No video :(

The Gaussian Integral // Solved Using Polar Coordinates

Подписаться 418 тыс.

Просмотров 102 тыс.

50% 1

The gaussian integral - integrating e^(-x^2) over all numbers, is an extremely important integral in probability, statistics, and many other fields. However, it is challenging to solve using elementary methods from single variable calculus. In this video we will see how we can convert it to multivariable calculus and then use tricks from multivariable calculus - in this case converting to polar coordinates - to solve this single variable integral. The crazy thing is that this integral ends up being in terms of pi, and if you didn't know about the polar trick you might wonder why pi shows up here at all! This proof is due to Poisson.
The previous video on double integration in polar: • Double Integration in ...
****************************************************
COURSE PLAYLISTS:
►CALCULUS I: • Calculus I (Limits, De...
► CALCULUS II: • Calculus II (Integrati...
►MULTIVARIABLE CALCULUS (Calc III): • Calculus III: Multivar...
►DIFFERENTIAL EQUATIONS (Calc IV): • How to solve ODEs with...
►DISCRETE MATH: • Discrete Math (Full Co...
►LINEAR ALGEBRA: • Linear Algebra (Full C...
***************************************************
► Want to learn math effectively? Check out my "Learning Math" Series:
• 5 Tips To Make Math Pr...
►Want some cool math? Check out my "Cool Math" Series:
• Cool Math Series
****************************************************
►Follow me on Twitter: / treforbazett
*****************************************************
This video was created by Dr. Trefor Bazett. I'm an Assistant Teaching Professor at the University of Victoria.
BECOME A MEMBER:
►Join: / @drtrefor
MATH BOOKS & MERCH I LOVE:
► My Amazon Affiliate Shop: www.amazon.com...

Опубликовано:

14 авг 2024

Ссылка:

Скачать:

Готовим ссылку...

Добавить в:

Мой плейлист

Посмотреть позже

Комментарии : 125

@wakeawake2950 3 года назад

We can also solve Gaussian integral by Laplace transform,but this method is really cool,I like this more, thnk u professor

@Sir_Isaac_Newton_ Год назад

Laplace transform is so much more elegant, I don't know what you're on about.

@claudeabraham2347 Год назад

Very good. I electrical engineering grad school 1979, my math professor solved this integral for us. I was fascinated with it since. The key is the dx dy = dA =r dr d(theta). In polar form the integrated possesses antiderivative. Great example of coordinate transformation being useful. In one coordinate system a problem which is very difficult becomes *easy peasy* in another coordinate system.

@jh-ij4by Год назад

Interesting thanks for sharing

@sbmathsyt5306 4 года назад

Awesome video the display is really nice and clear. I love the graphs helping to visualise the integrals.

@JUNGELMAN2012 5 месяцев назад

just 3 minutes into the video, and i'm already in love with the clean color commented animations. Keep up the good work. Your setting the bar for others.

@KM25263 Год назад

You have an awesome way of teaching, thanks. It is fascinating how cool is geometry sometimes when compared with calculus. It is also worth noting the physical meaning of 'sqrt pi' where calculus also touches statistics!

@ilkinond 3 года назад

Just discovered your channel today Dr. Trefor - awesome. Subscribed already.

@DrTrefor 3 года назад

Awesome, thank you!

@Saptarshi.Sarkar 3 года назад

Using the Gamma function is my favourite method to solve this

@atulkumars2095 8 месяцев назад

You can use gamma function x²=t 2xdx=dt And then the integral convert in Gamma(1/2) Which is √π QED Respect from india❤

@jaimanparekh4616 2 года назад

I did this out of my textbook today, and by the looks of it I got it right without using any outside sources to help. So happy. Thanks for the explanation with the visuals backing up my initial intuition

@GadgetGuyU.K. 4 года назад

Another brilliant and clearly explained video! Thanks for posting.

@elisabeth3254 7 месяцев назад

This finally makes sense, thank you so much! Greetings from a physics student from Austria! 😊

@dqrksun 2 года назад

My method of solving it is. convert it to the multivariable version. Then imagine it as infinitely many cylinder. then add up those cylinders. the radius of the cylinder is sqrt(-ln x) (the inverse of e^-x^2). Adding up them is just pi*r^2. where r is the function. So its just intergrating pi (sqrt(-lnx))^2. then You'll get -pi*-1=pi. Then take the sqrt of it

@andrewharrison8436 2 года назад

Yes, beautiful. So straightforward when you know how. Nicely presented, part way through I remembered this from years ago but enjoyed it to the end. Like watching a row of dominos topple, inevitable but satisfying.

