Next up we finally get to Fourier series, which will be the beginning of a turn in the series towards understanding the surprising depth and importance of exponential functions for differential equations. Stay tuned!

Hey 3bl1br! I've been a big fan of yours for a while, for a lot of the great videos you have. I did mathematics as my major in college, and your videos have always served to help me gain intuition on some areas that felt hazy. This video set that a level above for me. For context, I almost failed ODEs. I could not simply grasp what was going on, and had to resort to hoping I was applying the right method at the right time. My professor seemed to get these results out from nowhere, with e^(nt) popping up randomly, without really stopping to try and explain where things originate from (analytically, geometrically, or intuitively). For the first time, I feel like I can begin to grasp a lot of things I was already supposed to know. So, thank you! As a side note, I've also been reading Gleick's Chaos. This series of videos emerging at the same time feels like a perfect coincidence! Guess I'll attempt to learn more DiffEqu now.

Amazing video. You make differential equations seem easy thanks to your easy-to-follow explanations and your wonderful animations, which help to visualize the problem and make it very intuitive. Thank you, I learned a lot!

In 15 minutes you managed to explain PDEs better than anyone I've seen before. You really made everything just fall into place! Looking forward to Fourier series and hopefully their generalization in Laplace transforms in the next episode. Can't wait!

Hey! This is just an other enthusiastic comment! For me, your work is the most quality content on YouTube. Thank you very much! If I had a steady income, I'd be proud to be a patron… As I am just a high school student keen on math, I will just add this though, hopefully it gives you a shot of endorphin, at least =D

I do have one minor quibble, though: you can have solutions where dT/dx ≠ 0 at the ends by maintaining a constant temperature on each end of the rod. For example, the solution which "fails to describe actual heat flow" (7:17) actually describes the temperature distribution inside a wall where the interior is at one temperature and the exterior is at another temperature, and you can use it to figure out how much heat is flowing through the wall. We physicists & engineers do think about such solutions frequently. (I can understand why you didn't want to get into these weeds in an introductory video, though.)

Here's a simple toy to play with the heat equation https://math.berkeley.edu/~kmill/toys/heat/heat.html (showing the Fourier transform of the heat curve you draw).

Can you do a vid on that shape (I think it's the Hopfs fibration) that Eric Weinstein mentioned on Joe Rogan? It had something to do with gauge theory.

Absolutely fantastic video. As an EE Major I have studied a ton of PDEs, but learning about it through the lens of the heat equation is giving me a refreshed admiration and respect for it. Extremely well thought out video! Love it.

A question that does not exist with the video, if I have an exponential function f (x) = a ^(x) e h(x)=a^(x/n), I say that the function h(x)=a^(x/n) undergoes horizontal (1/n) times or (n) times dilation with respect to h(x)? Note: neIN and sorry if i said anything wrong, i´m Brazilian.

This explanation is a must for chemical engineers! This is how any system equilibrates and describes mass stranfer (Fick's law) and heat transfer. This is at the heart of any chemical engineering problem. For those interested: The Fourier number relates to the general solution of this equation. So, this is not only mathematically interesting. Our lives, quite literally, depend on it!

Very nice work again!! Loved it. However, I think you glanced over the boundary conditions too quick. There are conditions where for instance a side if the rod is kept fixed at a certain temperature or that the heat flow is determined by temperature. That is probably to detailed for this series but I would have appreciated a quick mentioning of that.

Grant, watch out for Flammable Maths! He has insulted other members of the math community with well known victims include blackpenredpen, Dr. Peyam and presh talwalker! Beware!

I have a B.S. In physics…in 1972 ! I have finally learned enough math to justify my degree ! Seriously,math and physics instruction is so much better on YouTube than it was in the 60's that even the few bucks it cost back then,that I am thinking of asking for my money back. With compounded interest of course. Then again that B. S. has me worth several mil at retirement so maybe I'll call it even.

I watched through your videos on linear algebra a while ago and thought to myself "man, if only I had seen these while taking my linear algebra course, it would have given me a lot more intuition for the subject a lot easier" and now this, in the middle of my course on integral transforms and differential equations 😀 Thanks a lot, you're doing a great job.

I really love your videos. They helped me a lot understanding complex math. Thanks to your explaination and animations I don't just see math as pure formulas anymore – I see vectors and functions in a transforming spatial space and this is really helpful. When this series is done I'd love to see an "essence of tensor calculus" or something like that. You have so much fun in creating animations – I guess tensors would be a good choice even for you 😉

would you say this is a high school topic? i need to do a research paper, and i am thinking of using this as my topic (the paper can be a thorough explanation of a proof / proving an equation), and I 'm not sure in terms of complexity if this is a viable option for me. I'm in Grade 11 (17y/o), any advice / opinions?

