Dude, you’re the voice of Bob Ross for math and science. We’ve got a happy little “differential equation” over here 🎨 and some excited molecules over here 🖼.
"by the time we actually start doing something meaningful and physical with it we just turn it right back into a discrete one". Very insightful, thanks!
Very nice. As always you could be the greatest math divulgator of our time. This remembers me of my Stochastic Processes exam. From a symmetrical random walk (in 1D means same probability to go left or right) one can derive the diffusion equation, and from the asymmetrical version you can derive the Fokker-Planck equation, aka the diffusion-with-wind-equation :)
Man your explanations cant stop surprising me! I can keep thanking you for explaining in such an intuitive way. You feel the passion and the love for what you are doing.
15:00 - It's a nice way to look at the diffusion equation! The rate of change of the variable (partial derivative) is directly proportional to the curvature of the curve (as a function of the space), with Diffusion Coefficient being the proportionality constant. The parts of the curve where the magnitudes of the curvatures are high experience higher time rate of change. This explains why the curve smoothens out, or in other words 'straightens out' as time goes by.
That's what Grant, the dude in the video, does. 3Blue1Brown is his TH-cam channel. One of the best math education/visualization channels on this whole site.
@@gileee Oh I know. I've played around with manim (the software library he uses to make the animations) myself, but Grant is still by far the king of making visualizations like these.
The "statistical noise" of 4:52 is not such, it isn't because the "random statistics" ( 4:38 ). It's actually because the implementation of the random walk explained at 3:52 : 50% hop to the left, 50% hop to the right The resulting curve would be smoother: - if we interpolate the values between steps - if we add a third option: 33% hop to the left, 33% stay, 33% hop to the right - Even better if it had 3 options with a discrete normal distribution, ie: 25% hop to the left, 50% stay, 25% hop to the right - The best would be a continuous normal distribution
When you start talking about the brownian motion and then later appears de squared of the delta_x I was waiting for you to start talking about how this thing are managed to be constant allowing the solution truly exist, introducing in this way you viewers into Ito's Calculus.... hope you can extend the video with this modern view of Brownian motions analysis. Thanks you beforehand!
Inspiring video. It made me to think about generalized Pythagorean theorem coming from duality in linear spaces. The divergence is another result of the duality that is coming from linearity it is keeping invariant and doesn't change with unitary transformations.
Hi! Why did you define the number of particles in the boxes as densities rather than concentrations? Are you assuming that the particles occupy all the volume of the boxes?
Dear Grant, What software do you use to create the interactive graphical animations for your videos? Is this commercial software? Perhaps, did you write the software? If so are you able to opensource for those interested in the code? Thanks as always from a keen learner from your work. Regards, Joshua
0:06 guess i now know why his channel is named "3blue1brown. and i got the same eyes as him, just the 3 blue is grey for me and i also have a smaller brown dot in my other eye
Anyone have any good resources (books/simple papers) for doing this kind of thing on a finite grid? I've been trying this with random walks on 8-neighbors and it keeps drifting and settling in the center and I'm not sure why that is.
hello,I am very happy that I found you, I am doing distant course in DSML in collaboration with MIT , I wish if i too had a teacher like you , any suggestion for me , i am very much interested in deep learning and ai , and i know your lectures will be very useful for me and thanks for making them available for free ....
Why would (delta x)^2/(delta t) be constant? (delta x)/(delta t) is molecules velocity and may be considered a constant value. But what about another delta x left?
Great question! If each particle is moving according to a random walk, then after many steps the distance traveled will not be proportional to the time. However, the *square* of the distance traveled will, in expectation, grow linearly with time. It's a fun little exercise to work out.
@@3blue1brown Please correct me if I'm wrong. Square of drift is proportional to time in the case of, strictly speaking, delta t (hence delta x) -> Inf. Here, though, we are dealing with small increments. Notwithstanding all this minutia, I am a big fun of your brilliant channel. Keep up your excellent work!
