Solving the heat equation | DE3

Boundary conditions, and set up for how Fourier series are useful.
Help fund future projects: / 3blue1brown
An equally valuable form of support is to simply share some of the videos.
Special thanks to these supporters: 3b1b.co/de3thanks
Home page: www.3blue1brow...
More about the heat equation, with a derivation in terms of slope corresponding to heat flow from MIT OCW:
ocw.mit.edu/co...
If you want to learn more about Fourier series, here are a few great videos/posts which I think you'll enjoy:
Mathologer's: • Epicycles, complex Fou...
The Coding Train: • Coding Challenge 125: ...
Jez Swanson: www.jezzamon.co...
Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld
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These animations are largely made using manim, a scrappy open source python library: github.com/3b1...
If you want to check it out, I feel compelled to warn you that it's not the most well-documented tool, and it has many other quirks you might expect in a library someone wrote with only their own use in mind.
Music by Vincent Rubinetti.
Download the music on Bandcamp:
vincerubinetti...
Stream the music on Spotify:
open.spotify.c...
If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then "add subtitles/cc". I really appreciate those who do this, as it helps make the lessons accessible to more people.
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3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with KZbin, if you want to stay posted on new videos, subscribe: 3b1b.co/subscribe
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[@3blue1brown]

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!

[@randomdude9135]

First Can you make more videos on 3D geometry?

[@davidadamic4322]

3Blue1Brown noooo, just a bit too late ☹️ i have differential ewuatuons exam tomorrow and i dont really get the Fourier series and Bernoulli method

[@Eltaurus]

I finally got, what the intro logo animation reminds me of. Is it in some way inspired by the ILM logo reveal? kzbin.info/www/bejne/api2hYN-d9VsmaM

[@prittbalagopal1105]

@@Eltaurus It's actually from Grant's heterochromia, his eye is literally a quarter brown and three-fourth's blue, like the logo. Google it.

[@dsdsspp7130]

Eltaurus there are hundreds of similar animations its not a rare idea You're just overthinking it

[@chriskindler10]

you know this is the best educational channel on KZbin when your maths professor recommends it.

[@CesarGomez-kp5lm]

Must be an awesome professor

[@tommihommi1]

my mother asked me if I knew 3blue1brown recently because she read about him in the newspaper.

[@jamesmnguyen]

My Calc 3 professor is a fan of him. Even has 3b1b's videos as a reference on the class website.

[@randomdude9135]

I discovered him myself.

[@brorelien8447]

I recommended this channel to one of my professors! 😂

[@pearceburns2787]

"Seek Idealized problems ->[Find] General solutions -> [Create] Realistic models" ( 3:20 ) resonates with me, and I would definitely love a poster with this mantra on it.

[@tdiaz5555]

Idealize. Generalize. Realize.

[@Thomas-er8xg]

I’d pay for that

[@nicolassamanez6590]

its essentially the cornerstone of mathematics

[@webx135]

@@nicolassamanez6590 Heck, it should also be a cornerstone of politics. Flawed political systems often result in revolutions which put an idealized system in place, which then falls apart, but the generalizable concepts remain and spread.

[@v44n7]

This, for me, is the key of why math is invented and not discovered

[@jibraniqbal9315]

This took a while but I know I'm going to enjoy it so the wait is most definitely worth it. Thanks for the content, Grant

[@feschber]

one could almost say we were _Granted_ access to this video (\____/) ( ͡ ͡° ͜ ʖ ͡ ͡°) \╭☞ \╭☞

[@satjanir1163]

It is amazing

[@hugblast1379]

worth nada cara ta loco

[@aniksamiurrahman6365]

To a math noob 3Blue1Brown videos are like heavenly inspiration. Intuition conveyed here often acts like the light at the end of the frustrating and treacherous tunnel. But as the course advances more and more things starts to make sense and the video wears well as one grows more experienced.

[@Mayank-mf7xr]

they are so beautiful because even the maths pros find it enlightening

[@Mayank-mf7xr]

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 .

[@YourThermalWorld]

I graduated BSME in 1978. ODE’s and PDE’s actually made sense to somewhat but I didn’t fully understand them. Now that I am retired, your channel has enriched my knowledge immensely. I thank you for stimulating this Senior Citizen’s mind!