@3blue1brown Hey Grant! Just a little feedback here. I think I found a jewel when I found your channel, for real. The reason why is because I have a tendency to understand things better when they're explained to me through graphs and animation, especially when it comes to math. I'm more of an artsy person myself, but I really love science. You see, I'm studying chemistry in Argentina, and despite the language barrier, I do think that it's worth the time and effort to try and understand the material you're producing, mainly because it's given me a deeper insight on the topic which I'm struggling to comprehend the most, that is, calculus. I'm currently doing the course for the second time… And last year my failure on every exam that I had was a bad hit to my self-esteem. But now, I'm starting to grasp more the concepts of divergence, curl, and all those things that seemed something that I had to memorize. Because of your videos, I do realise that I do have some kind of ingrained passion for math and physics, even though I do suck at them for the most part in terms of explaining my thoughts and reflecting it on paper. So, to wrap it up, thank you a lot for your passion and effort! I'm aware that the probability that you'll read this is rather scarce, but hey, a tiny dot in a tiny fraction of probability is not negligible, since it forms part of a bigger thing, right? Anyway, I'll go and try to watch the videos again. Listening to your lessons gets easier to understand the more time I let my brain traduce what you're saying, hahaha. Have a good one!

When you have exam from this in 2 weeks, there are no more 3Blue1Brown videos on the topic, and this is the 666th comment : This must be the work of the devil! 😛

A solution of T = Cx is a real world solution to the problem just corresponding to different boundary conditions than the adiabatic (lnsulated object) boundary conditions. But it's not important for the purpose of the video. This is a really interesting video because I never understood the connection of how Fourier transform could be used to solve heat equation. Thank you for making such a great videos!

A zero derivative at the boundary is only because you selected Neumann boundary conditions. We could have used sine waves if you wanted Diricelet Boundary Conditions

Walking into the bar I became aware of an exponentially decaying pencil of complex sinusoids flexing and writhing in the centre of the room, shedding colours across the watching conics. It was The Heat Equation.

Disregarded, a bored Lissajous pattern spun idly in the corner.

I like the fact that music from the Essence of Linear Algebra series plays during this particular video. Just to give a slight hint of how important the subject is for Differential Equations.

It sounds like a bell. Those wave functions describe the string on a guitar, or a bell, or the Moon itself, which also resonates like a bell in an specific frequency. Thanks! No wonder Tesla said that in order to understand we need to think in terms of frequencies, the Spectrum. Math is proof of it. You are a great professor. You manage to explain complex into simple. Such a gift from Logos.

Ohh la idea del seno modelando el eje x y el exponencial modelando el eje t me ayudó a entender porque se usa el método de separación de variables para resolver las edp

Every math class into the future will be infinitely more useful with these videos, man. Like… before, math was the domain of those few who could easily translate it all into animations in their heads, and everyone else just had to struggle to try to do so, never really knowing if they had it right up top. Showing these animations to someone once is enough to make it all click so much more reliably. I sucked monumentally in my math courses in HS & college. I spent 10x the time for 10% of the understanding of the kids who just "got it" out of the box. Watching all of these videos almost a decade later, so much of it finally starts to make sense. I don't have to just cram-memorize formulas the night before an exam, only to dump them from my head immediately afterward, never actually able to get any real-world use out of them, lol.

I get it that for t>0 the function have to satify the boundary condition, but nowhere in this video say it also have to at t=0. My question is for that condition to be satified, do the function at t=0 also have to satify it, or is it possible for us to generate a function such that the curves at boundaries is not 0 at t=0 and is 0 at all time t>0, and if not then why? Does anyone know?

Hmm, so for an arbitrary function of temperature as a function of x, the arbitrary function of temperature in terms of x is approximated by the fourier series as an infinite summation of trigonometric curves. I wonder, would it have been easier if the temperature as a function of x be approximated by a taylor or maclaurin series? Would an infinite summation of monomials satisfy the boundary conditions?

Hmm, so it seems that cosine functions (with zero derivative at boundary) makes sense for unbounded rods, where the rods are not subject to any temperature constraint. What would happen if we set the temperature of the rod at some constant temperature?

