana Sampaio .

[@joem8251]

Would this be an interesting problem? -- Explore the Hamiltonian Variational Principle in terms of stochastic processes by expanding Euler's equation using stochastic differentiation for the integrand, integrated over a Lebesgue measure to find a (computationally convenient) functional's stationary function of a stochastic random variable... perhaps a martingale.

[@luischinchilla-garcia4840]

Absolutely incredible work! Super clean and clear

[@LucasDimoveo]

What book should I read to learn variational calculus?

[@vcubingx]

I don't have a book to recommend, I do recommend Faculty of Khan's video series on variational calculus

[@throwaway692]

Dude... from where do you get your T shirts?

[@danbakana]

I do not understand why the functional is considered to depend on y(x), y˙(x) , and x. Is there a reason for this ?

[@sebastiangudino9377]

Same question here

[@Blastgun1]

Getting 3b1b vibes from the animations

[@vcubingx]

Yeah, I use the same animation engine as him (that he created)

[@si48690]

Great video man..Can u do a video on applications of langrange-euler eqn?

[@vcubingx]

Great suggestion! If I find a really interesting application that isn't already done by a lot of people eg. brachistochrone, I'll do it.

[@si48690]

@@vcubingx Sure,thanks!!

[@Isinlor]

​@@vcubingx I would really want to see optimal control for a rocket car that has rockets at both ends and needs to move from point a to point b along one axis. Or optimal control for landing a rocket where you want to minimize fuel usage - basically what SpaceX is doing to land Falcon 9 boosters. Mu understanding is that to do it, you need to fire engines at full throttle precisely at the very last moment. Seems to be the only way to land if you can not throttle below the weight of the rocket making hoovering impossible. Due to precision required the landing maneuver is often called a suicide burn :) .

[@insomniaReigns]

Would you even need to calculate d/dx(dF/dy') at the end? We know that dF/dy' is constant and from that we get y' = c_1 and integrating once results in the same solution.

[@vcubingx]

I don't understand how dF/dy' = k results in y' = c

[@insomniaReigns]

@@vcubingx dF/dy' = y'/sqrt(1+y'^2) = k. Rearranging for y' results in y' = sqrt(k^2 / (1 + k^2)), which is again just a constant which we can call c_1. Maybe I am missing something here.

[@insomniaReigns]

Correction: y' = sqrt(k^2 / (1 - k^2)), sorry I missed that somehow. It is still the same idea though.

[@angelmendez-rivera351]

insomniaReigns How is it true that dF/dy' is a constant? y'(x)/sqrt[1 + y'(x)^2] can be anything. For example, if we had not known from Euclidean geometry that the shortest distance between two points was a line, then one could have hypothesized that y'(x) = Ax + B instead, and then dF/dy' is clearly not a constant.

[@angelmendez-rivera351]

insomniaReigns In other words, there is nothing supporting the statement that dF/dy'(x) is a constant.

[@nikhiilrajesh3484]

Could you share the resources you used to piece together the manim library??

[@vcubingx]

github.com/vivek3141/videos

[@magiccarpet9172]

What is your background? Your explanations are very clear.

[@mohammedbelgoumri]

Great video, did you ever consider making a manim tutorial??

[@vcubingx]

I'm in the process of making a few with someone else (idk if im supposed to say who)

[@NPCooking69]

@@vcubingx Sure :p

[@Phi1618033]

Has anyone tried to count the number of times the word function is said?

[@amitbagde1979]

Love your videos From India

[@peterhunt1968]

At 2 mins 30 - how is the second equation derived please? I understand integral of ds from x1 to x2 but how does this become the functional given below? Also why f’(x)2 and not just f’(x)? Don’t both of those then just give straight lines?

[@ahmedzannaahmed]

I can explain if you're still interested.

[@atticusjones]

I have never done a day's calc in my life, but this... I understand this.. Somehow.

[@nadiyayasmeen3928]

19:34 Chemistry? But where?

[@pitreason]

Quantum chemistry

[@omarino99]

This is actually a great video, I’m just getting started on functionals and this is a proof I’ve been wanting to see. I’m a little dubious about one passage though, why did papa say that the integral is equal to zero iif the argument is equal to zero?

[@NathanThomasisepic]

Because the integral of zero is zero?

[@vcubingx]

^

[@bramlentjes]

This is not trivial. It is called the "vanishing lemma" in one of my analysis books. This is true when the integrand is a continous function. The proof of this is quite nicely. Let the integrand be denoted f as a function of x. Suppose by contradiction f > 0, then there exist a point in the domain, let's say x0 such that f(x0) > 0. Since f is continous, for a given epsilon > 0, there exist a ball around x0 of radius delta > 0 such that |f(x)-f(x0)| < epsilon. Now choose epsilon like f(x0)/2 > 0, hence. 0 < integral over the ball of f(x0)/2 < integral over the ball of f(x) < integral over domain of f. A contradiction, since we assumed the integral over the domain was zero. 😀

[@omarino99]

Nathan Thomas yes of course but it is not an “if and only if”. The integral can evaluate to zero even if the function is not, e.g. integral from 0 to 2pi of sin(x), so I’m not sure why you’d pick that kind of solution specifically.

