Understand the idea of coordinate systems, go from rectangular to polar and apply change of variables theorem The concept is a rectangle of (-inf, inf) x (-inf, inf) = the entire plane You can describe this using circle: you need radius and angle. Let the radius go [0, inf) and rotate that line around [0, 2pi) the full circle covers the entire plane! Which is the equivalent formulation! Lastly, you also need the Jacobian.
For those interested in the mentioned video for the polar change of coordinate, that's Peyam's video named The Jacobian (Part 2) and in the exercise 4.
This is my favorite proof of all time, i think it’s the coolest application of fubinis theorem, change of variables theorem and algebra It’s also a fundamental result (normal distribution)
There's actually no need to use multivariables in this case; let t=x^2 then dt = 2xdx = 2u^(1/2)dx so dx = 1/2*u^(-1/2)*du our original function was even so we can express it as: I= 2* integral of e^(-x^2)dx from 0 to infinity after plugging substitution we get: I = 2 * integral of 1/2*u^(-1/2)*e^(-u)du from 0 to inf 2 and 1/2 cancels out so we get: I = integral of u^(-1/2)*e^(-u)du from 0 to inf and that's just gamma of (1/2 ) which we can easily calculate using for example this property: gamma of (1/2) * gamma of (1-1/2) = π/sin(π/2) so (gamma of (1/2))^2 = π and it's positive so I = gamma of (1/2) = π^(1/2)
Filip Pozar the property he used is B(1/2,1/2)=gamma(1/2)^2 (you can prove the relationship between beta function and gamma function by substitution and switching the order of integration) B(1/2,1/2) can be easily integrated as an elementary antiderivative exists for B(1/2),(1/2). Also, if PAX used Euler’s reflection formula instead(as plugging 1/2 into Euler’s reflection formula results in the same equation that PAX wrote), a proof for Euler’s reflection formula doesn’t involve the evaluation of the Gaussian integral or gamma(1/2)... Thus, PAX’s solution is not circular. Dr. Peyam has two videos about the identity he used and I think you should check them out before stating that gamma(1/2) cannot be evaluated without knowing the evaluation of the Gaussian integral first...
Great classic proof! However could you make a video where you solve the Gaussian using differentiation under the integral sign. It also is a great proof. I’m so happy you are back
The polar coordinate party! I wish they had one of those at the ice hotel in Jukkasjärvi. They could have special cocktails and get creative with names: Pi punch, or Sine fizz, or maybe they can come up with a beautiful Zeta function cocktail, mmmm.
Dear, Dr. Peyam.. here is what you need to change! the placement of the camera. the lighting. the cleanliness of the white board ( there is some black stuff that makes the video FEEL kinda blurry... ) placement: camera needs to be closer. lighting: its kinda grey ish, just point a flashlight at the board... xd cleanliness ( of the white board ) : basically get the white board to be more like BPRP's !
Dear San, Unfortunately I can’t change most of the things you mentioned. The whiteboard is the one in my office, and I can’t replace it, and the lighting is also the one in my office and I can’t change that either, nor can I change the volume. I’ll try to fix the positioning of the camera, though
Nice presentation. He does have a video on the Jacobian and I don't discourage anyone from watching it, but there are other presentations of the Gaussian integral solution, that convert the integral to a polar system quite easily and are easily solved without the Jacobian, which gets slightly messy with matricies of derivatives. The polar coordinate approach makes the integral solvable in a straight forward manner and you WILL see the "r dr" come out of the 2πr which comes out of integrating polar solids of revolution. 2πr is the circumference of the infinitesimals "tubes" that come out of polar integration. The Jacobian is a detailed transformation from Cartesian to polar coordinates. Here is his link: ua-cam.com/video/Ilb-moEtJcY/v-deo.html
Yay!!!!!!!!!!
You have an unbearable amount of energy and it is adorable. And you’re talking about calculus, which makes things even better.
I guess sleep can wait 7 more minutes
Everything is completely clear for me, except that crucial part when you change to polar coordinates. Great video, btw.
Watched that explanation video, now that is clear too! Yay!
Understand the idea of coordinate systems, go from rectangular to polar and apply change of variables theorem
The concept is a rectangle of (-inf, inf) x (-inf, inf) = the entire plane
You can describe this using circle: you need radius and angle. Let the radius go [0, inf) and rotate that line around [0, 2pi)
the full circle covers the entire plane!
Which is the equivalent formulation! Lastly, you also need the Jacobian.
For those interested in the mentioned video for the polar change of coordinate, that's Peyam's video named The Jacobian (Part 2) and in the exercise 4.
This is my favorite proof of all time, i think it’s the coolest application of fubinis theorem, change of variables theorem and algebra
It’s also a fundamental result (normal distribution)
This is madness! THIS. IS. SQUARE ROOT PI!!!
Happy to see you again online.
This is my first time commenting on a youtube video and YOU ARE AWESOME.
Thank you!!!
What a nice result! Great explanation, thanks!