@@maharajxx also consider that as dx->0, the number of cells in a given volume of space goes to ∞. It's the granularity of x and t, not their magnitudes, that makes the random walk statistically a normal distribution, which has the squared relationship in the expectation that's at issue here.
@@3blue1brown Is the constant of proportion just an arbitrary choice of the person creating the mathematical model or is there any connection to the real material that is diffusing? It seems like this is the only link missing between mathematics and physics in your presentation. I wonder if it can be filled in a mathematically rigorous way.
![ILLSLICK - WINTER IS COMING [Official Video]](http://i.ytimg.com/vi/dwTMeM1zIhQ/mqdefault.jpg)
i see grant in the thumbnail, i watch. simple man and such...
yeahhh ! haha
we all do.!!
Likewise. He has that soothing voice.
@@jesselybianto8650 kinda like math asmr
Yes, no idea about diffusion, just wanted to watch him explaining it
Dude, you’re the voice of Bob Ross for math and science.
We’ve got a happy little “differential equation” over here 🎨 and some excited molecules over here 🖼.
Check out his channel, 3Blue1Brown. Lots more of this.
There's another channel who does Bob Ross styled math explanations. Check out Tibees
And a sad calculus student over there.
fun fact: Grant’s voice is more famous than his face
I disagree. Everyone recognizes that plushy pie expressive face
I had never seen his face before, I literally kept thinking "This guy sounds like 3 blue 1 brown" lmao
@@vikranttyagiRN lol
@@greenstatic9294 +++++
Khan academy as well?
Grant you NEED to do a series on PDEs. You are the best teacher I have ever encountered.
Me: procrastinates studying heat and mass transfer all day
also me: oh look a grant sanderson video *click*
"by the time we actually start doing something meaningful and physical with it we just turn it right back into a discrete one". Very insightful, thanks!
The intuition and derivation for the heat equation was superb! It felt very natural
Excellent. I feel less dense already.
Me as a high school student: I can understand this equation.
Me after university: Let me open my calculator app and check again if 6+7=13.
damn multi-variable cal in high school lol
Clicked for Grant, was surprised by intuitive explanation of stochastic calculus...this man is the gift that keeps on giving.
Very nice. As always you could be the greatest math divulgator of our time. This remembers me of my Stochastic Processes exam. From a symmetrical random walk (in 1D means same probability to go left or right) one can derive the diffusion equation, and from the asymmetrical version you can derive the Fokker-Planck equation, aka the diffusion-with-wind-equation :)
Once you've seen this derivation, you'll always remember the unit of the diffusivity coefficient. Very cool stuff!
Waaaait a minute! Isn't that the 3blue1brown guy?
This is the most intuitive explanation of the diffusion equation. I finally get a rough idea of it this time. Thanks, Grant
I think Grant is the feynman of 21st century when it comes to teaching.
+
Nah, he's the Grant Sanderson of 21st century teaching
Jawad this
agree
Oh boy did I not continue deep enough into Calculus. I learned a lot, and will save this to my favorites for rewatching. Give this man some bongos.
Man your explanations cant stop surprising me! I can keep thanking you for explaining in such an intuitive way. You feel the passion and the love for what you are doing.
Very high quality material. Just a small remark from a physicist after listening the first minute: heat is the one that flows, not the temperature.
Probably the best 20 minutes I've spent on TH-cam. I could just rant on the insight and intellectual value. Cheers! 👏👏👏
15:00 - It's a nice way to look at the diffusion equation! The rate of change of the variable (partial derivative) is directly proportional to the curvature of the curve (as a function of the space), with Diffusion Coefficient being the proportionality constant. The parts of the curve where the magnitudes of the curvatures are high experience higher time rate of change. This explains why the curve smoothens out, or in other words 'straightens out' as time goes by.
Wow, that 3D graph of the surface was incredibly well visualized.
That's what Grant, the dude in the video, does. 3Blue1Brown is his TH-cam channel. One of the best math education/visualization channels on this whole site.