[@philipvarghese7204]

It boils down to something a bit subtle. Clearly, if you integrate a single function over a certain domain, like sin(x) from 0 to 2π, it can integrate to 0. But let's say we multiplied it by a non-zero function: let's call it g(x). If we integrate g(x)*sin(x) over the same domain, it won't necessarily equal 0 {e.g. if g(x) = x, it integrates to -2π}. The real question is this: what function, f(x), do I need such that f(x)*g(x) integrates to 0. As it turns out, this can only be achieved when f(x), and hence the integrand, equals 0. This is why it's known as the "vanishing lemma", and a more formal proof can be seen in an above comment.

[@rayvianasampaio379]

Am I wrong or the professor on the green board is german ??? His light deutsch accent just reveals his origin !!! Sehr gute erklärungen Untericht von beiden professor !!! Ray Viana Sampaio .

[@vcubingx]

Yep! He's German!

[@uva1312]

very cool

[@saiprasad3584]

anybody who knows which vids of 3blue1brown and vsauce are those please link em if uk

[@sohailtabarhossain6096]

It seems we're looking for a function derived out of a BVP; what if there were two initial values instead!?

[@underfilho]

what are the videos of vsauce and 3brown1blue?

[@vcubingx]

3b1b: kzbin.info/www/bejne/eZ3HYaNplplmnLc VSauce: kzbin.info/www/bejne/qZzZn51sbL56o9k

[@underfilho]

@@vcubingx I really love all vsauce videos, but this one is simply incredible

[@vcubingx]

@@underfilho I agree! That was was really good.

[@douglasstrother6584]

"The Principle of Least Action" ~ The Feynman Lectures, Vol. II, Ch. 19 www.feynmanlectures.caltech.edu/II_19.html Richard Feynman inserts a "WOW! That's cool!" lecture in the middle of electromagnetism. I came across this before getting introduced to Lagrangian Mechanics.

[@THEMathHacker-121]

Thanks, enjoyed the read.

[@douglasstrother6584]

@@THEMathHacker-121 That chapter and Ch. 28 on "Electromagnetic Mass" are my two favorites.

[@thebeerwaisnetwork8024]

All hail lord Lagrange.

[@redaabakhti768]

could be made more explicit or rigorous I have always disliked how that euler lagrange / variational calculus has been presented by physicists if you see some inconsistencies and I see a lot of them you can refer to a rigorous functional analysis course requires more work but less confusing and actually makes sense

[@NovaWarrior77]

It's da boyz.

[@sweetsweetbiswal4677]

Thanks sir

[@markginsberg5854]

I hate to complain. The math on the board disagrees with the verbal descriptions given. The second line is a good example. Either we are talking about the perturbed function, and the LHS is correct. Or we are talking about optimality where the RHS is correct. To equate the two is wrong. Therefore, the rest is sloppy at best. 😢

[@thangaraj6411]

I have a doubt. For a function, if we need to find a minimum distance between its two end point, we will find it's shortest distance as you said. Then why we derive the lengthy equation to find it's minimum distance.

[@ArthurSchoppenweghauer]

Me: "Mom, can we get 3Blue1Brown?" Mom: "No, we have 3Blue1Brown at home." 3Blue1Brown at home:

[@T3WI]

Oh hey its my dad

[@vcubingx]

He's my dad too

[@PapaFlammy69]

Mine too

[@Jirnyak]

flammable math part is not very clear but your is good

[@meowwwww6350]

Papa flammy is best!!!

[@miguelaphan58]

try to find ...a destiny , to your hands..when ..explaining something ...to the public...to much erratic movement...chassing your hands, and trying to predic,...the next location...becomes ..an atractor

[@rahmanayan3383]

unbelievable

[@freddyfozzyfilms2688]

6:15 I got so confused cause I thought t was a parameter lol

[@javierr1755]

Herr Jens Fehlau, Ich respektiere Ihre Arbeit und Ihr Land, also respektieren Sie bitte meine. (see 12:56). Hyenas are not part of the native wild mammal species in Mexico. Stick to the math and before you criticize a country, first travel to it and explore its culture and people. Keep up the good work vcubingx and Flammable Maths.

[@markerpullus5830]

Not gonna lie, your video is a lot like 3blue1brown’s (also in terms of explanation skills)

[@vcubingx]

I use his animation tool!

[@erebology]

I want to learn his animation tool! You mastered this style! I'm gonna watch 3b1b's video on it, but I would appreciate your perspective on it, too.