The Gaussian Integral

[@GogiRegion]

I love how the video makes you feel smart by hinting at where it’s going so that you figure a lot of it out on your own.

[@NazriB]

Lies again? Dark X

[@spiderjerusalem4009]

yea, pretty much most math books. "as the readers should verify", "it is indeed trivial and shall be left as an exercise to the readers", "the argument will be outlined in exercise (insert number)"

[@hashimabbas3977]

No word for this great video. Spectacular.

[@David_F97]

According to you, there are actually 7 words.

[@jadeblades]

@@David_F97 ok

[@DiegoMantilla]

But you said no word.

[@BurningShipFractal]

That 7 words doesn’t include

[@JPiano]

"Might ring a bell"... I see what you did there

[@wiseversa5369]

hehehehehehe

[@newwaveinfantry8362]

*Laughs in Bell curve*

[@saurabhshukla4900]

You have transformed the integral into polar coordinates. This is a classic example of how sometimes complex cartesian coordinate integrals can be simplified in polar coordinates.

[@rohitjain1455]

Do also read about cylindrical coordinate system

[@williamswang7052]

@Kappa Chino laughs in confocal eliptical coordinates.

[@chaahatsingh6733]

Hey can you please explain why was the height taken as the function itself ( -e power R2 squared) m

[@danipent3550]

One integral is used for area, two for volume, what’s the use of triple integral?

[@sample8289]

@@danipent3550 density

[@abhishekbhatia6092]

Dude integrated so hard that his voice changed at 9:35

[@newwaveinfantry8362]

"I talked to Barzini."

[@matthewheydon1591]

@@newwaveinfantry8362no way I watched the godfather yesterday for the first time and coincidently just understood this reference

[@newwaveinfantry8362]

@@matthewheydon1591 Haha, great! I forgot about this scene entirely.

[@gamechep]

I am blown away by the way you simplified it. Pure awesomeness.

[@slender1892]

I mean he didnt invent anything its in every calc textbook about integral calculus right

[@petachad8463]

@@slender1892 stfu bit*h , you probably scored 0 in maths 😂😂

[@logasimpso8274]

That is stunningly beautiful. One of the best mathematical explanations I’ve ever seen. Well done, sir

[@MichaelMiller-rg6or]

This is easily the best video explaining the Gaussian Integral I have ever seen!

[@ananthakrishnank3208]

This is no random math as your channel name suggests. The geometrical proof is always the most elegant way. I can't skip ads for you. Great content!

[@rajendramisir3530]

I saw Professor Christine Briner from MIT used polar coordinates, double integrals and change of variables to evaluate the Gaussian integral. Now, I see you find the volume under the bell curve by summing tubes(hollow cylinders) whose radii varies from 0 to infinity. I like the geometrical approach to finding the volume of each tube. I find the visual aids intuitive. I think this is a fresh and intuitive way of evaluating the Gaussian integral. Thanks.

[@x.invictus6597]

I'll just echo what others have typed; this is probably the best explanation on math I've ever had.

[@markolazarevic4209]

thx a lot man. finally a good explanation

[@laser4887]

no kidding this is probably the best explaination i found on youtube

[@grinreaperoftrolls7528]

THIS IS BEAUTIFUL! This is the best video I’ve seen that talks about this integral.

[@laman8914]

I have watched a number of clips on the Gaussian Integral, but I like the practical way this has been explained.

[@celsio2230]

Came back to this video after taking calc 3 and seeing you explaining the gaussian integral without getting into jacobians and double integral mess is absolutely stunning

[@citiesinspace]

Probably the best video on KZbin explaining this problem. Thank you!

[@SlingerDomb]

i find this is the best way to understand what's going on and what is the concept of doing such things with this problem. thanks for making this video.

[@1131295]

This video randomly popped up on my feed and I am glad that it did. Have been out of touch of mathematics for about 8 years. Brought back a lot of memories.

[@WillHeydinger]

This video is AWESOME! I think they skipped this in all my calc classes in college. He makes it so intuitive. I'll def be coming back to this channel

[@neuroxplorein]

you have an entertaining teaching style (like a short movie), to the point that I was excited about what's next 🍿🍿🍿

[@ЧингизНабиев-э2г]

OMFG, I hope i could get that video when I started to learn Calculus ... Special Functions(Non-elemental) were such a pain for and after all I just used abstract rules to get them, but omg this interpretation of Gaussian Integral is awesome...