[@otisarmyalso]

Yep class of 77, nice graphics... I like Vissim numerical integrators

[@TebiByyte]

I'd like to point out that this is something that is done in Quantum Mechanics quite a lot! Being able to exploit the linearity of a PDE, and then finding solutions using a Fourier Transform is immensely powerful. In my opinion, it's some of the most beautiful mathematics that I've ever come across.

[@freakyfrequency2530]

The Schrodinger equation is basically the heat equation with and imaginary diffusion constant

[@HaramGuys]

​@@freakyfrequency2530only if V = 0

[@diaz6874]

I'm genuinely waiting for the Laplace Transform one.

[@piman7319]

That will be a glorious, much anticipated/requested day. I predict it will happen three videos from this one

[@raulgda121]

Really in need of that one, god of math hear my pray and provide us with your knowledge

[@bpetersson5024]

I thought he did but it was the Fourier Transform..

[@arunprakash6508]

👋, I'm a 10th grade student in India, can someone please tell me some of the most interesting topics like this series, So that I'll be able to get more career ideas?

[@vutruongquang3501]

@@arunprakash6508 it's the Basel Problem and colliding blocks compute Pi

[@AaronFresh09]

You should know how important this channel and the videos you make are to the online learning community. Nothing else, not even your early work at Kahn academy, comes close to how comprehensive and professional the videos on this channel are. It is an invaluable source for self learners and I hope to see many many more.

[@gonzalezm244]

“This makes for a good stopping point.” No, no it doesn’t 😢

[@aok76_]

I know right.

[@hoodedR]

Yeah 😢

[@justcarcrazy]

We want MOAR! We NEED MOAR!! NOW!!!

[@howardbaxter2514]

Agreed

[@corpsiecorpsie_the_original]

Group hug *sniffles*

[@AaronPM55]

I started watching your videos in high school and I'm currently doing signal processing and modeling at my current internship where I look at PDEs all day and I still find every video you make extremely enlightening. Thank you.

[@bobtivnan]

This is so rich with content that I find myself pausing every few seconds just to appreciate the imagery and your artistic approach. Simply Beautiful.

[@newkid9807]

Yo bob, you reading the right book?

[@bobtivnan]

@@newkid9807 , not sure what you mean.

[@newkid9807]

Bob Tivnan It’s okay man...

[@張海鷗-t2t]

所有參政言論活動中破壞性者提交違規行為摧毀性系統工程滅族性操作方式執政者，都必要革職移除頻道，終身不得登入帳號密碼遊戲經商使用，移送法辦監獄服刑處分

[@DiracComb.7585]

Given that I’ll have to take this course in my college curriculum, I appreciate the well done and extremely insightful resource. Pls don’t cancel the series

[@toebs_]

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

[@keyyyla]

Your animations are just beautiful. It is so nice being able to SEE your explanations and at the same time hearing them.

[@FF-wl1oo]

Hey Mister, I just wanted to let you know how grateful I and probably many many others are for these kind of videos including the visualization. You know, sometimes when you are learning some new things, it can be very hard to imagine something just out of a line in a book or article. I can imagine these videos take hell of a time to make, but they are superb. I often times spent even days thinking about only a few pages in a book trying to understand something I read, meanwhile googling for all sorts of videos, until I made my mind clear and this is a perfect shortcut to understand. Here goes my thanks as well as thanks of many other students and possibly many other people who love maths.

[@lucasnicoara7400]

I actually got excited seeing this in my subscription box!

[@amyshaw893]

could you not have posted this series before my exam :D

[@BudskiiHD]

same haha

[@HerbaMachina]

Lolol this was me after finding his calculus series and realizing it posted a year after I had already finished high school 😭.

[@filmweaver2013]

My semester final results are also out!

[@myfestivities8095]

😄😄😄

[@jwingo7257]

You have an amazing gift to explain visually as well as through your narration the abstractions of higher level mathematics. If I only had your excellent videos when I was in graduate engineering school.

