Introduction to Variational Calculus - Deriving the Euler-Lagrange Equation

[@mathjitsuteacher]

In your drawing you tagged the red curve as eta, but I think this is not correct because in this case you would have eta(x1)=y1 and eta(x2)=y2 and you wanted these nunbers to be zero. From what I understood what you call eta in the picture is actually y+epsilon eta, where eta is a curve which vanishes in the boundary. Am I right?

[@Freeball99]

Yes, you're correct. A few others have asked about this, so I'll pin this comment to the top of the discussion section in the hope that others see it. The red line should be labeled y_bar. I started out drawing one thing and it evolved into something slightly different. Unfortunately, since KZbin no longer allows annotations, I am unable to correct this on the video.

[@mathjitsuteacher]

@@Freeball99 Thanks for answering so quickly. Your video was fantastic.

[@lioncaptive]

Good catch 💯

[@Mechanical_360]

The red curve represents ybar(x).

[@michaels333]

I probably also would have switched y_bar and y. y is arbitrary and can be written as the sum of the optimal path plus some arbitrary path. Maybe I’m knit picking(?)

[@serchuckseyonski9908]

That is, without doubt, the best explained and cleanest derivation of the Euler-Lagrange equations on the Internet.

[@brunoazevedo6380]

insightful

[@petermason7799]

Why oh way didn't I know this 50 60 years ago. There is nothing here that anyone with an engineering degree could not understand. Thank you

[@kvasios]

Exceptional! Absolutely exceptional! Only someone with deep understanding could deliver as such. Extra credits for the historical intro... these couple of minutes for providing a timeline of people, events and facts are helping tremendously in putting things into perspective.

[@Michallote]

Yeah it helped a lot to know we where heading to the generalized form of what Laplace described earlier. Just by adding historic context it unconsciously help you to organise the ideas... Brilliant!

[@mohankrishnan2022]

Yes! The historical introduction at the beginning - succinct but comprehensive - was a great table setter!

[@ako8205]

I too, liked the historical part. Regarding the gallantry of Euler, I read somewhere ("The Music of the Primes"?) that Euler took several weeks to get to Russia where he was invited to work because he was loaded down with creature comforts requested by colleagues already working there.

[@AbhishekSachans]

Most in-depth and elaborate illustration I've seen on the topic. A lot of aha moments. Thank you!

[@akbarahmed3078]

Almost everything I learn, I learn from the internet. It's been like this for the last 5 years and I can confidently say that this is the finest and the most well explained video on this topic I have watched so far.

[@jonathanmarshall2518]

This is beautifully explained. I’m an practical engineer - my brain responds very well to understanding the motivation behind the mathematics. Thank you!

[@motherisape]

most people teach this topic by starting with integral and showing that this integral is stationery. which doesn't makes sense what does it even mean to be stationery. every explanation I see on internet doesn't makes sense this is clearest explanation .

[@NithinGoona]

More than 10 years of confusion in my head cleared in 10 mins. Thanks a lot.

[@cheeseinmypocketsvelveeta2195]

Thank you for doing what others couldn't do for me in helping me understand this beautiful principle. As someone who has the calculus tools and has been interested in classical mechanics for longer, discovering the lagrangian is like finding buried treasure in your backyard. Who has been keeping this from me!

[@adamconkle4042]

As someone who has taken Intermediate Mechanics and has gone through this material, this has been the most thorough explanation of the derivation that I have seen. This is just phenomenal.

[@GustavoOliveira-gp6nr]

Man, this is the best explanation EVER of euler lagrange equation! You were very meticulous in explaining the important details (that was holding me back from fully understanding it) that most videos skip through, and you even explained the history behind it! It was perfect! Congratulations!

[@evanhagen7084]

I knew from the instant I heard his voice that this was going to be an absolute banger of an explanation. This video is incredible. Very hard to find content this high quality even from the biggest names on the internet.

[@ayushtaylorsversion1253]

Im 16 but this is far better than any ecstasy out there

[@ylmazcemalunlu3429]

Maybe I watched more than 15 videos and read various papers on this subject, but mate, this one is far better than the rest you can find on the internet. Why does it always take this much to find quality content? Not sure but this might be my first comment on the platform as well.

[@stephenhicks826]

Thanks so much for this. You've shone a bright light on the Euler-Lagrange equation for me. Thanks. I'm 67 years old but still learning.

