"I'm going to be a Chad and do the integration by hand." Of course you are! Thanks for providing detailed examples of how to solve Isoperimetric problems using Euler-Lagrange. Take care!
Love the whole playlist but I have one general question that pops into almost every problem. Very often we reach a point where y'^2 = something, where we square root and continue. Why do we get the positive square root every time? I am confused because the slope of the paths at different points have different signs and that confuses me a bit. Other than that the proofs have been very straightforward for me and your videos have helped a lot!
sqrt(y'²) = |y'| If y' < 0 then y'² > 0 if y' > 0 then y'² > 0 And thus sqrt(y') > 0 ; forall y in R It doesn't matter if y' is negative. Hope I could help :)
@@yasyasmarangoz3577 I don't think your top expression is correct- the square is always positive but the square root can be negative. For example sqrt(5^2) = sqrt(25) and -5 is a valid answer. since -5 * -5 is indeed 25.
Can you show calc of variation problems where you have to solve for a vector valued function or a multivariate function? Not many examples are available in such cases
What software/hardware do you use to write on the computer? Whenever I try to write using my mouse it looks awful - do you instead use a tablet, or something similar?
There is a [relatively] cheap drawing tablet by drawing device company Wacom called the intuos, you can get it for under $100. There are cheaper options like XP-pen artist products and Huion products, and these work almost as good but at a much lower price.If you have a windows PC microsoft one note can do everything he shows here and has good options like choosing if your page is lined, or has square grid, and you can change the colours. I don't know if it's free or comes with microsoft office however.
This Euler Lagrange equation can't really prove whether the critical point found is max, min or saddle? Also, is it a local extrema or global extrema? Might need some clarification on that
What are some references that get this in-depth into the calculus of variations? Most of the references I can find are for Lagrangian and Hamiltonian mechanics...
This is the sort of math where Alessio Figalli used the optimal transport. Maybe you could explain something about this advanced topic in a future video of yours. But thank you for bringing into youtube ▶️ the real mathematics that is worth learning. Not the kind of mental confusion of the followers 👣 of homotopy type theory, infinity categories and similar madness.
Well this is a problem that was solved in the 19th century and infinity categories don't even have a fully developed theory, so it would make sense this problem is better organized by today, as the explanation had almost 200 years to mature.
@@jure4846 At the level of philosophical opinions, most mathematicians (if not all of them) would agree that researching means/implies 90% risk of failure. But when it comes to practically admitting that a specific branch of research (for example infinity category) they followed is indeed an over complicated, useless, inconclusive attempt to rewrite the math foundation, then no one is honestly willing to say that. On the other side you always have the possibility to review an old problem in a new challenging fashion, that is still open nowadays, because it's not important the specific, original problem, but the math techniques and tools you can use to solve it are relevant.
mmm I don't see a proof of why you can do this. You do as if this is about a functions but this is about functionals so you can't just apply lagrange multipliers like this.
The first 1000 people to use the link will get a free trial of Skillshare Premium Membership: skl.sh/facultyofkhan01211
what for? )))
This channel is a shining beacon in the era of closed universities.
Very interesting problem! I think I'll need to watch a few times to fully understand the solution, but clearly the tools used are quite powerful.
What a chad
"I'm going to be a Chad and do the integration by hand." Of course you are! Thanks for providing detailed examples of how to solve Isoperimetric problems using Euler-Lagrange. Take care!
5:25 LOL
Love the whole playlist but I have one general question that pops into almost every problem.
Very often we reach a point where y'^2 = something, where we square root and continue.
Why do we get the positive square root every time? I am confused because the slope of the paths at different points have different signs and that confuses me a bit.
Other than that the proofs have been very straightforward for me and your videos have helped a lot!
sqrt(y'²) = |y'|
If y' < 0 then y'² > 0
if y' > 0 then y'² > 0
And thus sqrt(y') > 0 ; forall y in R
It doesn't matter if y' is negative.
Hope I could help :)
@@yasyasmarangoz3577 I don't think your top expression is correct- the square is always positive but the square root can be negative. For example sqrt(5^2) = sqrt(25) and -5 is a valid answer. since -5 * -5 is indeed 25.
@@sammysanjayagrawal7430 🤦♂️.
Thank you, you are right.
I disliked my own comment 😂.
Can you show calc of variation problems where you have to solve for a vector valued function or a multivariate function? Not many examples are available in such cases
I'm gonna survive an exam thanks to you.
Can anyone explain to me why the circumference being less than 2pi means that C2 is positive?
Good stuff
What software/hardware do you use to write on the computer? Whenever I try to write using my mouse it looks awful - do you instead use a tablet, or something similar?
There is a [relatively] cheap drawing tablet by drawing device company Wacom called the intuos, you can get it for under $100. There are cheaper options like XP-pen artist products and Huion products, and these work almost as good but at a much lower price.If you have a windows PC microsoft one note can do everything he shows here and has good options like choosing if your page is lined, or has square grid, and you can change the colours. I don't know if it's free or comes with microsoft office however.
It seems to me that he made the slides on a tablet beforehand and uses masks to reveal what he has written
This Euler Lagrange equation can't really prove whether the critical point found is max, min or saddle? Also, is it a local extrema or global extrema? Might need some clarification on that
What are some references that get this in-depth into the calculus of variations? Most of the references I can find are for Lagrangian and Hamiltonian mechanics...
I recommend nonstandard analysis, it simplifies it greatly and gives non physics examples
You made this video by which apps or side or device?
hi, is it mean that, Between closed curves of constant length, the circle has the largest area?
The isoperimetric Problem my friend!
area/circumference is maximized by the circle.
This is the sort of math where Alessio Figalli used the optimal transport. Maybe you could explain something about this advanced topic in a future video of yours. But thank you for bringing into youtube ▶️ the real mathematics that is worth learning. Not the kind of mental confusion of the followers 👣 of homotopy type theory, infinity categories and similar madness.
Well this is a problem that was solved in the 19th century and infinity categories don't even have a fully developed theory, so it would make sense this problem is better organized by today, as the explanation had almost 200 years to mature.
@@jure4846 At the level of philosophical opinions, most mathematicians (if not all of them) would agree that researching means/implies 90% risk of failure. But when it comes to practically admitting that a specific branch of research (for example infinity category) they followed is indeed an over complicated, useless, inconclusive attempt to rewrite the math foundation, then no one is honestly willing to say that. On the other side you always have the possibility to review an old problem in a new challenging fashion, that is still open nowadays, because it's not important the specific, original problem, but the math techniques and tools you can use to solve it are relevant.
mmm I don't see a proof of why you can do this. You do as if this is about a functions but this is about functionals so you can't just apply lagrange multipliers like this.
Functionals are functions of functions.
@@yasyasmarangoz3577 True, but the theory of Lagrange multipliers applies to function from R^n to R and not to functionals.
@@aj-uo3uh Could you serve a source for what you mean?
Btw I had a not so good day, my brain capacity did fall ^^