@AbhishekKumar-jg7gq 3 года назад

You are showing the beauty of mathematics 🥰🥰

@Noone-wz1ys Год назад

I want to understand how u defined the limits for theta... Sir,I need help here,if u can.

@JigsaW-goat 3 года назад

Thanks bro... really helped me understand this...already liked and subscribed :)

@DrTrefor 3 года назад

Thanks for the sub!

@davidm9442 Год назад

Really good explaination, thanks Dr. Bazett!!

@jewulo 3 года назад

I am new to your channel and I have watched all day today. It is awesome. You are awesome.

@DrTrefor 3 года назад

Welcome aboard!

@212ntruesdale Год назад

There’s another video claiming that Laplace solved the Gaussian integral without needing to switch coordinate systems. However, the nuts and bolts all look the same. The claim is that a parameter, t, avoids it. However, r is also just a parameter, not a function of theta, when it comes to converting dxdy to the tiniest area in the polar coordinate system. Start with S=rtheta. Differentiate with respect to theta, treating r as a parameter now. dS=rdtheta. Now multiple both sides by dr. You get dA = rdrdtheta = dydx. Took me a while to work it out starting with length of a sector of a circle, which is where my intuition starts.

@pamodakoggala 3 года назад

Wow, the way is cool. And the way you teach is very clear.

@DrTrefor 3 года назад

Thank you! 😃

@ycombinator765 3 года назад

Respect from Pakistan! Just pure respect!!!

@DrTrefor 3 года назад

Thank you!!

@emilwang8818 2 года назад

Very helpful video!!! I was trying to use complex analysis, but it didn't quite work out as expected :/

@vikramnagarjuna3549 4 года назад

I'm waiting for wonderful topics in Vector Calculus

@SuperWiseguy3 Год назад

Thank you! This helped make sense of verifying the pdf of a normal distribution!

@212ntruesdale Год назад

Another master class. My brain thanks you!

@declanwk1 2 года назад

this is a brilliant short video

@kebman 2 года назад

I think those who have worked with 3D modelling and rendering have a more intuitive grasp on these things. Especially if they used something like POV-Ray, which is a fully scripted and Turing complete modelling language.

@divishthamalik309 3 года назад

You are an amazing prof I wish you were my instructor

@akhilkrishnan824 2 года назад

You make it simple.... 👍

@salimismail6859 7 месяцев назад

made it really easy to understand thanks

@ilproko3689 Год назад

GENIUS

@michaelwise5089 3 года назад

I just came across this. Thanks for helping me understand this! I especially liked how you showed the 3D function and linked it to polar coordinates with spherical symmetry.

@forresthu6204 Год назад

that's insane and amazing mathmatica tirck.

@neilliang4209 2 года назад

My personal favorite

@POLYMATH_RAGHU Год назад

Great explanation. Thank you

@continnum_radhe-radhe 2 года назад

Sir , can you made a video on centre of gravity....multiple integral??

@sr.tarsaimsingh9294 2 года назад

Thanks a lot lot Sir, Watching this Video ; I instantly becomes yours subscriber. I had seen multiple videos, but I didn't get it whole. I am student of +2 class from India, Use of polar coordinates is not there in our curriculum; But help me to provide yours video regarding polar coordinates in description; Thus to enjoy this fun. 🙏🏻👍🏻👍🏻

@mohammadjaveed7404 Год назад

Very cosy methodprofessor thanks.

@user-sz9pf4st9h 3 года назад

Nice work!

@yogitakukreja2296 3 года назад

Thanks man!

@DrTrefor 3 года назад

No problem!

@walterwhite28 3 года назад

I had a question- In some beta or gamma integrals, after substituting some variable as sin(theta) or cos(theta) we get intehrals in terms of theta. So, in that case, integrating from 0 to 2pi, gives the integration value 0 if the answer is in terms of sine. So to get non-zero answer, we need to break integral according to symmetry as 4×integral(0 to pi/2). Why do we get 2 different answers then? Shouldn't the answers be same, if we are equating one thing to another for solving.

@JigsaW-goat 3 года назад

Well I guess due to discontinuity of that particular func^ at some points..so we need to split them...one example - integral (dx/(2+sin2x) limits 0->2π..

@adresscenter 4 года назад

Great teacher 💪👍

@ilong4rennes Год назад

thank you so much !!!!! this video saved me!

