Solving the Gaussian Integral the cool way
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- Опубліковано 25 чер 2024
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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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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!
That's so kind of you to say, good luck!
Wish I had UA-cam and channels like this when I was an undergraduate student. Unfortunately that was 30 years ago.
man i have a similar story
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@@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.
I really liked this. Feynman's technique has always been obscure to me. Your description was revelatory! Thank you very much!
Thank you! Isn't it cool?
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!
so clever!
I have an extreme love for math and I just discovered this channel. This is an absolute goldmine. Thank you Dr. Trefor
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❤
Same im 13 and in algebra 1
@@ryemiranda6800❤
I am 9 year old
Wonderful proof. Thanks for sharing.
Wow, up until today I only knew the polar coordinates way of solving the Gaussian integral!
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?
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!
Absolutely, and we often get stuck only knowing the "normal" way things are done
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.
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
This is multivariable..
Nice
Hi, Really nice video! I was wondering if I can help you edit your videos and also make highly engaging shorts out of them.
Before the video started: “Feynman technique?”
Video starts: Feynman technique.
Shouldn't it be −arctan 1 + arctan 0 + C at 7:26?
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.
Oops, good catch. Yes irrelevant for the end result:)
What A WAY😮😮😮😮
Can somebody explain why we introduced y and why when we introduced the new variable y, we set the top border to one?
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.
@@DrTrefor thank you so much, it seems so easy right now, you must have a gift for maths
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.
Ha ya it really is tbh, so many cool tricks to learn
Grazie.
Thank you so much!
7:25 That signal it's wrong? Don't changed because is zero, but should be +arctan(0)?
That Englishal it's wrong too lol
This guy here is so nice
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.
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.
I remember solving this using power series and differential equations
How 😮😮😮😮
How 😮😮😮😮
Really fun if you understand imo
😂😂😂😂😂
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.
Why?
Gozeean integral.
Hey look, it's Dr. Chreethaw, he's gonna us Mothimautecs
Sir i dont understand one thing
.. that is how did you assume that f(t) is a square of that integral initially?????
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 😅😅
yeah... i understood.. thankyou :)
@@umairbutt1355
What is the need of taking partial derivatives?
It's just a partial because the integrand is a function of both x and t
@@DrTrefor Dear Sir, don't we concern with 't' only?
How?😮😮😮
11 12 22 4..
disappointed you didn't prove the limit was 0 and instead justified it by looking at a graph!
WHY BUST YOUR BRAIN TRYING TO REFIGURE OUT SOMETHING THAT'S ALREADY BEEN SOLVED. AHAHAHA AHAHAHA LOL
No offence but Polar coordinates were much better 🫠
ha, I like both!
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 :)
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
Such an evil way to solve it
1st
Haha
@@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 😀
ha, nice one!
So, if you're technically first, is every other comment under your first comment first also? ...
It's not true that the gaussian integral's value is √ π/2!
I can prove it rong 👩🚒
I enjoyed this video. You might like some of mine too. Just saying.
The hand movements are really distracting
You like to talk.