@@gileee Oh I know. I've played around with manim (the software library he uses to make the animations) myself, but Grant is still by far the king of making visualizations like these.
Thanks Grant ! One take away for me : Directly-Unsolvable PDE, Numerical simulations to the rescue !
This video just made me less dense. Loving it, thanks Grant!
loved watching the smooth transition from a simple idea to an equation
So beautifully and lucidly illustrated! I thank the TH-cam algorithm for bringing me to this channel.
So beautiful! We also use this equation in population genetics to study how allele frequency changes along with time.
I'm new to the field. Could you recommend a reference with implementation?
for the 1st time in my life i see the hope of understanding laplacian... it freaked me out every time when i try to understand graph convolution
It is a revolutionary little equation. Basically, what it states is - the higher the inequality, the higher the incentive for change.
The "statistical noise" of 4:52 is not such, it isn't because the "random statistics" ( 4:38 ).
It's actually because the implementation of the random walk explained at 3:52 :
50% hop to the left, 50% hop to the right
The resulting curve would be smoother:
- if we interpolate the values between steps
- if we add a third option: 33% hop to the left, 33% stay, 33% hop to the right
- Even better if it had 3 options with a discrete normal distribution, ie: 25% hop to the left, 50% stay, 25% hop to the right
- The best would be a continuous normal distribution
19:00 oh wow, this really opened my eyes regarding why exactly the mean value formula for harmonic functions works!
The temperature do not just GENERALLY flow from the warm part to the cold part. It's the first law of thermodynamics.
When you start talking about the brownian motion and then later appears de squared of the delta_x I was waiting for you to start talking about how this thing are managed to be constant allowing the solution truly exist, introducing in this way you viewers into Ito's Calculus.... hope you can extend the video with this modern view of Brownian motions analysis. Thanks you beforehand!
If I had his voice, I’d never stop talking! 😂
Gorgeous way of teaching diffusion equation!
How does this man teach everything I learn?
And I think that if you give the time dependent side an imaginary number, you get the schrodinger equation from quantum mechanics.
Inspiring video. It made me to think about generalized Pythagorean theorem coming from duality in linear spaces. The divergence is another result of the duality that is coming from linearity it is keeping invariant and doesn't change with unitary transformations.
20:30 In some cases you can solve this equation exactly (Green functions, etc.). So you can kind of check your computer code against these solutions
i really like your style and appreciate it that you explain julia
Finite differences, it makes more to use finite volume schemes, specificly when D is not a constant
I like the approach (thinking about random walk) and the clarity of the explanation :)
It already clicked when you increased to dots to 10,000. That's why you're a great teacher, Grant.
P.s. missed 3b1b.
A bit unclear why we have dX^2 in denominator when we take difference of differences 😢
Frikkin wizard-man, this guy!
HEY I KNOW THIS VOICE :O
Ohhhh nice. I'm in a semiconductor course and was wondering how ficks law came to be.
I liked this video, but why the Schrodinger Equation is a diffusion equation ? Perhaps the Wave Function is diffusing in space, any idea ?
Ahhhh, so this is the '1brown'. Love your videos so much!
11111
x=x+1
Let ε < 0.
area over curve
AoE2?
Suppose 1=2
first time here and worth it, i see grant's handwriting
No idea how this ended up in my Recommended at 01:15 am but I did like it.
Wow, TH-cam has never shown me these. And I never noticed them on Twitter I guess.
It was great. Your videos are amazing. I hope you continue to upload these kind of videos.
Brilliant animations as always!
I'll be Happy If He writes a book on Mathematics.
Then we wouldn't be able to hear him talk...
Bonus: the diffusion equation plays a central role in Grover's function inversion algorithm in quantum computing.
Excellent channel
It'd be nice to see some Finite Volumes Method solving the diffusion problem...
Everyone is commenting about him, but no one is talking about the topic. Probably because it was explained clearly.
