Solving the Gaussian Integral the cool way

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  • Опубліковано 12 чер 2024
  • Check out MAPLE LEARN ►www.maplesoft.com/products/le.... Try out the limited trial version of Maple Learn
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    My video on the "normal" way to solve the Gaussian Integral using a double integral in polar coordinates ► • The Gaussian Integral ...
    In this video we're going to see a different trick to be able to solve the Gaussian Integral utilizing Feynman's trick of introducing a parameter.
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КОМЕНТАРІ • 76

  • @bryangough6424
    @bryangough6424 11 місяців тому +47

    Hello Dr. Trefor! I am an undergraduate student who loves your videos a lot. I have been struggling in my current major and really not felt confident in my studies. I think your videos have helped convince me that my passion lies in mathematics instead of the subjects I have pursued so far. Thank you so much for your content!

    • @DrTrefor
      @DrTrefor  11 місяців тому +12

      That's so kind of you to say, good luck!

    • @grahaml6072
      @grahaml6072 11 місяців тому +4

      Wish I had UA-cam and channels like this when I was an undergraduate student. Unfortunately that was 30 years ago.

    • @ShanBojack
      @ShanBojack 11 місяців тому

      man i have a similar story

    • @agutilinus2828
      @agutilinus2828 11 місяців тому

      ​@@DrTreforhi

    • @fordtimelord8673
      @fordtimelord8673 10 місяців тому

      @@grahaml6072
      Same here. These UA-cam professors that share their knowledge, intuition and joy of a subject or problem are an incalculable blessing to mankind.

  • @caryfitz
    @caryfitz 11 місяців тому +24

    I really liked this. Feynman's technique has always been obscure to me. Your description was revelatory! Thank you very much!

    • @DrTrefor
      @DrTrefor  11 місяців тому +4

      Thank you! Isn't it cool?

  • @212ntruesdale
    @212ntruesdale 11 місяців тому +7

    Who in the world is brilliant enough to come up with this?! I feel really smart just being able to understand what’s happening! There’s clever, and then there’s CLEVER. Know your station!

    • @DrTrefor
      @DrTrefor  11 місяців тому +2

      so clever!

  • @ajb16384
    @ajb16384 4 місяці тому +1

    I have an extreme love for math and I just discovered this channel. This is an absolute goldmine. Thank you Dr. Trefor

  • @slavinojunepri7648
    @slavinojunepri7648 10 місяців тому +1

    Wonderful proof. Thanks for sharing.

  • @Valori_4
    @Valori_4 8 місяців тому +14

    Hello! I'm currently 12 year's old studying in 6th grade but I'm a very curious person in learning mathematics and physics so I chose you to teach to solve this infamous Gaussian Integral Hope when I get older I'll find a new invention like any other scientist's ❤ Thanks to you for your video's You got a new sub❤

    • @ryemiranda6800
      @ryemiranda6800 8 місяців тому +3

      Same im 13 and in algebra 1

    • @Valori_4
      @Valori_4 8 місяців тому +2

      ​@@ryemiranda6800❤

    • @kasg8427
      @kasg8427 10 днів тому

      I am 9 year old

  • @johnchessant3012
    @johnchessant3012 11 місяців тому +6

    Wow, up until today I only knew the polar coordinates way of solving the Gaussian integral!

  • @umairbutt1355
    @umairbutt1355 7 місяців тому +3

    This was my first proof of the gaussian that I came across and it blew my mind (I didn't study multi-variate calculus yet).
    Does anyone know the original author of this proof?

  • @aweebthatlovesmath4220
    @aweebthatlovesmath4220 11 місяців тому +8

    Nice video! it's always good to solve a problem in more then two ways. i mean all and all it's not the answers that make math interesting it's the way we do math!

    • @DrTrefor
      @DrTrefor  11 місяців тому +4

      Absolutely, and we often get stuck only knowing the "normal" way things are done

  • @user-cg7gd5pw5b
    @user-cg7gd5pw5b 2 місяці тому

    Thanks, I've done multiple exercises where they give the developed derivative and you have to climb back to the Gaussian integral, and was always wondering how people found this kind of expressions in the first place.