@johnnisshansen Год назад

squareroot of pi is also the sidelength of a square with the same area as a unitcircle.

@physicslover1950 4 года назад

Your way of presenting the content in a vsual ways witg animations is great 💚💚💚💚. You make hard things easy. Can you please make a video series on complex analysis?

@physicslover1950 4 года назад

@@DrTrefor Ha ha Ha but thanks You again Sir !

@kebman 2 года назад

So what's your _least_ favourite integral then?

@terasathi8699 Год назад

Sir love from the ❤️ 💙 💜 💖

@barrytaylor2265 Год назад

There are many of us who could not solve this equation, but have enough background to enjoy watching the solving. To ignore us is to leave a large fraction of potential subscribers on the sideline. You may think that the content of a prior video is all you need to refer to. That is not true. When I watch a video, I want you to take me from start to finish. Don’t shorten your video simply because you made a video about the process previously.

@samhobbeheydar5969 Год назад

Check his channel. This video is part of a full course on multi variable calculus. Expecting him to cover all the fundamental ideas again in this video is like showing up to a class for the first time on the 15th day and being mad you don’t know what’s going on

@yongmrchen Год назад

Nice idea 💡

@muhammadumarsotvoldiev8768 2 года назад

thank's a lot. Very good explanation.

@DrTrefor 2 года назад

Glad it was helpful!

@lebdesmath2510 Год назад

no music, perfect

@tonireyes844 Год назад

How did you convert the bound of integration ? I mean how to write that mathematically ?

@adw1z Год назад

I came across a really abstract way to solve this integral, to obtain an ODE from two different integrals of multi valued function and using Fenyman’s Integration Technique of differentiating under the integral sign to obtain a differential relation

@k_wl Год назад

or you could do the way laplace did it

@wakeawake2950 4 года назад

Nice video!

@yoavgolan4916 2 года назад

Hey, thanks for your video Prof. It was all clear to me, except for one step. What theorem did tou use in order to justify combining the multiplication of the two single variable integrals to one double integral?

@DrTrefor 2 года назад

Fubinis theorem

@rfmvoers 2 года назад

I think it's the constant factor rule... because both integrals are constants w.r.t. each other.

@sergiolucas38 2 года назад

nice trick, i didnt know of it :)

@suhailawm 4 года назад

amazing explanation prof. tnx alot

@samuelfoin5531 2 года назад

is there a way to compute the gaussian integrale without going in the polar coodinate ?

@carultch Год назад

Infinite series.

@ogunsadebenjaminadeiyin2729 3 года назад

Super super

@victoraguiar3489 2 года назад

Thank you for the explanation, Dr. Trefor. I would like to ask what would have changeg if instead of integrating from -inf --> inf, you integrated from say xo --> inf, where xo is a a point in the curve. Cheers

@ridazouga4144 Год назад

That's an interesting question, but the answer doesn't exist unfortunately, in other words this integral from xo to infinity can be obtained only numerically and not algebraically

@onionbroisbestwaifu5067 Год назад

This is an example of a non-elementary function, in other words, there is no writeable combination of sines, cosines, polynomials, logarithms, or exponents that can give your answer in general. It can only be solved given bounds through non-elementary methods (like this trick or feynmans trick or laplace transforms)

@adelyoutube7530 2 года назад

Why the (r)dr was not transferred to udu ? As the extra r is there once you change to polar ??!!

@geektoys370 Год назад

How can you change the variable

@tintinfan007 Год назад

now what happens if we differentiate the root of pi

@mohamedirshaathm32123 Год назад

SIR I am still confused why the LIMIT OF THETA is 0 to 2pi and why not 0 to pi /2

@monzirabdalrahman4573 Год назад

You're the best

@iaaan1245 Год назад

awesome!

@SHAHHUSSAIN 4 года назад

♥️♥️SUPERB💝💝♥️

@slendrmusic 3 года назад

Awesome

@joaomattos9271 Год назад

Great!!!!!

@FiboYT 2 года назад

I still wonder,why you allowed to merge the squared integral

@JP-re3bc Год назад

If I did that mighty hand waving in a test I guess my grade would be bad indeed.

@johncrwarner 4 года назад

Is this how Gauss solved it originally?