[@anthonyesquire9830]

Oh the gods truly blessed us today. My goodness. How long have I been waiting for this. Thanks you as always.

[@davidsonjoseph8991]

E X A C T L Y.

[@webx135]

@3Blue1Brown I had a great professor from Australia who taught us this stuff in my second-semester differential equations course. But when I watch your videos, it makes it all click so much better. You really are a quality channel. Specifically, a channel I would recommend as a supplement to the courses to get a more goal-oriented understanding. I'd say the #1 problem with math teachers and professors is that they tend to give you a bunch of information without giving you a motivation to learn them in the first place. I'd say it's one of the biggest reasons math "clicks" with some while never resonating with others.

[@piman7319]

Well said. I agree completely.

[@李明彰-z9h]

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

[@michasz4297]

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

[@prydin]

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!

[@WhompingWalrus]

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.

[@zacn9159]

Completely blew my mind when you were talking about using sine waves for an function and I realised it: the graph’s second derivative will be the reflection of the the graph in the x-axis 🤯

[@owensilberg2966]

I watched this a while back without having any real knowledge of differential equations, and now that I'm a decent fraction of a semester into dealing with them it's awesome to see this video again having dealt with the tools that're mentioned!

[@jmvt3]

And this is all for modeling a one dimensional Bar...

[@snookerbg]

there was an illustration by Nathen W Pyle along the lines of: Science is so hard it sometimes makes me sad

[@mikhailmikhailov8781]

I think this should make you understand why numerical methods are used

[@Royvan7]

a 2D plate isn't too much more complicated just more boundaries and sinh/cosh starts getting involved.

[@張海鷗-t2t]

提交參政言論活動中違規破壞性者侵入行竊違反政策內容罪嫌移送法辦處理。執行移除頻道革職處分，資金賠償終身不得登入帳號密碼遊戲規則經商使用。移送遣送出境海外市場強制性系統工程處理刑責責任債務償還貸款銀行金融界集團旗下品牌形象廣告所有的虧損各類型態債券投資產業資源損譽虧損金額理賠處理

[@Royvan7]

@@張海鷗-t2t your little rant doesn't seem to have much to do with the video or anything else said. care to elaborate?

[@tonybobay6276]

Amazing! Geometric interpretation of mathematics = 3Blue1Brown. Nobody in the world has helped us simpletons to appreciate the beauty and intuitiveness of higher maths better than you. Thank you so much for your efforts!

[@chroma9848]

Im about to take DE in 2 months man I wish this series could finish sooner. Thank you for the legendary content btw!

[@lewebusl]

Your videos are a great tool for understanding difficult to visualize math concepts. I have watch many of your videos and all of them have expanded my knowledge in several ways. They are helping a lot of young and not so young people in embrassing the love for science. Just want to say that and encourage you to continue on...

[@tech2077]

Probably another thing to mention about using only working on "one example" is that most introductory PDE courses exclusively cover heat and wave equation due to their properties and difficulty alone. Working with equations beyond these requires the framework that these well studies problems give. Differential equations, and especially PDEs rarely occur ideally in nature, and understanding ideal cases and generalizing is effectively the only means we have to deal with what we find in nature. I love the series and can't wait for the next video!

[@chucklindgren8914]

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 KZbin 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.

[@jessrevill1852]

You do a really great job of explaining. I understand things now that I never knew, plus a lot of things I thought I knew but was totally wrong... Much appreciated.

[@michaeljburt]

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.

[@masterkeyforfun]

Those Boundary cond. are also called: Neumann boundary condition. Other examples of Boundaries are Dirichlet or even mixed boundaries :).

[@CyclesAreSingularities]

this was very visually pleasing to watch and I actually feel like the gears are starting to turn in my head. these seemingly impossible to comprehend formulas are quite simple in reality is what I'm starting to see. I think why it's hard to grasp these things is that when you look at everything at once there's just too much to try to make sense of. while in reality you can just do a whole bunch of simple equations over a time period to not alone get there but also describe how something is changing and using that information for your modeling. this, has been the most satisfying part about diving into maths thus far. like you said, these videos give you the confidence to try to tackle problems. and the more i'm learning about maths the more I realize it's not really about trying to make sense of numbers. it's about thinking of a practical way to solve problems and built up a toolkit for doing so

[@amarroy193]

Dear grant, You have no idea how much we love this channel. You just happen to know what we need right now. That's why we feel a conection beyond this much distance.