[@Jason-mr8tp]

Really great video! Thank you so much for this, I had a really hard time understanding how to integrate wave functions that included this exact integral. Thank you so much!!

[@Peter_1986]

One of my math teachers at university showed this exact method when I studied Applied Mathematics (which was basically "math used for all kinds of real-life things", so it was like a big test of all the students' previous knowledge in other courses), and I was blown away by it.

[@johanneshagel3609]

Sehr raffinierte Transformation, die das Problem entscheidend vereinfacht. Das könnte man sogar in einem motivierten Leistungskurs bringen! Super!

[@bisky-4967]

this channel is like sent from god, great video! you deserve a lot more!

[@ahadamin7361]

I am not good at English but this video is one of greatest explanation videos I have ever seen. Thank you very much

[@Bearman5]

1:25 "might ring a bell" nice pun haha

[@GamerTheTurtle]

An extremely well done video for an extremely beautiful integral.

[@monke9865]

The best intuitive and clear video ive seen on this topic!

[@miso-ge1gz]

pff i love how you just turned one integral into the same integral+another

[@antoinegaillard9440]

I listened 9:35 to 9:36 at 25% speed, and I enjoyed it very much!

[@glaswasser]

ouuuhhhhfff paaaaaaaiiii

[@Miguelgil-bb4oz]

WOW THIS IS AMAZING THIS IS SO CLEAR!!!!!

[@todabsolute]

The substitution at 8:44 is spooky, I mean, it comes out so smoothly. It's easy to see that this integral comes down to pi but man I'm still scratching my head on that.

[@mattialuca100]

it wasn't necessary tho. You just need to know that integral of f'(x)*e^(f(x)) = e^(fx). So in this case you have -pi * integral of -2x * e^(-x^2) which is -pi*e^(-x^2)

[@TheDigiWorld]

I have seen a couple of videos where this integral is solved by substituting x^2 + y^2 = r^2 and letting dy dx = r dr d(theta) and another one where instead of polar coordinates stuff it goes by substitution y = xt. This is the first time I have seen a visual approach and it's really really good... Much appreciated :slightly_smiling_face:

[@danidino1645]

Oh dammit. This is exactly what i needed to answer a question on a paper that was due to 2 days ago. Glad that I understand it now, even if its a little late. Good video!

[@leonardobarrera2816]

Fantastic It is like... you have to use properties of complex numbers to end out with the error funcion That was very funny to think!!! Thanks a lot for the video!!!

[@annaisabanana6848]

I always knew to do this by converting to polar form integrating from 0 to 2pi and then 0 to infinity, but this explains where the extra r comes from very well! thank you so much

[@MuitaMerdaAoVivo]

In a more mathematical way, the 2pi*r is the jacobian of the substitution and you can calculate it solving the determinant of the jacobian matrix.

[@gulshanmustafayeva1707]

Thank you for providing with detailed explanation not just formulas, great job!

[@1mfikri]

One of the best explanations I've ever seen...well done!

[@traiancoza5214]

I stopped at 4:07 and tried doing the problem with stacked circles and it gave a very easy answer. Was kinda disappointed that you would go on and do it with cylindrical shells, but it works both ways.

[@imreallyhatebufferin]

Finally i can understand this without polar things. Thank Youuuu. Very Logical and easy to understand. Big much thanks

[@surendrabarsode8959]

Absolutely brilliant.... never seen such an explanation. Thanks a lot.

[@AymanSussy]

That video make me remember my first year in college when we were studying electrostatic and electromagnetism 😊😊

[@vincentdublin3127]

Is it the electric field of a uniformly charhed disk?

[@AymanSussy]

@@vincentdublin3127 Yes those things XD

[@mladengavrilovic8014]

this is a very great video that helped me understand the gaussian integral and I hereby recommend it to everyone

[@kirstenwilliams9246]

Bravo! Best explanation of this integral ever!!!

[@pinklady7184]

My first time in your channel. You describe well everything, so I have to subscribe.