I prefer using conservation of mass in integral form and the divergence theorem tô deduce the equation
when will you come back to 3blue1brown :(
He's producing lectures for MIT... obviously that's much more prestigious than a TH-cam channel.
Hi! Why did you define the number of particles in the boxes as densities rather than concentrations? Are you assuming that the particles occupy all the volume of the boxes?
I took this class last semester wish i would have waited for this one!!!
this makes a math nerd happy 😭
Awesome! Thanks Grant
Is it a Blue Yeti microphone?
2:54 if you are watching on phone and full screen and jiggle the phone a bit while looking at the dot It'll make a nice effect
3 Blue 1 Brown is DIFFUSING!
Very beautiful explanation video. Now you can blink
your voice sounds like 3 blue 1 brown. good lecture but was surprised to hear that exact voice cadence.
Dear Grant,
What software do you use to create the interactive graphical animations for your videos? Is this commercial software? Perhaps, did you write the software? If so are you able to opensource for those interested in the code? Thanks as always from a keen learner from your work.
Regards,
Joshua
He uses a library called Manim in Python. As far as I know, he is the one who founded this library.
Great ! Makes me feel mathematical young, but I’m not young I did these exams 30 years ago, I wish I had this fun at the time
This is Pewdiepie if Pewdiepie is a mathematician.
Temperature does not flow. Its eat.
Amazing explanation. Thanks a lot.
3blue1brown has a second channel? I had no idea.
you had me at "hot rod"
Rho, rho, rho your boat
Gently down the slope
Merrily, merrily, merrily, merrily
Life is but a delta
Absolutely perfect.
So that's why cityfolks where I live can't understand the keep your distance signs. Because they're too dense. Which means they don't keep distance...
0:06 guess i now know why his channel is named "3blue1brown. and i got the same eyes as him, just the 3 blue is grey for me and i also have a smaller brown dot in my other eye
Brilliant content, thank you!
Incredible!
Anyone have any good resources (books/simple papers) for doing this kind of thing on a finite grid? I've been trying this with random walks on 8-neighbors and it keeps drifting and settling in the center and I'm not sure why that is.
You have good looking, good voice and good teaching!
Is there a course for differential equations
Somebody buy this man a nice pen and squared paper!
why did the uppercase D disappear
Great class, professor.
What if the fraction of the density variation is not 1/2?
Sir,please teach us also about advection..please😥
hello,I am very happy that I found you, I am doing distant course in DSML in collaboration with MIT , I wish if i too had a teacher like you , any suggestion for me , i am very much interested in deep learning and ai , and i know your lectures will be very useful for me and thanks for making them available for free ....
Hi, I'd like to know how did you make the graph animation
thanks
it's his own library called Manim ("math animations"); it is open source
Very well explained !!
The best equation ver 🥰
Why would (delta x)^2/(delta t) be constant? (delta x)/(delta t) is molecules velocity and may be considered a constant value. But what about another delta x left?
Great question! If each particle is moving according to a random walk, then after many steps the distance traveled will not be proportional to the time. However, the *square* of the distance traveled will, in expectation, grow linearly with time. It's a fun little exercise to work out.
@@3blue1brown Please correct me if I'm wrong. Square of drift is proportional to time in the case of, strictly speaking, delta t (hence delta x) -> Inf. Here, though, we are dealing with small increments. Notwithstanding all this minutia, I am a big fun of your brilliant channel. Keep up your excellent work!
@@maharajxx also consider that as dx->0, the number of cells in a given volume of space goes to ∞. It's the granularity of x and t, not their magnitudes, that makes the random walk statistically a normal distribution, which has the squared relationship in the expectation that's at issue here.
@@3blue1brown Is the constant of proportion just an arbitrary choice of the person creating the mathematical model or is there any connection to the real material that is diffusing? It seems like this is the only link missing between mathematics and physics in your presentation. I wonder if it can be filled in a mathematically rigorous way.
Wow this guy should open his own youtube channel 😂
I guess genius doesn't always correlate with madness after all :)
This was helpful! Thank you!
amazing explanation !