  • @monke9865
    @monke9865 11 місяців тому +4

    Interestingly there is another way to compute this integral using single variable calculus, with strong geometric flavor.
    Consider F(r)= int _{x^2+y^2

  • @monsterhunter8595
    @monsterhunter8595 11 місяців тому +1

    Nice

  • @marylinebentzinger7378
    @marylinebentzinger7378 11 місяців тому +4

    Can somebody explain why we introduced y and why when we introduced the new variable y, we set the top border to one?

    • @DrTrefor
      @DrTrefor  11 місяців тому +5

      It's just a change of variables y=x/t. Specifically when x=t for the upper bound, in the new variable y=t/t=1. The point of this change of variables was to get the parameter t out of the limits of integration and into the integrand.

    • @marylinebentzinger7378
      @marylinebentzinger7378 11 місяців тому

      @@DrTrefor thank you so much, it seems so easy right now, you must have a gift for maths

  • @foysalsahriar5608
    @foysalsahriar5608 10 місяців тому

    Hi, Really nice video! I was wondering if I can help you edit your videos and also make highly engaging shorts out of them.

  • @MurshidIslam
    @MurshidIslam 11 місяців тому +6

    Shouldn't it be −arctan 1 + arctan 0 + C at 7:26?

    • @balubaluhehe2002
      @balubaluhehe2002 11 місяців тому +2

      It should've been, because there was a negative sign at the front, but -0 or +0 didn't change the final result at all.

    • @DrTrefor
      @DrTrefor  11 місяців тому +2

      Oops, good catch. Yes irrelevant for the end result:)

  • @josafajunior5425
    @josafajunior5425 11 місяців тому +4

    7:25 That signal it's wrong? Don't changed because is zero, but should be +arctan(0)?

  • @alebaldus9229
    @alebaldus9229 11 місяців тому +1

    Grazie.

    • @DrTrefor
      @DrTrefor  11 місяців тому +1

      Thank you so much!

  • @user-rq6gd8yy2t
    @user-rq6gd8yy2t 11 місяців тому +1

    What A WAY😮😮😮😮

  • @andrewharrison8436
    @andrewharrison8436 11 місяців тому +8

    50 years ago I felt that integration was a black art. If anything videos like this confirm it.
    Love the way that the annoying C that crops up in any integration turns out to be the part that carries the result from the doable integral to the unknown one.
    I will probably watch this again to see exactly when you hid the rabbit in the top hat.

    • @DrTrefor
      @DrTrefor  11 місяців тому +4

      Ha ya it really is tbh, so many cool tricks to learn

  • @airman122469
    @airman122469 11 місяців тому +1

    Before the video started: “Feynman technique?”
    Video starts: Feynman technique.

  • @tg0406
    @tg0406 3 місяці тому

    I remember solving this using power series and differential equations

    • @mosarof_cp
      @mosarof_cp Місяць тому

      How 😮😮😮😮

    • @mosarof_cp
      @mosarof_cp Місяць тому

      How 😮😮😮😮

  • @ahmedyacine5661
    @ahmedyacine5661 11 місяців тому

    This guy here is so nice

  • @strikerstone
    @strikerstone 7 місяців тому

    Really fun if you understand imo

    • @mosarof_cp
      @mosarof_cp Місяць тому

      😂😂😂😂😂

  • @samtux762
    @samtux762 Місяць тому

    Neat approach. But I wouldn't dare to repeat it and would go for the classic switch to polar coordinates.
    But, I am not in math (was in chemistry), so this isnot my field.

    • @samtux762
      @samtux762 Місяць тому

      Integrals arenot my jam and are not a part of my work. But I added your channel to my watch list.
      This "I will just use Feinman approach and square" came out of the blue.
      Brain massaging. Love it.

  • @egor.okhterov
    @egor.okhterov 11 місяців тому

    Why?