[@AnmolSingh-ig3ji]

Wowa💝

[@barehill100]

77

[@SenzeniNxasane]

Wow you did better than my mechanics lecturer, you made it so simple and understandable, you did what my lecture would never do even if they gave him an entire year to explain, to think our mechanics lecture is for 3 hours but still I did not understand, with you it took 25 min, bravo.

[@RellowMinecraftJourney]

To think our lecturer (let me not speak names) couldn't explain it better 😂😂

[@tshilidzisibara8805]

@@RellowMinecraftJourney Poor Warry. But let's thank him for leading us to this teacher here

[@SenzeniNxasane]

🤣🤣🤣🤣🤣ahah you two

[@henryparker3420]

I was reading Landau Mechanics and I couldn't follow the logic. I finally understand it from this perspective, and I was able to work backwards to figure out what Landau was saying too. Thank you very much!

[@gauravkanu2823]

Great video and explanation. Very grateful for the history of classical mechanics and for keeping the concept simple without complicating it.

[@hakankarakurt1100]

You are on fire! One of the best educational YT channels I’ve encountered so far. Way underrated but I guess when you go deep into detail you somehow sacrifice being mainstream. Nevertheless, even though the view counts are low, the appreciation of the viewers are high. Thanks for the content. Stay safe!

[@augustowanderlind7963]

completely agree

[@David-mm6nx]

Words cannot describe the brilliance of this presentation. Best one yet.

[@jamestucker1126]

Only one of the best explanations of the Calculus of Variations that I have ever seen or heard.

[@theo-zj7dm]

I am a french student and I had trouble finding good mathematical explanations in French, and then I found your video. This is amazing, very well explained and rigorous. You made my day !

[@vychuck]

Absolutely delightful delivery in less than half an hour, thank you.

[@jeissontoscano1477]

Thank you A LOT, I really mean it! So much useful information is only a few tens of minutes! It's so difficult to find videos of even simple document explaining those concepts in a simple, yet comprehensive and entertaining way... so thank you for you contributions not only for this video but all of them. This channel is truly a gold mine!

[@ultimatedarktriforce]

Phenomenal explanation I've seen on the internet, no stutters, no delays, no questioning their work, just pure art.

[@dwinsemius]

Great stuff. It's the first time I have heard the word "brachistochrone" actually pronounced. The perspective that the goal is to calculate a function rather than a scalar leads into the need for operators rather than definite integrals very nicely. I wish that I had been prepared for quantum mechanics with this framework.

[@dwinsemius]

@22:37. "I know this must be setting your mind spinning". Right. I still remember when Dr. Katz laid this out at the very beginning of the sophomore course that I took in the summer of 1968 at the University of Michigan. It was rather unsettling, but once the fog in my brain distilled and I could see its wide applicability it became such a wonderful elixir.

[@eleanorterry-welsh7784]

I'm taking a graduate level classical mechanics course and needed a review of calculus of variations because I had gotten rather lost in a recent lecture. This was an incredibly clear explanation and made the whole lecture I had been totally lost in completely make sense. Definitely going to be watching through more of these as my mechanics class covers more of the types of minimization problems mentioned in the beginning of the video.

[@copernicus633]

The best derivation of the Euler Lagrange QE I have seen. Very concise, yet fills in details missing in most other explanations, written or animation.

[@sonyaraman]

This is the gem, I’ve been struggling to find a good video on derivation of this equation, and there it is. Simply the best 🤝🏻 Additional kudos for bringing in the historical overview of how that used to look like back in time😊

[@gouravhalder1256]

I find myself lucky to have found these lecture series on KZbin...😊

[@giuseppecerami1764]

This video is a gold nugget for self-learners. Thank you so much!

[@Freeball99]

You're so welcome!

[@fawgawtten9515]

The best and cleanest on all internet. Thank you

[@johnhalle6404]

Beautifully done. One of the most lucid and insightful lectures I have heard on any subject. Thank you for investing the time and energy to produce it.

[@xhonshameti1749]

This video makes me happy. It’s is obvious you understand the heart of this theory. And it’s obvious that you are genuinely passionate about mechanics. You know know it like an old school watch maker knows it’s watches!

[@wargreymon2024]

Good editing, Intuitive and comprehensive. Your voice is soothing. This is the best explanation on Larangian mechanics, no one on KZbin even comes close.