This is the point of my life, where I realize how idle have I been 🙁 I have a PhD. in Maths and don't understand what he said about the flat thing in the boundaries (I could solve the equations as a student, yeah, but didn't understand the meaning); I will watch the video again… be dilligent and start again if necesary 🙁 Thanks a lot for the very illustrative videos

Next up we

finallyget to Fourier series, which will be the beginning of a turn in the series towards understanding the surprising depth and importance of exponential functions for differential equations. Stay tuned!how do u make that animations?

they are amazing

I am not able to give you infinite likes please solve this problem :-))

Hey 3bl1br! I've been a big fan of yours for a while, for a lot of the great videos you have. I did mathematics as my major in college, and your videos have always served to help me gain intuition on some areas that felt hazy. This video set that a level above for me.

For context, I

almostfailed ODEs. I could not simply grasp what was going on, and had to resort to hoping I was applying the right method at the right time. My professor seemed to get these results out from nowhere, with e^(nt) popping up randomly, without really stopping to try and explain where things originate from (analytically, geometrically, or intuitively). For the first time, I feel like I can begin to grasp a lot of things I was already supposed to know. So, thank you!As a side note, I've also been reading Gleick's Chaos. This series of videos emerging at the same time feels like a perfect coincidence! Guess I'll attempt to learn more DiffEqu now.

Amazing video. You make differential equations seem easy thanks to your easy-to-follow explanations and your wonderful animations, which help to visualize the problem and make it very intuitive. Thank you, I learned a lot!

Your videos is amazing ..do you teaching maths somewhere?

Make a video for the unsolved Greek problem..If N a natural number how many first numbers smaller than N exists?

Cool, as always !

I like your chanel…good job..thank you..from morocco.

In 15 minutes you managed to explain PDEs better than anyone I've seen before. You really made everything just fall into place! Looking forward to Fourier series and hopefully their generalization in Laplace transforms in the next episode. Can't wait!

“Sine = Nice”

Im not a genius, but im damn close ~ my math teacher.

Your animations get more ellegant with each new video you make!

Hey! This is just an other enthusiastic comment! For me, your work is the

mostquality content on YouTube. Thank you very much! If I had a steady income, I'd be proud to be a patron… As I am just a high school student keen on math, I will just add this though, hopefully it gives you a shot of endorphin, at least =DOmg I just realized there’s 3 blue pi’s and 1 brown one 😭

normal people: I hate math

nerdy people: I love mathSSSS

How do you make the animations?

AWESOME

in first semester , i had calculus and you had your series on it going on . awesome .

i had linear algebra in 2nd semester and oh god you had a series just then . awesomely awesome .

and now . i will be starting course on differential equations for my 3rd semester . and guess what … a golden series on DEs is going on .

this is not fake and is a very very good motivation for me .

Bloody brilliant!

Just in time for my mathematical physics exam

Have anyone seen tetration?

The series means a lot to me. Thank you for making these brilliant videos.

pLEASE MAKE VIDEOS ON Advanced calculus and tensors.

Nice work as always!

I do have one minor quibble, though: you can have solutions where dT/dx ≠ 0 at the ends by maintaining a constant temperature on each end of the rod. For example, the solution which "fails to describe actual heat flow" (7:17) actually describes the temperature distribution inside a wall where the interior is at one temperature and the exterior is at another temperature, and you can use it to figure out how much heat is flowing through the wall. We physicists & engineers do think about such solutions frequently. (I can understand why you didn't want to get into these weeds in an introductory video, though.)

Please, please, please don't make us wait 8 more weeks. That was hell.

Very nice. If you get into the finite element method (e.g. Galerkin variational method), I will probably implode.

Can u do a series of videos on mathematics of machine learning

5:02 AlBSiO

Here's a simple toy to play with the heat equation https://math.berkeley.edu/~kmill/toys/heat/heat.html (showing the Fourier transform of the heat curve you draw).

Can you do a vid on that shape (I think it's the Hopfs fibration) that Eric Weinstein mentioned on Joe Rogan? It had something to do with gauge theory.

I have a test this Wednesday, will I be able to see the next chapter? 🙁

I know you are very busy but could you do a video on Airy's differential equation?

Awesome

Absolutely fantastic video. As an EE Major I have studied a ton of PDEs, but learning about it through the lens of the heat equation is giving me a refreshed admiration and respect for it. Extremely well thought out video! Love it.

I don't know if here is the right place to ask, but I'd like a video on Simplex Algorithm in Linear Optimisation.

A question that does not exist with the video, if I have an exponential function f (x) = a ^(x) e h(x)=a^(x/n), I say that the function h(x)=a^(x/n) undergoes horizontal (1/n) times or (n) times dilation with respect to h(x)? Note: neIN and sorry if i said anything wrong, i´m Brazilian.

Please consider doing a lecture series on complex analysis. Heck, while you're at it, please do all of mathematics 🙂

thanks so much for all the great vids！I really learnt a lot from you.