[@stevestarcke]

In Chemical Engineering, heat transfer is extremely important. For the first time I understand where these thermodynamic equasions come from. The world is even more beautiful than I thought. Thank you for the insight!

[@babathec9438]

I just discovered this channel a week ago and i would like to tell you that even for a French the video is understandable (I'm quite good at English but still) and also, as a 17 yo guy, you comfort me in my will to become a mathematician si i would like to sincerely thank you pal ; )

[@oswaldbourayne2594]

tu l'as découvert comment sa chaine ?

[@ewout256]

First off, I'd like to congratulate you, this video is absolutely stunning in both an aducational and visual aspect. The way you introduce boundary conditions is really clear and makes me wonder if you'd consider doing a full video on them at a later point in time, perhaps at the end of this series to really bring PDE's into practice with some examples. Applying Dirichlet/Neumann/Robin/Mixed boundary conditions all to the same general heat equation (or any other PDE you might discuss in the future) would be great to show how widely useful it actually is. I'm sure your animations and simple ways of explaining would help a lot in understanding what the different boundary conditions actually represent. Love the content!

[@pedromayorga874]

Wow, those three dimensions curved surfaces representations are precious

[@joni5028]

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

[@creativeclub2023]

I was just thinking about 3B&B video. Thanks

[@declanmercer2587]

I was lectured on this stuff during a first year physics course on waves because my prof was really overqualified for his job. I ended up learning a lot of it but because i was rushed into the understanding i never ended up grasping the beauty of the boundary conditions. thank you so much for this video.

[@stefanocrema5146]

A mind-blowing clarity! I think it might be difficult to thank you enough for your contribution in spreading the love of (mathematical) beauty. I really agree beauty is the language of the universe.

[@olgierdgrzyb1142]

Thank you so much for these videos. You are not only making math understandable, you are making it accessible. I don't study math and probably would never entertain these subjects if I had to read papers using overintellected language and letters outside the unicode basic multilingual plane.

[@xiaoyuliu6642]

I've been waiting for this video for a month! Love 3Blue1Brown! tmr is my Physics exam

[@itareu95]

Good luck my dude

[@aloAlopi]

When I read Fourier analysis in engineering, I was hooked. Until then, I always wondered what beauty in math means. I think this opened that door to the beauty that holds! This was 15 years ago. And I still remember the thrill I got when I fell in love. I work now in computational biology, but this channel brought back that thrill!

[@manuelschonborn3269]

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 :)

[@hawaslianas]

You are soooooo underrated man! You should have billions of views

[@dontsmackdafish3771]

top 10 cliff hanger endings off all time

[@noonesperfect]

its never been so easy to understand , his smooth and to the point. Such a great work always. Thanks

[@GigaDavy91]

I love so much your videos, I study electrical engineering and I lack some of this explaination in some of my courses, and this elegant explaination really help. Thanks

[@Diaryofaninja]

GigaDavy91 why did you have to mention you’re an electrical engineer?

[@Diaryofaninja]

I’m r

[@rs-tarxvfz]

I am wondering how much of editing and creativity this requires to c=visualize this math stuff which are just plain equation. This is the power of animation and visualisation!

[@piman7319]

He has said that he estimates it takes him 40 hours of work to research, script, program, edit, and narrate 10 minutes worth of video. (I'll bet it's even more than that, though.)

[@georgeluckyanchikov353]

as a programmer myself i was thinking through the video about difficulty and beauty of this simulation truly amazing :)

[@strayedaway19]

truly said !!

[@piman7319]

Agreed. I think another aspect to the "secret sauce" of this channel is the particularly tight scripting. If you pay close attention, he makes about every word count. Every sentence earns its way in.

[@MaxxTosh]

I was checking KZbin every day for the next 3B1B video, I’m so glad it’s here!

[@taylorross5722]

My friends : what kind of series do you watch? Me : that's kind of complicated..