[@Freeball99]

🙏 I'm telling my wife what you said about my voice! 😇

[@yaokay7585]

i’m confused about equation 6, shouldn’t eta(x1) = y1 and eta(x2) = y2? You say eta(x) starts at point 1 (x1, y1) and ends at point 2 (x2, y2). It would make more sense to have eta(x) start at (x1, 0) and end at (x2, 0) (like in equation 6) and then the orange line @ 11:09 would be y bar not eta. Thanks!

[@Freeball99]

The red curve should be labelled y_bar instead of η(x). I started off drawing one thing and it evolved into something else. I think this is the source of the confusion. So, the red line is the varied path, y_bar and the difference between the two paths is the variation. Consequently, the variation is 0 at point 1 and point2.

[@yaokay7585]

@@Freeball99 ah yes thank you!

[@jesusfuentes7589]

'... and that's it, we're done!' Brutal, absolutely brutal! Many, many thanks - great lesson!

[@moussadiaw1682]

Un sujet très rare sur KZbin and well explained. Thank. If possible a video of Euler-Lagrange applied to image processing

[@manmis007]

People who have some depth to the interest they have would love this......grt job sirji. .....

[@vinodgopinath7837]

Most complete, thorough and clear explanation of EL equation with its background history on youtube! You are a very inspiring teacher.. Lot of respect from India

[@pedrocolangelo5844]

I definitively need to watch your other videos. Your way of teaching is by far one of the best on KZbin! I was trying to understand properly calculus of variations for a long time and you are the one who made it possible for me to understand! Thank you so much, professor! The funny part is that I'm not even a physics student, I'm an economics student. Your video is helping several areas of knowledge.

[@squirepegg6157]

You have my vote for clarity; it's a great presentation.

[@charleshudson5330]

Excellent presentation. I especially enjoyed the introductory historical perspective.

[@bird5119]

This was such a good explanation in a college lecture format that it triggered a Pavlovian reflex: at 22:25 i felt the itch to put everything away in my bag and start to walk out the lecture hall while the professor is still talking

[@Fishtory]

Excellent stuff! Love the history tour in the beginning as well!

[@yuthikasenaratne7250]

the best derivation of the eular larange equation seen so far( espeacialy about that apsolone) others just skip over that

[@MrSlowThought]

You have made clear so many thoughts I've been having on the history of mathematics and physics and the importance of (in hindsight) such simple concepts. You have sketched in some historical connections that I was unaware of, and provided the clues that opened my mind to the Lagrangian and Hamiltonian.

[@EconJohnTutor]

The best introduction into this concept ever. Thank you so much!

[@fisicayquimicahoy]

That's completely and utterly great!! it's the best lecture on Euler-Lagrange equations I ever saw. Thank you very much

[@Ikbeneengeit]

Thanks for the history at the beginning, really helps put the concepts into perspective.

[@theonionpirate1076]

I've never seen this before but now feel I understand it completely. Thank you!

[@euereren]

This is pure art

[@GuidoNagel-h1q]

I dont really comment much in videos, but you deserve one. Really good explanation, clear, concise and also you speak really smooth and easy to understand (im not a native english speaker). i didnt know anything of calculus of variations like 20 minutes ago but now i know how to start it, Thanks For the video Man!!. Hope you have a great day.

[@miaoshang7732]

I learned this equations from Landao's book and i really appreciate your mathsmatical derivation. They are clear and easy-understand.

[@beauanasson3570]

Damn, this content is great. So concise yet so clear, cheers.

[@jevaughnclarke6174]

I am a PHD student in Economics. While I passed the classes utilizing Lagrange and Hamiltonian optimization I always struggled with the 'why'. Thank you sooooooo much as I now got an intuitive idea as to the why. Please do a full course on Variational Calculus. I will pay to be a part of such a class with you if that is what it takes. Please consider doing a course on VC. Thanks.

[@moart87]

You get THIS level math in Economics? Seems more like Econometrics.

[@jevaughnclarke6174]

I had to utilize both principles for Macro and little less so in Micro

[@moart87]

What are the types of problems in economics that you use this on?

[@jevaughnclarke6174]

I had not used hamiltonian nor Lagrange in my econometrics class. Time series models were stressed econometrics along with GLS models. The Lagrangian was used to minimize/maximize utility/ profit functions etc in Micro. The Hamiltonian was used similarly for continuous systems that require optimization with certain constraints on the system variables.