[@matthiasdjveitmanndajenth5365]

Very easy-to-digest explanation of a very complex topic. Excellent video!

[@Vidrinskas]

Best explanation I've seen for this integral. Well done sir.

[@taksin0214]

โอ้วพระเจ้าจ้อดดดด คุณเป็นคนไทยรึนี่ นั่งดูจนจบ สำเนียงโครตเทพ วิชาการ เทพสัด สุดๆครับ ดีใจ กับ อนาคตของชาติจริงๆ

[@Matteo-sw5xc]

It’s easier too. If you know the gaussian formula, you can moltiplicate and divide that expression for sqrt(2*π*1/2), and drag out the numerator from integral. Now the integral represents the area of a gaussian with mean=0 and variance=1/2 that’s equal 1 and the amount out of integral is sqrt(2*π*1/2), that’s actually sqrt(π). Great video anyway

[@MrG0olden]

Thank you. This was helpful. Even though I came with somewhat of an understanding of both the Gaussian function and Integrals, my background isn't math-heavy. This was very pleasant.

[@luigibeccali2840]

The f ing best integral ever

[@jugsma6676]

Super explanation, this should have more likes and views.

[@madeyedexter]

Thank you. You just earned a subscriber. Looking for more such great content.

[@zildijannorbs5889]

Fantastic video. Kept me entertained with great explanation and graphical representation.

[@nickr7437]

This is a great video.... very understandable. A lot of videos go through what to do, but this really helped to visualize it. Thanks, and great job!

[@AlejandroGomez-yx1sg]

Beautiful! Congratulations for such a wonderful video and demonstration.

[@afrika-karibianaestudionan4050]

[2024] Greetings from Curaçao, an Island Nation in The Caribbean. Great clear video explanation.

[@meraldlag4336]

First time I understood this integral. Incredible video

[@RiverMtn21]

my jaw dropped when it came out to be square root of pi!

[@eitanethan]

This is so beautiful. Amazing explanation, thank you

[@АлиАскеров-е5г]

I have watched a lot of videos , but understood nothing. But you explained it perfectly!!

[@roygalaasen]

Your drawings are absolute **** but they are really doing its job and your explanation is absolutely ace! Really a brilliant video and I am actually here to praise your work, not diss it, because it is simply brilliant! (In case it failed to show, I am actually just making a joke while my real goal is to praise this video. The drawings are crude but they are very successful in conveying the message and your explanation is on 3b1b level in terms of teaching and being understood.)

[@rachalaishram9699]

Wow.. great presentation sir

[@MikeB3542]

A neat solution with a remarkable result...and with profound real world applications.

[@Coldgpu]

I am stunned and amazed with your explanation

[@bonbonpony]

So in the integral of `e^(-x²)` the simple lack of `·x` is what makes it (almost) impossible to solve? And the whole idea of translating the problem to polar coordinates it what helps to bring that `·x` (or `·r` in this case) back?

[@NoobLord98]

Exactly, by multiplying it with itself, but with a different variable, you can turn it into this integral over the entire plane of the function f(x,y) = exp(-x^2-y^2). It is then indeed a smart idea to do a coordinate transform to polar coordinates, doing this transformation then changes the shape and infinitesmal area of the d-bit (no clue what it's called, the dx dy) by an amount according to the jacobian of the transformation (look it up, it's the determinant of a transformation matrix and can be used for any set of coordinates you'd want), this jacobian just happens to be r for polar coordinates, which then makes the entire integral evaluable.

[@AuroraNora3]

@@NoobLord98 Alternatively, think of dxdy=dA, a tiny area in the xy-plane caused by changes in x and y. In polar coordinates, a change in r and θ will cause a tiny area of approximately rdθdr which will be equal to dA=dxdy in the limit as they become smaller and smaller. So dxdy=rdθdr

[@raterix2]

1:26 "Might ring a bell" I see what you did there

[@AJ-et3vf]

Just spectacular! Well explained and visualized. Speechless

[@KidNamedVashin]

You explained it very well

[@danlurny4129]

You put a lot of effort to make this video!! A great work done!!!!!!!!!!