@visualgebra 3 года назад

What technique you use for this kind of animation

@DrTrefor 3 года назад

I make everything in MATLAB

@mustafaakyol7440 3 года назад

I didn't understand how you can split double integral by multiplying two integral and vice verse.Iwill be glad ifyou can write an explanation abot this step. Thanks. MUSTAFA AKYOL

@carultch Год назад

Given I = ∫ e^(-x^2) dx. Make a copy of I, and change the variable to y: I = ∫ e^(-y^2) dy Multiply it with itself squared: I^2 = ∫e^(-x^2) dx * ∫e^(-y^2) dy The same way that we can pull constants out of an integral, we can add constants back in to the integral. Thus: I^2 = ∫(∫e^(-y^2) dy)*e^(-x^2) dx Since the differential terms are really just implicitly multiplied by the integrand, we can relocate dy to the end, and collect the integral signs at the beginning: I^2 = ∫∫ e^(-y^2) *e^(-x^2) dx dy Consolidate the exponents: I^2 = ∫∫ e^(-x^2 - y^2) dx dy Then transform to polar coordinates to carry it out.

@continnum_radhe-radhe 2 года назад

🙏🙏🙏

@stvayush 4 года назад

Hey, where is the url of video that you mentioned in the current one. I couldn't find it in description. Please help, i wanna learn

@stvayush 4 года назад

@@DrTrefor Thanks Prof! 🙂

@l.h.308 Год назад

What if the interval were from a to b, would it be feasible?

@carultch Год назад

Unfortunately no. Otherwise we could find the cumulative distribution function in elementary functions, and not need to define the erf(x) function, or use infinite series to evaluate it.

@atriagotler 2 года назад

Wow this was 3b1b kind of beautiful

@habacuuq 4 года назад

This is used as the main way of proof that the pdf of the gaussian distribution integrates to 1

@youssefdirani 2 года назад

Super

@ozgurhamsici9293 Год назад

brain storming but how dx and dy and x and y are matching and making a polar couple. with what major sense. out of scope of my head

@DerejeNegash-bu4vo Год назад

it is best if your vedio suported by animation

@githika5935 Год назад

king

@jrfutube2013 2 месяца назад

When is The Gaussian Integral used in real world applications? 🌎

@RF-fi2pt Год назад

The bell shape created by e^(-x^2) is so beautiful as that created by other base constants instead of 'e'. At excel drawing ,eg, 12^(-x^2), gives that shape, with less variance, centered at 0 and same maximum value 1. Sure integrating this with the same tricks showed by Dr. Bazett, will obtain other finite razonable constant . So is very interesting explain why Gauss uses base 'e' instead of any other number (except the obvious not useful like base 1 or 1/2 or other between 1 and 0). Tradition ? Because 'e' is the "natural" base although irrational? Advantages of this? ...

@carultch Год назад

The calculus is more elegant when you use base e. Using another base ultimately is the same thing as having a constant grouped with -x^2 in the exponent, because B^x in general is the same thing as e^x. Thus, B^(-x^2) is the same thing as e^(-ln(B)*x^2). Let L=ln(b). When you integrate ∫∫B^(-x^2) * r dr dθ, your inner integral will become: ∫B^(-r^2) * r dr = ∫e^(-L*r^2) * r dr = -1/(2*L) * ∫e^(-L*r^2) * (-2*L*r) dr = -1/(2*L) * ∫e^(u) * du = -1/(2*L) * e^u + C= -1/(2*L) * e^(-L*r^2) + C Apply r=0 to infinity, recall L: 1/(2*ln(b)) Then evaluating the outer integral from 0 to 2*pi, we get: pi/ln(b) If you use base e, then all of those L's and ln(b) terms will equal 1, which greatly simplifies this. The actual Gaussian distribution contains constants all over the place, to adjust it for mean, standard deviation, and to force it to have an area of 1. Knowing to put sqrt(pi) in the leading constant, to normalize its total area, is an application of this knowledge.

@user-pb4jg2dh4w 2 года назад

Wwwwwoooooowwww thank youuu

@amaljeevk3950 10 месяцев назад

❤

@edanarator7716 Месяц назад

Three analysis classes and I still can't solve it, worst integral

@prashanthkumar0 4 года назад

pi shows up all unexpected places..hehe

@prashanthkumar0 4 года назад

@@DrTrefor yes... math is always cool...and amazing 😁...

@prashanthkumar0 4 года назад

@@DrTrefor do you know about manim lib?? its amazing for making animation for math ...its made by 3blue1brown... github.com/3b1b/manim/tree/master/ its in python btw .....

@prashanthkumar0 4 года назад

@@DrTrefor you have done really great job sir... the green screen and the editing 👏👏👏👏👏... really amazing channel...it deserves more IMO...