[@xadir]

@@moart87 consumption functions, production functions, growth functions etc. To be fair, proper variational calculus is usually taught at postgraduate level of macro and microeconomics --I had to do it in my MSc course back in the day. Although, I still remember Euler and Lagrange equations from my BSc Econ course as well. It is a common misconception where economics is placed in line with "business studies". Truth is economics is a mathematical science, implementing applied mathematical methodology in both theoretical and empirical research.

[@AbhishekSachans]

Most in-depth and elaborate illustration I've seen on the topic. A lot of aha moments!

[@GoutamDAS-ls1wb]

Thank you very much for a presentation of extraordinary clarity! One of the best expositions on the topic on KZbin!

[@Freeball99]

Glad you enjoyed it!

[@jaafars.mahdawi6911]

Not yet done watching but couldn't resist pausing to throw a word of appreciation and gratitude. Keep it up, sir.

[@luffis1985]

"Euler case you weren't aware was quite the mathematician of his time" Quite the understatement. I'd say he was quite the mathematician of any time.

[@Freeball99]

Agreed...or quite the mathematician of ALL time.

[@moatazabdelrahman5691]

In love with the history part, gets me really interested! and 19 Yo!!.. goodness!!

[@FranFerioli]

Thanks a lot. The fact that you pass from y_bar(x) to y(x) when eta is small is key. A good intuition for this is considering that eta parametrises a whole family of y_bar(x) curves all similar (proportional) to each other, but at different "distance" from y(x). When eta ==> 0, Int [y_bar(x)] ==> Int [y(x)] so you can make the substitution.

[@Cherem777]

Excellent video. As someone watching for the first time, I liked how you pointed out some areas where other’s explanations fell short. Thank you!

[@Freeball99]

Glad you enjoyed it!

[@alonsosainz5214]

Impressive video. I have been looking for a good explanation for a while, yours was the best by far.

[@avatar098]

Thank you for this! My background is in computer science, but recently decided to go back and self study some more mathematics just as hobby. Your explanation truly has put things into perspective for me. Thank you again!

[@hugo_kruger]

excellent presentation, only discovered your channel know, as a civil engineering who work with finite element analysis as they apply to nuclear structures, I really appreciate this explanation.

[@johnmosugu]

You simplified this subject. God bless you

[@ai_serf]

My calculus teacher made me fear the concept of variational caculus, that it was so advanced and abstract. You make it comprehensible and logical. Maybe it's because I'm older and have a lot more experience, but I absolutely treasure the historical background.

[@brandongammon6978]

Great refresher, perfectly explained !

[@chenweizhi8609]

Very very easy to follow, nice video!

[@alvaros9038]

The best explanation I have seen so far! Thank you

[@horacioguillermobrizuela4295]

Excellent video. Thank you so much for your effort to keep it clear and simple. The historical briefing at the beginning was quite enlightening for me

[@garvinmugala7003]

Mathematical and scientific beauty. Wonderful presentation of the lesson Sir. Just what i needed for the morning.

[@erdi749]

...looking for a path that minimizes a function. What is a path? It's a function. So we are looking for a function that minimizes another function.. voooov! wonderful explanation, never thought of variational calculus like that!

[@markgoretsky766]

There is an inconsistency at 17:09. The integral "I" in eq (9) is independent of epsilon "e"; whereas right-side integral in eq (9) does depend on "e". So equating both sides seems to be unnecessary for the derivation of the resulting equation.

[@Freeball99]

Eq 9 does contain ε because I have written it in terms of the varied parameters (y_bar and y_bar_prime which contain ε). So in a single step I have both written the integral in terms of ε and I have indicated that I need to find the result as ε --> 0. So we have created a fictitious parameter and then eliminated it. We do this because we are interested in observing the behavior of the functional in the vicinity of an extremal - namely that it has a stationary point.

[@markgoretsky766]

@@Freeball99 Integral "I" in left side of eq (9) is defined by eq (8) in which it is independent of epsilon (e) and therefore dI/de equals zero for any value of (e), (no need for ε --> 0) Whereas the right-side integral equals zero only when ε --> 0. Right?

[@jaideepganguly]

Excellent presentation, crisp and succinct! Thank you!