[@gameplayer1378]

Perfect explanation. Big thanks from Germany 👌👌

[@bongusbing6719]

I’m so happy that I watched this video and knew everything he was going to say and exactly what was going on :,) never thought I would be here lmao

[@TheBreeze976]

I didn't see any error function video on youtube which depicts closer picture of the phenomenon than yours. It is the best one the approaching perfection asymptote.

[@ahmedaghadi8281]

5:51 we were supposed to find volume under the curve, therefore we should have taken solid cylinder rather than taking hollow cylinder. And if you had taken hollow cylinder then you should find volume under hollow cyling including it's volume but you had just find volume of cylinder itself and not volume under the cylinder. Why???

[@alexismiller2349]

He used tubes (with 0 volume) instead of solid cylinders (with >0 volume) because by integrating an area you get a volume. It’s a bit confusing and if you want to do it with solid math than you do it like this: ∫ ∫ f(x,y) dx dy (with x and y going from -inf to +inf) = ∫ ∫ f(r,θ) * J dr dθ (with r going from 0 to +inf and θ going from 0 to 2*pi) J is the Jacobean coefficient from cartesian to polar, in this case J = r This is a bit to complicated for the video but voila

[@naysay02]

Wow this is just so bloody mystical. Pi doesn’t even appear in e^(-x)2 and yet the volume is pi. this is such a mindfuck of a number.

[@souravpaul_per]

Nicely done sir👍. we can also find this integral using Gamma function.

[@zyplocs]

Fantastic explanation of a somewhat abstract idea!

[@UTUBDZ]

Very Good explanation ! Keep sharing such valuable content !

[@shagunshukla8490]

This integral is basically everywhere where there is kinetic theory of gases is

[@mathalysisworld]

This is such a good explanation btw

[@lucidhydra2446]

Ive only taken Calculus 1 but somehow this all makes sense haha! wonderful video!

[@mostafatarek5088]

give this man a medal

[@TheGeckoIsKing]

Doing this is calc 3 was so amazing for me to see

[@vardaanchaphekar]

I have a doubt... Why do you think of the infinitesimally small divisions of the function as tubes and not disks?

[@Aditya_196]

Infinitely small tubes = discs ..😅 why not just use discs because discs are 2d you can't make up 3ds from 2ds

[@amogh5427]

First video from ur channel...now I am feeling guilty... Bcz I found ur channel to late...I loved it a lot

[@sid6645]

Reminded me of my god damn electrostatics classes haha. Finding the integrands of shapes and stuff... Still have nightmares about it to this day.

[@assassin_un2890]

very nice video, i like how you explain things, its like how my mind process things exactly thank you

[@RigoVids]

I was literally just in a lecture for 3D calculus and they introduced polar 3d integrals for the intersecting areas and volumes of sheets and cylinders.

[@Byt3me21]

How fun was that! Thanks!

[@The_Aleph_Null]

Isn't this just using cilindrical coordinates directly into the double integral to solve it and that's about it?

[@GamerTheTurtle]

no it's polar coordinate switch

[@gslim7337]

I would also like to add that if you take the 3D image at 3:57, rotate it 90 degrees so that the tip of the cone is pointing towards you and imagine that the point of the cone is a stack with smoke steadily streaming out from against a steady wind, that cone would be a concentration profile of the smoke as it travels from the tip of the stack to infinity. We have much to credit Gauss for.

[@passer2by]

Simply elegant and brilliant!

[@spyrex3988]

Truly brilliant and magnificent

[@초록초록-g6u]

This is so beautiful and awesome

[@lucyhaddant1303]

Great teacher! Thank you very much. Keep making maths videos...

[@michaelmacdonald2907]

Dude . . . Where have you been hiding ?! I've been lookin' everywhere for You.

[@jayalakshmi4355]

I love this video and the way you explain

[@hossainahd]

Wow! Great illustration!

[@Eniro20]

Great video! I was thinking if it'd be simpler to calculate the integral of I^2 by using circles (cylinders) that make up the bell shape (parallel with the x-y-plane). The 3-D bell shape would be z = e^-r^2 so r^2 = -ln(z). Now A(z) = pi*r^2 = -pi*ln(z). Then just integrate from 0 to 1 A(z) with respect to z. Yields pi.