I spent 2 hours watching this video without ever not understanding something- a testament to the mulling potential portrayed by this video.

How do you make those beautiful graphs? With Python?

This explanation is a must for chemical engineers! This is how any system equilibrates and describes mass stranfer (Fick's law) and heat transfer. This is at the heart of any chemical engineering problem. For those interested: The Fourier number relates to the general solution of this equation. So, this is not only mathematically interesting. Our lives, quite literally, depend on it!

Do langrangian and hamiltoniann (animated 3b1b)

What's – 3² ?

Me:

9

3Blue1brown:

[5464364643_5466£4464-73743433,65965¢%868-868⅞+3²>55522+5578556866/8686×e=mc²2855556+555

=9,0000000314

I love the way you visualize the topic where you gain a good intuition for why doing all this math. Thanks Grant!

Very nice work again!! Loved it. However, I think you glanced over the boundary conditions too quick. There are conditions where for instance a side if the rod is kept fixed at a certain temperature or that the heat flow is determined by temperature. That is probably to detailed for this series but I would have appreciated a quick mentioning of that.

You should do a series on stochastic processes – things like Brownian motion will probably be really interesting to the viewers 🙂

Grant, watch out for Flammable Maths! He has insulted other members of the math community with well known victims include blackpenredpen, Dr. Peyam and presh talwalker! Beware!

I have a B.S. In physics…in 1972 ! I have finally learned enough math to justify my degree ! Seriously,math and physics instruction is so much better on YouTube than it was in the 60's that even the few bucks it cost back then,that I am thinking of asking for my money back. With compounded interest of course.

Then again that B. S. has me worth several mil at retirement so maybe I'll call it even.

I watched through your videos on linear algebra a while ago and thought to myself "man, if only I had seen these while taking my linear algebra course, it would have given me a lot more intuition for the subject a lot easier" and now this, in the middle of my course on integral transforms and differential equations 😀

Thanks a lot, you're doing a great job.

wish I had a better math teacher in highschool, whenever I see formulas I just memorize them and skip the math. This was very educational, thank you

Hi Grant,

I really love your videos. They helped me a lot understanding complex math. Thanks to your explaination and animations I don't just see math as pure formulas anymore – I see vectors and functions in a transforming spatial space and this is really helpful. When this series is done I'd love to see an "essence of tensor calculus" or something like that. You have so much fun in creating animations – I guess tensors would be a good choice even for you 😉

Thank you very much for your work 🙂

can you please do series on innerproduct and vector spaces?

would you say this is a high school topic? i need to do a research paper, and i am thinking of using this as my topic (the paper can be a thorough explanation of a proof / proving an equation), and I 'm not sure in terms of complexity if this is a viable option for me. I'm in Grade 11 (17y/o), any advice / opinions?

@3blue1brown Hey Grant! Just a little feedback here. I think I found a jewel when I found your channel, for real. The reason why is because I have a tendency to understand things better when they're explained to me through graphs and animation, especially when it comes to math. I'm more of an artsy person myself, but I really love science.

You see, I'm studying chemistry in Argentina, and despite the language barrier, I do think that it's worth the time and effort to try and understand the material you're producing, mainly because it's given me a deeper insight on the topic which I'm struggling to comprehend the most, that is, calculus. I'm currently doing the course for the second time… And last year my failure on every exam that I had was a bad hit to my self-esteem. But now, I'm starting to grasp more the concepts of divergence, curl, and all those things that seemed something that I had to memorize. Because of your videos, I do realise that I do have some kind of ingrained passion for math and physics, even though I do suck at them for the most part in terms of explaining my thoughts and reflecting it on paper.

So, to wrap it up, thank you a lot for your passion and effort! I'm aware that the probability that you'll read this is rather scarce, but hey, a tiny dot in a tiny fraction of probability is not negligible, since it forms part of a bigger thing, right?

Anyway, I'll go and try to watch the videos again. Listening to your lessons gets easier to understand the more time I let my brain traduce what you're saying, hahaha.

Have a good one!

When you have exam from this in 2 weeks,

there are no more 3Blue1Brown videos on the topic,

and this is the 666th comment :

This must be the work of the devil!

😛

Are you a teacher?

What font do you use for your videos?

Could you solve the heat equation for a microwave burrito?

Hey mate, I was sure i'd find a perlin noise video on your channel

A solution of T = Cx is a real world solution to the problem just corresponding to different boundary conditions than the adiabatic (lnsulated object) boundary conditions. But it's not important for the purpose of the video. This is a really interesting video because I never understood the connection of how Fourier transform could be used to solve heat equation. Thank you for making such a great videos!