[@hamsterdam1942]

no it's not

[@ashtonsmith1730]

yeah, you have finite friends

[@mikefayc]

Man, I'm so happy I fount this channel. I'm really struggling with math currently, as my lectures are heald in a language I can only understand so much. But this channel really helps me put things into place! Thank you so much for your content

[@macronencer]

Another exquisite work of pedagogical art! I don't care how long we have to wait for these - it's always worth it. When I see a new video from you, I postpone viewing it until I'm about to go to bed, so that beautiful maths is the last thing I consume from the internet that day, as opposed to most of the other stuff, which doesn't help me sleep soundly AT ALL. Thank you for healing my troubled soul :-\

[@hydrokloricasid]

I cannot thank you enough ! The best explanation and visualization of differential equations and their solutions.. I have been doing a master after 12 years of my graduation and I couldn't remember the subject at all, I needed this.. Thank you. You are an amazing teacher, much much better than I have currently :))))

[@AlejandroBravo0]

If there were any kind of award to best educational content, this channel should absolutely win.

[@SrtaCifer2012]

My Computational Heat Transfer Class are way more understandable with your explanations. I'm falling in love all over again with the things that I'm learning. I cannot thank you enough for your work ♥

[@noahbertholon1076]

I'm french and in the fifth class (i think.. i'm 16 so do the maths) and i dont fully understand those video But they're so well animated and so interesting that i could watch your video all day long !

[@colt45caliber]

America needs to spend more time with educational content like this. That is the best antidote to dump.

[@vinayakvadlamani2261]

The timing of this is amazing, I am currently solving a 2-D heat PDE!

[@GarrettDore]

I took a whole class about applying the Fourier Transform and Fourier Series to heat and vibrations, and I only learned how to bash through it to get an answer. Only now, about two years later, because of your Fourier video series do I actually intuitively understand what it does and why.

[@miketian5348]

Yesterday, I was just watching Chapters 1&2 of this series, hoping the next one was going to happen soon. Thank You 🙏 Grant!

[@xavierlebeuf3061]

I thought I couldn't love math more than I already did.... You proved me wrong!

[@Sonickart]

you mean maths, all the cool people are saying it like that.

[@thexabi123]

Damn, just as I'm studying for my PDE exam, here comes another differential euqations video? I'll be glad to translate it into Polish as soon as I'm finished!

[@freddytonnesen4099]

I starter with the Finite Element Method course yesterday at my university and were chocked by the heat equations. Now after watching your series on differential equations, I feel better equipped for the course! Thank you!!

[@mohaamer6866]

for the first time in my life I feel like the little Topology I learned through Calc 2 is finally making sense and when you said I should be happy if I understood what was happening I had the biggest shit eating grin of my life

[@ArneHude]

Thanks for this great video @3Blue1Brown ! It also gives a huge benefit when plaing around with analog signals. What would be a nice to have for the next video were also descrete functions such as DFT for example.

[@jeffgalef121]

3:57 Can you explain why, after some tiny timestep, everything scales down by a factor?

[@3blue1brown]

Because the rate of change for each point is proportional to the size of that point. So for example, if the rate of change was -2 * the magnitude at each point, then after a time step of 0.01, each value T would shift to about - 2 * T * 0.01, so it would move to T - 2 * T * 0.01 which is (0.98) * T. Since this constant is the same at all points, the whole function T shift to 0.98 * T. That's if the time step was 0.01, but you can also think about what happens as this time step tends towards 0.

[@kumarshivang4431]

@3Blue1Brown but this is true only when second derivate with respect to time is zero, right. Are we assuming it to be zero, I am not sure here.

[@keeranparthipan2716]

@@kumarshivang4431 The rate of change of each point is proportional to the size of the point just before the current time step, not at t=0. So, the second derivative won't be 0 - if it was, then the graph at 5:50 would be a straight line and not curved.

[@kumarshivang4431]

@@keeranparthipan2716 exactly, then scaling bit doesn’t make sense, since by initial conditions we only know at t=0 time rate at any point to be zero it doesn’t rules out the possibility of second rate of time.. which means at any other instance there could be non zero time rate, then scaling deduction is wrong.

