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)"
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.
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.
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.
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!!
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.
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
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...
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.
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!
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!!!
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
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?
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.
@@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
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.
Lies again? Dark X
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)"
Dude integrated so hard that his voice changed at 9:35
"I talked to Barzini."
No word for this great video.
Spectacular.
According to you, there are actually 7 words.
@@David_F97 ok
But you said no word.
That 7 words doesn’t include
"Might ring a bell"... I see what you did there
hehehehehehe
*Laughs in Bell curve*
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.
Do also read about cylindrical coordinate system
@Kappa Chino laughs in confocal eliptical coordinates.
Hey can you please explain why was the height taken as the function itself ( -e power R2 squared) m
One integral is used for area, two for volume, what’s the use of triple integral?
@@danipent3550 density
I am blown away by the way you simplified it. Pure awesomeness.
I mean he didnt invent anything its in every calc textbook about integral calculus right
@@slender1892 stfu bit*h , you probably scored 0 in maths 😂😂
That is stunningly beautiful. One of the best mathematical explanations I’ve ever seen. Well done, sir
This is easily the best video explaining the Gaussian Integral I have ever seen!
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!
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.
Probably the best video on RUclips explaining this problem. Thank you!
I have watched a number of clips on the Gaussian Integral, but I like the practical way this has been explained.
THIS IS BEAUTIFUL! This is the best video I’ve seen that talks about this integral.
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.
thx a lot man. finally a good explanation
no kidding this is probably the best explaination i found on youtube
Thank you for providing with detailed explanation not just formulas, great job!
The best intuitive and clear video ive seen on this topic!
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!!
One of the best explanations I've ever seen...well done!
Very easy-to-digest explanation of a very complex topic. Excellent video!
An extremely well done video for an extremely beautiful integral.
Absolutely brilliant.... never seen such an explanation. Thanks a lot.
Best explanation I've seen for this integral. Well done sir.
Beautiful! Congratulations for such a wonderful video and demonstration.
Fantastic video. Kept me entertained with great explanation and graphical representation.
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.
Bravo! Best explanation of this integral ever!!!
Just spectacular! Well explained and visualized. Speechless
This is so beautiful. Amazing explanation, thank you
Finally i can understand this without polar things. Thank Youuuu. Very Logical and easy to understand. Big much thanks
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
Sehr raffinierte Transformation, die das Problem entscheidend vereinfacht. Das könnte man sogar in einem motivierten Leistungskurs bringen! Super!
this channel is like sent from god, great video! you deserve a lot more!
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...
Simply elegant and brilliant!
I am stunned and amazed with your explanation
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
Thank you. You just earned a subscriber. Looking for more such great content.
Perfect explanation. Big thanks from Germany 👌👌
A neat solution with a remarkable result...and with profound real world applications.
My first time in your channel. You describe well everything, so I have to subscribe.
I'll just echo what others have typed; this is probably the best explanation on math I've ever had.
First time I understood this integral. Incredible video
Very Good explanation ! Keep sharing such valuable content !
You explained it very well
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.
Excellent! A very clear explanation, thank you!
This is so beautiful and awesome
Super explanation, this should have more likes and views.
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!
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!
I have watched a lot of videos , but understood nothing. But you explained it perfectly!!
You put a lot of effort to make this video!! A great work done!!!!!!!!!!
Amazing, now I know the proof of error function very clearly
very nice video, i like how you explain things, its like how my mind process things exactly thank you
Wow! Great illustration!
How fun was that! Thanks!
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!!!
Truly brilliant and magnificent
1:25 "might ring a bell" nice pun haha
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
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.
WOW THIS IS AMAZING THIS IS SO CLEAR!!!!!
I am not good at English but this video is one of greatest explanation videos I have ever seen. Thank you very much
this is a very great video that helped me understand the gaussian integral and I hereby recommend it to everyone
Fantastic explanation of a somewhat abstract idea!
I love this video and the way you explain
Brilliant video. Loved it
The f ing best integral ever
Thank you so much! This is beautiful!
Great teacher! Thank you very much. Keep making maths videos...
Wow.. great presentation sir
Simple and elegant.
Such a wonderful integral 👍
Very nicely done! Thank you!
This is such a good explanation btw
Pretty clever way to solve!
This is a great video! Thank you so much!
pff i love how you just turned one integral into the same integral+another
Got a subscribe for this! Excellent video!
That video make me remember my first year in college when we were studying electrostatic and electromagnetism 😊😊
Is it the electric field of a uniformly charhed disk?
@@vincentdublin3127 Yes those things XD
I love this.Thanks a lot
Amazing visualization!!!
Ive only taken Calculus 1 but somehow this all makes sense haha! wonderful video!
Beautiful
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?
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.
@@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
Beautiful!
Beautiful.
fantastic video
Brilliant video
Congratulations! Very good!
give this man a medal
Appriciated! Thanks for the video!
that is so awesome explanation
Nice! But one thing, how and why is related de Area of the curve with the volume?
yea exactly what I said. exactly. Thanks for reaffirming my answer
Great video!
Doing this is calc 3 was so amazing for me to see
Amazing!
my jaw dropped when it came out to be square root of pi!
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