There's actually no need to use multivariables in this case;
let t=x^2
then dt = 2xdx = 2u^(1/2)dx
so dx = 1/2*u^(-1/2)*du
our original function was even so we can express it as:
I= 2* integral of e^(-x^2)dx from 0 to infinity
after plugging substitution we get:
I = 2 * integral of 1/2*u^(-1/2)*e^(-u)du from 0 to inf
2 and 1/2 cancels out so we get:
I = integral of u^(-1/2)*e^(-u)du from 0 to inf
and that's just gamma of (1/2 )
which we can easily calculate using for example this property:
gamma of (1/2) * gamma of (1-1/2) = π/sin(π/2)
so (gamma of (1/2))^2 = π and it's positive so I = gamma of (1/2) = π^(1/2)
PAX yes but afaik proving the gamma(1/2) uses the fact that Gaussian integral is √π. So your method is circular and thus not correct
Filip Pozar the property he used is B(1/2,1/2)=gamma(1/2)^2 (you can prove the relationship between beta function and gamma function by substitution and switching the order of integration) B(1/2,1/2) can be easily integrated as an elementary antiderivative exists for B(1/2),(1/2). Also, if PAX used Euler’s reflection formula instead(as plugging 1/2 into Euler’s reflection formula results in the same equation that PAX wrote), a proof for Euler’s reflection formula doesn’t involve the evaluation of the Gaussian integral or gamma(1/2)... Thus, PAX’s solution is not circular. Dr. Peyam has two videos about the identity he used and I think you should check them out before stating that gamma(1/2) cannot be evaluated without knowing the evaluation of the Gaussian integral first...
A day with Peyam video - a happy day!
Great classic proof! However could you make a video where you solve the Gaussian using differentiation under the integral sign. It also is a great proof.
I’m so happy you are back
whoaaa I didn't know that was possible.
www(.)phys.uconn(.)edu/phys2400/downloads/gaussian-integral(.)pdf
It had to be intentional. Pi || e = pie, they're intrinsically related.
Whoa 😮 I actually never realized that!!!
So that's where the video went...
Just woke up in a cold sweat convinced it was gone for good
this is beautiful....
I remember this from the last day of multivariable calc! something OMG right? awesome as always!
Yesss, exactly!!! 😄😄😄 Awwww, I miss this class already, it was so fun to teach and the students were great!
The polar coordinate party! I wish they had one of those at the ice hotel in Jukkasjärvi. They could have special cocktails and get creative with names: Pi punch, or Sine fizz, or maybe they can come up with a beautiful Zeta function cocktail, mmmm.
Just love the infinteeee to infinteee
Link to the video I mentioned, about why you get that extra factor of r when you change to polar coordinates:
ua-cam.com/video/Ilb-moEtJcY/v-deo.html
Impresive!
after watching all your great videos, I just noticed that you're left handed! Go lefties!
Excellent video, you’re awesome 👏🏻
At least! We missed you.
Amazing dear Professor
I loved this video
Can you link to the jacobian video? I'm lazy....
The Jacobian Part 2 Ex 4 ua-cam.com/video/SFLMNvJ7R5E/v-deo.htmlm37s
So cool 💫
Dr. Peyam, I’d like to you do a video doing the ln derivative of the gamma function, show the meaning of the digamma and polígamma’s function
Thank you! This was very helpful :D
Welcome :)
love you man..
Bruh you forreal gonna make me watch another video simply to know why you added an r at the end of the expression? I cant find this video you refer to
Polar coordinate integral
ua-cam.com/video/Ilb-moEtJcY/v-deo.html
Thanks You! It's very helpfull
No mr P ,this multivariable madness calculus .
I'm new here. I don't mean to offend but, what's behind his distinct way of talking?
Could you please do the integral from e^(-1/2) to 1 of arctan(sqrt(-2ln(x))) it is very hard.
Best.....
could you link your video on the description??
the video of the jacobian
ua-cam.com/video/Ilb-moEtJcY/v-deo.html
4:21 could you put a link in the description to that video?
Polar coordinates in a Gaussian Integralua-cam.com/video/Ilb-moEtJcY/v-deo.html
thanks
Dear, Dr. Peyam..
here is what you need to change!
the placement of the camera.
the lighting.
the cleanliness of the white board ( there is some black stuff that makes the video FEEL kinda blurry... )
placement: camera needs to be closer.
lighting: its kinda grey ish, just point a flashlight at the board... xd
cleanliness ( of the white board ) : basically get the white board to be more like BPRP's !
Dear San,
Unfortunately I can’t change most of the things you mentioned. The whiteboard is the one in my office, and I can’t replace it, and the lighting is also the one in my office and I can’t change that either, nor can I change the volume. I’ll try to fix the positioning of the camera, though
Nice presentation. He does have a video on the Jacobian and I don't discourage anyone from watching it, but there are other presentations of the Gaussian integral solution, that convert the integral to a polar system quite easily and are easily solved without the Jacobian, which gets slightly messy with matricies of derivatives. The polar coordinate approach makes the integral solvable in a straight forward manner and you WILL see the "r dr" come out of the 2πr which comes out of integrating polar solids of revolution. 2πr is the circumference of the infinitesimals "tubes" that come out of polar integration.
The Jacobian is a detailed transformation from Cartesian to polar coordinates.
Here is his link: ua-cam.com/video/Ilb-moEtJcY/v-deo.html
Oh, but polar systems implicitly require the Jacobian, anything else is non rigorous :)
Dr. PEYAM!
A Peyam Integral
Its a meeee fubini!!!
he reminds me of mark ruffalo bruce banner
π=3
Here is the video you reference to: ua-cam.com/video/SFLMNvJ7R5E/v-deo.htmlm53s
Also... I am still waiting...
Cute guy:)
I still think we should redefine π = 6.28!
Euler's fallacy