[@akshaychandandey3551]

The best privilege of my life has been to watch and enjoy the videos of this channel. That's too for free!

[@Josh-wb7ii]

Probably your most clinical and "lecture" like video yet, but understandably so given the nature of PDE's

[@johnchessant3012]

Wow, the explanations here were extremely clear. Thanks!

[@СтепанНестеров-р2р]

I've had my courses in Analysis and Differential Equations a couple years ago, but though I had already formally studied everything in the 'essence of' series, I'm always soooooo hyped for the next episode. Because I know that these topics can be so easily explained in a bad way, like in my classes, where you just do a lot of integration by parts to ensure something converges and you don't even know why you care about PDE's in the first place. Grant is just a thousand times better than your average university prof, I literally could not believe that math I thought was nerdy and hard is actually pretty accessible.

[@dogbiscuituk]

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.

[@drbitanmaity]

Sir, I genuinely thanks you from the bottom of my heart. I, despite being in medical profession, try to learn math for joy, in my leisure time. your videos help a lot.

[@TheToxiss]

Last week I had an exam with heat equation as one of the questions... great timing! But seriously, great video, keep it up.

[@Valthyrion]

You're amazing! The "camera" pan when showing the graphs seems like an awesome way to convey what's happening to people who have trouble visualizing it from a still. Keep up the awesome work!

[@halbeard2996]

Wish I had such clear explanations at university

[@kemsekov6331]

The next step is to watch this video again. Thanks for your hard work!

[@gsuberland]

I understand Fourier transforms quite well from the audio space, so it was nice to hit the point in this video where I suddenly understood why they're relevant here! Looking forward to the remaining video(s) in this series.

[@rrequelme7196]

From the bottom of an ecuadorian subscriber's heart, tranks for your effort :)

[@Mageling55]

We've started here with relatively simple boundary conditions. A particularly common in practice (albeit so common that we've pre-solved it for a range of variables and use simple approximations or lookup tables) is fluid flowing in an unisulated pipe. You have forced convection on the inside of a pipe: dT/dt|r=r_i = -kv_fluid (T(r_i) - T_fluid), free convection (ugly perturbation theory equation at r=r_o, and steady state if your lucky (dT/dt = 0). There are lots of interesting boundary conditions that radically change the solutions, so having a tool this flexible is insanely powerful

[@francesco5254]

Me: what a miserable day I had 3blue1brown: makes another video Me: this is one of the best days of my life ❤️

[@meganton9417]

The last animation of the set of cosines is definitely one of your most beautiful animations ever!

[@reckz420]

0:26 "where there's curvature in space, there's change in time" can you or someone please explain it a bit more

[@3blue1brown]

By all means, take a look at the previous video :)

[@MattyMatiss]

the animations here are so beautiful and make so much sense!

[@rafaeldias5059]

Just Learned how to solve this Stumm-Liouville tipe of equation on my post-graduation

[@vaibhavdev974]

Firstly I like the grant's etude tune, secondly, believe me you are an amazing and best teacher ever ever ever on youtube

[@xyzct]

Life is cruel. Where were these videos when I was studying physics at Berkeley in the early 80s?

[@gaboqv]

you were studying physics at berkeley, we can switch positions if you want

[@xyzct]

@@gaboqv, yeah, Berkeley was AWESOME ... but so many of the 3Blue1Brown visuals would have made the pressing workload just that much easier. That said, it is super fun to be able to enjoy all of his and so many others' amazing videos about mathematics and physics on KZbin.

[@ioannismavrogiannis151]

The cliffhanger in the end though!! Quite interesting take on the meaning of the heat equation, gives me a new perspective on solving the problem. Thanks a lot for your work

[@maitland1007]

These are sooo good! I just doubled my patreon amount.

[@kylebowles9820]

Visuals are getting hardcore! Nice work. I recently had to add up probability density functions like this to model a more complex distribution; fun stuff!

[@jayant9151]

Been waiting for this Sooothing Voice 💕

[@DirkdeZwijger]

extremely impressive video!! Being able to appreciate this makes math and physics feel so rewarding to study