Brain.exe stop working

can you tell me which software you use to make the animation?

A zero derivative at the boundary is only because you selected Neumann boundary conditions. We could have used sine waves if you wanted Diricelet Boundary Conditions

Can the temperature distribution of a klein bottle over time be modeled with PDE's without using boundary conditions? 😉

Waiting anxiously for the next part..😢

Great video!!!

Your 3D animations look freakin great dude!!!

Why demonetized video?

You should do a vídeo only on bondary conditions.

what software do u guys use!?????

Walking into the bar I became aware of an exponentially decaying pencil of complex sinusoids flexing and writhing in the centre of the room, shedding colours across the watching conics.

It was The Heat Equation.

Disregarded, a bored Lissajous pattern spun idly in the corner.

do you forget the code number of bilibili? we are waiting for your course

If there are heat sinks on both sides of the rod, we don't need the "flat at ends" condition, right?

ugh I HATE messy nature why can't rods be infinitely long and then they'd DEFINITELY be like a sine wave forever :V

I like the fact that music from the Essence of Linear Algebra series plays during this particular video. Just to give a slight hint of how important the subject is for Differential Equations.

It sounds like a bell. Those wave functions describe the string on a guitar, or a bell, or the Moon itself, which also resonates like a bell in an specific frequency.

Thanks!

No wonder Tesla said that in order to understand we need to think in terms of frequencies, the Spectrum.

Math is proof of it.

You are a great professor. You manage to explain complex into simple. Such a gift from Logos.

I just can't believe he never said "Sturm–Liouville" even one time! what an outrage!

This is like trying to understand aliens. I tried!

Does this mean high frequency waves decay after a shorter distance?

Ohh la idea del seno modelando el eje x y el exponencial modelando el eje t me ayudó a entender porque se usa el método de separación de variables para resolver las edp

Im so sad that I can't understand with my language, Korean.

I need more translation for this video.

Every math class into the future will be infinitely more useful with these videos, man. Like… before, math was the domain of those few who could easily translate it all into animations in their heads, and everyone else just had to struggle to try to do so, never really knowing if they had it right up top. Showing these animations to someone

onceis enough to make it all click so much more reliably. I sucked monumentally in my math courses in HS & college. I spent 10x the time for 10% of the understanding of the kids who just "got it" out of the box. Watching all of these videos almost a decade later, so much of itfinallystarts to make sense. I don't have to just cram-memorize formulas the night before an exam, only to dump them from my head immediately afterward, never actually able to get any real-world use out of them, lol.I get it that for t>0 the function have to satify the boundary condition, but nowhere in this video say it also have to at t=0. My question is for that condition to be satified, do the function at t=0 also have to satify it, or is it possible for us to generate a function such that the curves at boundaries is not 0 at t=0 and is 0 at all time t>0, and if not then why? Does anyone know?

just finished a degree in ECSE and did not have an intuition for boundary conditions until just now, thank you for your content

What are the functions that cannot be represented with a fourier transform? :/

Looking forward to the video on Laplace Transform!

Thanks for the video!

I'd like to ask how when there is a curvature i.e. change in space you say that there is a change in time as well?

Hmm, so for an arbitrary function of temperature as a function of x, the arbitrary function of temperature in terms of x is approximated by the fourier series as an infinite summation of trigonometric curves. I wonder, would it have been easier if the temperature as a function of x be approximated by a taylor or maclaurin series? Would an infinite summation of monomials satisfy the boundary conditions?

GOOD

Hmm, so it seems that cosine functions (with zero derivative at boundary) makes sense for unbounded rods, where the rods are not subject to any temperature constraint. What would happen if we set the temperature of the rod at some constant temperature?

OMG, it's so frustrated, WHY do the slope of the solution at x=0 and x=L have to be 0 even for t=0?????

Pure Joy..<3 Thanks for the amazing video!

Saw E4 about Fourier advertised, started at E1, just to have a snowball's chance to understand. 🙂

Will you do a series on Optimization as a continuation of the Linear Algebra series?

This is the point of my life, where I realize how idle have I been 🙁 I have a PhD. in Maths and don't understand what he said about the flat thing in the boundaries (I could solve the equations as a student, yeah, but didn't understand the meaning); I will watch the video again… be dilligent and start again if necesary 🙁 Thanks a lot for the very illustrative videos

Man, this guy is the reason behind sapiosexuality studies in universities…

god bless this man

This behavior is not reminiscent of my rod….

I thought I was a good teacher. Crazy video.