Peyam goes PUTNAM: Destroying a Monster Polar Integral (2021 Putnam A4)
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- Опубликовано: 30 июл 2024
- Peyam goes Putnam: Destroying a monster polar integral. Calculating a hard math competition integral that appeared on the 2021 William Lowell Putnam Exam A4, using calculus techniques such as polar coordinates, averages, change of variables and jacobians, rotations, weierstrass substitution, and inverse trig substitution. In simplest terms, the question is: What is the limit of the integral as the radius of the domain goes to 0. Watch a math professor get destroyed by this seemingly innocent integral. Can you handle the math heat?
0:00 The Integral
3:30 Rotation
6:00 Cleaning this up
9:50 Polar polar polar
18:50 Weierstrasssssssssssss
22:50 The answer
Big thanks goes to Kiran Kedlaya, for maintaining his Putnam exam webpage: kskedlaya.org/putnam-archive/
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This was great! Very fun to see all these techniques put together. If you still have the energy, hope you can post more of these in the future. Understandable though if you don't, ha. Definitely a monster!
"is anyone still watching?" Yes!!!
Wow 😮
This video left me speechless. One must have tremendous practice and insight to solve such monster integral, particularly in a math competition.
Wowww.well done for this great solution! Thanks for the detailed solution😃💯💯
What a great problem! We can review all the techniques of change of variables and integration by substitution!
Especially, it is the coolest to convert the integration area to an annulus!
I love the little sound effects at the beginning of each video!
Thank you 😊
How long did it take you to come up with that!?🤣 love your videos!
I wouldn't have guessed the transformation of x, y to x-y, x+y (equivalent to pi/4 rotation).
I could probably solve from 12:55 by myself but there's no way I'd be able to come up with the steps before that
same , although the polar substitution is pretty straightforward you shouldn't do it right off the bat because of the lonely x's and y's in the nominator
17:40 I was like yes I am still watching!!
Salute
Once we got everything in terms of x^2+ y^2 , we could have transformed that into polar cordinates and integrated over the region
But that’s what I did
@@drpeyam okay then i should probably watch the complete video before commenting something.. sorry for that. 😂
Amazing
I got to the end!
It is really great mathematician
now let's do this integral on a manifold!
damn your videos interest me even though im like 14 years old trying to learn all of this stuff by myself.
17:47 Can you actually solve without changing the limits from 0 to 2 pi to -pi to pi ? or if we need to , why actually?
"did you guess that. probably not". uuhh no, you lost me already before the endless series of u-subs :)
Well I assumed from the start it was going to have some multiple of pi.
Arrrrgh! It is hard! 🤕
I thought this was some exercise for a University entrance exam or something of the like.
I believe that step 4 requires little bit more explanation- why we can actually swap limit and integral. This is often very troublesome. I was trying to solve this problem little different way, and i end up with limit as R-> inf of integral where actually i was NOT allowed to swap limit with integral as i would end up with divergent one.
So you want this video to be even longer? 😂
We love rigor :D
@@drpeyam actually yes since i enjoy such math problems and your way of presenting them :)
@@drpeyam Rigor is good.
This is why I'm recording philosophy videos instead of math videos. Congratulations. Mathematician.
Hahaha
Now see me on figure out the relation between magnitude and angles as inverse measurements on my game dev. Haha!
Seems a tiny bit dodgy around 12:40 where you simplify the function to be the dominant terms and then just integrate that.
In the solution it mentions uniform bounded to bring the limit inside.
So you want the video to be even longer? 😂
pretty trivial tbh
Haha I got nowhere on this one when I took it last year
oh nice. which university were you representing?
Very tasty!
First
You could spend time solving problems that are not cooked up to work. Your tricks are not much use in reality.
Nonsense. How can you solve real world problems without knowing how to apply all the fundamental tools and techniques?
Dr Peyam has plenty of videos about more practical things anyway