  • @joelchristophr3741
    @joelchristophr3741 8 місяців тому

    Sir i dont understand one thing
    .. that is how did you assume that f(t) is a square of that integral initially?????

    • @umairbutt1355
      @umairbutt1355 7 місяців тому

      It's not an assumption. It's a definition for f(t). Now, the question of WHY such a function f(t) was defined in order to get to the gaussian is the really interesting question 😅😅

    • @joelchristophr3741
      @joelchristophr3741 6 місяців тому

      yeah... i understood.. thankyou :)
      @@umairbutt1355

  • @MooImABunny
    @MooImABunny 11 місяців тому

    Gozeean integral.
    Hey look, it's Dr. Chreethaw, he's gonna us Mothimautecs

  • @sandorszabo2470
    @sandorszabo2470 11 місяців тому

    Tricky solution! However it is not the "Feynman's trick". This method was known earlier. Only Feynman was the person who used a lot to simplify the calculation of many complicated integral.

  • @mosarof_cp
    @mosarof_cp Місяць тому

    How?😮😮😮

  • @user-xo5th3cm5k
    @user-xo5th3cm5k Місяць тому

    11 12 22 4..

  • @ViralShorts_2323
    @ViralShorts_2323 11 місяців тому +2

    What is the need of taking partial derivatives?

    • @DrTrefor
      @DrTrefor  11 місяців тому +2

      It's just a partial because the integrand is a function of both x and t

    • @ViralShorts_2323
      @ViralShorts_2323 11 місяців тому

      @@DrTrefor Dear Sir, don't we concern with 't' only?

  • @gnarlybonesful
    @gnarlybonesful 11 місяців тому +3

    disappointed you didn't prove the limit was 0 and instead justified it by looking at a graph!

  • @mariostelzner4530
    @mariostelzner4530 11 місяців тому

    WHY BUST YOUR BRAIN TRYING TO REFIGURE OUT SOMETHING THAT'S ALREADY BEEN SOLVED. AHAHAHA AHAHAHA LOL

  • @avinashyadav8314
    @avinashyadav8314 11 місяців тому +2

    No offence but Polar coordinates were much better 🫠

    • @DrTrefor
      @DrTrefor  11 місяців тому +2

      ha, I like both!

  • @Ben-rd3mg
    @Ben-rd3mg 2 місяці тому

    Such an evil way to solve it

  • @billcook4768
    @billcook4768 11 місяців тому +1

    No, no, no. In order to truly be “cool”, you have to use Taylor series. It’s like a law or something. Then again, for the life of me I don’t see a Taylor expansion so maybe there is no official cool method :)

    • @DrTrefor
      @DrTrefor  11 місяців тому +1

      Taylor series is all good, but the question becomes for definite integrals what does that series sum to? We might get that it converges but harder to know in general if there is a nice answer for the sum

  • @grahaml6072
    @grahaml6072 11 місяців тому

    1st

    • @monsterhunter8595
      @monsterhunter8595 11 місяців тому

      Haha

    • @grahaml6072
      @grahaml6072 11 місяців тому +1

      @@monsterhunter8595 I don’t know why people do that and post they first but this just happened to be the only time I have ever been first so I thought what hell I will look like an idiot like all the others 😀

    • @DrTrefor
      @DrTrefor  11 місяців тому +2

      ha, nice one!

    • @diezelvh4133
      @diezelvh4133 5 місяців тому

      So, if you're technically first, is every other comment under your first comment first also? ...

  • @mosarof_cp
    @mosarof_cp Місяць тому

    It's not true that the gaussian integral's value is √ π/2!
    I can prove it rong 👩‍🚒

  • @Electronics4Guitar
    @Electronics4Guitar 3 місяці тому

    I enjoyed this video. You might like some of mine too. Just saying.

  • @xl000
    @xl000 11 місяців тому

    The hand movements are really distracting

  • @nickzadeh7082
    @nickzadeh7082 11 годин тому

    You like to talk.