[@quantusmathema]

you described this very eloquently thank you

[@Eigenbros]

Excellent video. Really high quality and touched upon many things that typically get glossed over

[@ducciom.gasparri9727]

Best. Explanation. Ever. Now my plan for preparing for the intermediate mechanics exam is to watch all of your videos... and then go back to the Goldstein for the details :)

[@NeelDhar]

I have honestly watched so many videos before this on this topic, and I swear that in 6 minutes you have explained the concept much better than all those videos. All the other videos spent far too much time on the math before breaking down the concept. Love this video.

[@jwilliams8210]

Wow! That was an absolutely extraordinary presentation! Just awesome!!

[@tusharmadaan5480]

Reignited my passion for calculus of variations and optimal control. Beautifully explained!❤

[@KazeReload]

Fantastic video. I just have a question, and it's been a while since I last had calculus stuff in my hands so it may be stupid, but here it is: at 20:53, we write the "du" for the integration by part, so we're writing d(dF/dy'). How does that become "dF/dy - d/dx(dF/dy')"? I really don't understand. I tried applying the differential of fractions of functions but I don't get this. Thanks to anyone who will spend his time to answer me.

[@russellsmart32]

Hey Kaze! Imma try. The parts formula is only being used to manipulate the second term in the integral. n' is like writing dn/dx. So the term is d(n)/dx times dF/dy'. The parts formula switches the d/dx from the 'n' to the dF/dy' but makes the term negative. So now the n is not n' and it can be factored from both terms on the integral.

[@russellsmart32]

Oh and the v and u are functions and so the parts formula (with $ as an integral sign) could be $[u*dv/dx]dx = (cancelled term) - $[v*du/dx]dx I love this stuff if you want more explanation 🙂

[@KazeReload]

@@russellsmart32 Thank you for your answer, even though maybe I didn't explain my doubt well enough because actually now I rewatched the video and got what I didn't get. When he makes the substitution at 20:53 he's actually skipping a part of the calculation. He started like he was only writing the parts integration result, but actually on the second line he's rewriting the whole equation in the first line using the substitution he got from the integration by part and gathering "eta" all in one passage. I thought $(dF/dy-d/dx(dF/dy')) was the "-$vdu" part of the integration by parts, but it actually isn't. Maybe it was stupid to not see it but I hope it helps someone else who didn't see the skipped passage in which the integral by part was substituted into the equation in the first line and then eta was gathered. Thank you ruscle for your help anyway!

[@russellsmart32]

@@KazeReload awesome. Yeah obviously the ‘n’ is my keyboard’s best ‘eta’ haha.

[@KazeReload]

@@russellsmart32 yeah, KZbin should implement LaTex in comments hahahaha

[@thescientist7753]

taking a class on lagrangian mechanics next semester, can't wait!! also hearing about how Lagrange discovered this stuff at only 19 makes me feel bad abt myself lmao. same w hearing about Eulers work, but its inspiring. I think part of the problem is that it seems many of the students in my classes like to take formulas at face value and go off using them with no solid understanding of what any of it means but I dont like to move on until I have a complete conceptual understanding of the topics enough to derive them myself, maybe it will serve me well later in life but for now at least I can see the beauty in some of it that makes it all worth it. Seeing things like this make me so excited because I just know that once I really have a thorough understanding of all this ill be able to see the poetry within the math as I apply it. Still trying to figure out why it must be a function F[x,y,y'] with the y' explicitly included. I also think the eta(x) on the graph should be y bar, not sure. Fantastic video though!! it was my first introduction to the topic and it was better explained than anything I've seen in university and I can tell its definitely not the simplest thing I've learned so kudos!! :) thank you

[@Freeball99]

You are correct, the red line in the figure should be labeled y_bar rather than η. F can be extended to higher derivatives of y, i.e. F = F(x, y, y', y'', y''', y''''). F can also be extended to include additional independent variables (this is what we do when we introduce the parameter ε). I didn't extend it too much in this video because it gets very mathematically tedious and I didn't think it would add anything. Still, I wanted to show how the derivatives of y are treated i.e. we integrate them by parts. Higher order derivatives are integrated by parts additional time depending on the order of the derivative. We use these derivatives in calculating the strain energy (as I have shown in some subsequent examples). Good luck next semester!

[@chiragkshatriya9486]

Sir, One of the best video on Euler-Lagrange Equation on KZbin till date. Could you please make a whole series on ‘General Theory of Relativity’ from scratch to the final equation and it’s solutions like this video.

[@devsutong]

history... motivation... derivation. perfect 🔥

[@workerpowernow]

wow-by far the best explanation of calculus of variations i have seen in undergrad or now in graduate school. This is the first time the concept really made sense. Beautiful idea and great explanation. Also, you have an excellent voice for these types of narrations. Could be a professional narrator haha

[@paaabl0.]

Very good lecture, thank you. Love the historical intro!

[@aryadebchatterjee5028]

u are the best teacher I never had actually well I am an eighth grader and I started learning calculus in grade 7 and none of my teachers supported me and helped me when I faced problems I wish I had a teacher like u to help me out back then I would have way easier and much less frustrating If I had a teacher like u keep up the good work man !! love your videos

[@aniketsengupta9137]

It's great that you are working hard from such a young age. Kudos to you. If you are learning calculus from such a young age you must be brilliant because I couldn't even understand basic trigonometry at that age. Teachers won't support you for such things, you need to take advanced coaching for that advanced stuff.

[@nihilisticboi3520]

Beautifully explained! This is elegance at its best. Thank you so much for this lecture!

[@Freeball99]

Glad it was helpful!

[@yamsh638]

I would give thousands thumps up to this

[@AA-gl1dr]

Wow this is art. I’ve hated math my whole life and you’ve made it digestible and palatable. You’re a skilled teacher

[@adityabaghel1270]

Thank you so much for this wonderful video! Beautifully explained

[@jonathanaarhus224]

The fact that we can minimize any arbitrary functional integral with a single first order differential equation is mind-blowing.

[@damian.gamlath]

My gosh this is so great! Wonderfully explained and made so many things very clear!

[@rangamurali7667]

Beautiful, word for word, line by line, breaking down the mathematical poem, syntax ..speechless! Brings back memories of college days I wrestled with trying to figure. Can you plz do Maxwell equations? Am sure there are many to catch up, we ask for more and more. Our sincere thanks! Awesome!

[@matthewjames7513]

Great video but maybe there is a mistake at 12:25. In the graph you imply that y(x) + eta n(x) = n(x) which contradicts what you write on the left that y_bar(x) = y(x) + eta n(x). I believe in your graph you meant to write y_bar(x), not n(x) at the top.

[@Freeball99]

Yeah, you’re exactly right! Should be y_bar(x). Thanks for catching that.

[@ps200306]

@Matthew James I noticed this too, though I think you mean epsilon where you say eta. In other words, _ȳ(x) = y(x) + ε η(x)._ That means that the actual _η(x)_ is an arbitrary shape function (not shown in the diagram) with the constraint that _η(x₁) = η(x₂) = 0,_ resulting in _ȳ(x₁) = y(x₁)_ and _ȳ(x₂) = y(x₂)._ Great video! -- I read a whole book on analytical mechanics and didn't really get it until I watched this.

[@matthewjames7513]

@@ps200306 yep, thanks! How did you manage to write math equations in KZbin? :O

[@ps200306]

@@matthewjames7513 , they're all just unicode characters -- y_bar, epsilon, eta, subscript 1 and 2 etc. It's a pain having to look up each one, but the result is worth it for something like this. The final touch is to italicise them which you can do in yt comments by surrounding with underscores. Gotta make sure the underscores are bounded by spaces though or yt screws up, so include any punctuation such as periods within the italics, e.g. ȳ(x) = y(x) + ε η(x). Btw, I noticed that some other treatments do away with the scaling constant _ε_ and replace _η(x)_ with a perturbation function _ε(x)._ For instance, see en.wikipedia.org/wiki/Hamilton%27s_principle#Euler%E2%80%93Lagrange_equations_derived_from_the_action_integral . I'm working through the video to check that would still make sense, as it seems that it would be simpler as long as it works out the same. (EDIT: Elsewhere on Wikipedia it gives the same approach as in the video, e.g. see the "Derivation of the one-dimensional Euler-Lagrange equation" section of en.wikipedia.org/wiki/Euler%E2%80%93Lagrange_equation . I think perhaps having the _ε η(x)_ formulation allows it to separately specify the constraints that _ε_ is small and _η(x)_ is differentiable. I think I'll stick with that, partly on the basis of "don't mess with stuff you don't understand").

[@russellsmart32]

Thanks for these comments!! lol. I was getting frustrated.

[@classictutor]

Best! It fits my brain perfectly! I love the historical background too!

[@wuyizhou]

extremely well explained, please keep making great videos like this!