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The heat equation, as an introductory PDE.

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Supported by viewers: 3b1b.co/de2thanks

Infinite powers, by Steven Strogatz:

amzn.to/3bcnyw0

Typo corrections:

- At 1:33, it should be “Black-Scholes”

- At 16:21 it should read "scratch an itch".

If anyone asks, I purposefully leave at least one typo in each video, like a Navajo rug with a deliberate imperfection as an artistic statement about the nature of life ;)

And to continue my unabashed Strogatz fanboyism, I should also mention that his textbook on nonlinear dynamics and chaos was also a meaningful motivator to do this series, as you'll hopefully see with the topics we build to.

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Animations made using manim, a scrappy open source python library. github.com/3b1b/manim

If you want to check it out, I feel compelled to warn you that it's not the most well-documented tool, and 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.bandcamp.com/album/the-music-of-3blue1brown

Stream the music on Spotify:

open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u

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 MRpost, if you want to stay posted on new videos, subscribe: 3b1b.co/subscribe

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Ayy, that's actually a 4D structure, cuz we are putting each infinitesimally small change in the curve shape one after the other along the time axis. Well in simpler words, you can say, that this is how a 2D curve look like changing over time, in 4D. The fourth axis is time axis.

Enjoying "Infinite Powers" on Audible. Thanks for the recommendation.

just enjoyed the graphics , great . thanks.

It seems like "Delta Delta T_2" would be more descriptive since we are discussing how the middle temperature (T_2) changes.

Those "Fourier" circles are amazingly mesmerizing! Kind regards

3B1B deserves a Nobel Prize for explaining math! Simply outstanding, yet again.

can't even find the words for how good you can make people understand such hard things ! as a student in theoretical mechanical engineering, I'm just amazed and so thankful that u gave some color and visuals to the letters and white boards of my teachers at university ! thanks ! aaaaaa LOOOOOOOTTTTT

Magnificent. Simply magnificent

What does the "alpha" stand for? I assume it's some constant?

do one for wave equation

Nice and easy explanation. Love it

Oh honey. You're brilliant, you know. Handsome voice. You MUST be handsome in person! Meteorology was my passion, but I majored in microbiology (a close second!) because I was afraid of graduate-level fluid dynamics and the very idea of thesis defense to this bashful gent was simply too much to bear. But I went to a state college famous for its engineering program and as such took the most rigorous math classes I could find. At a community college in Durham in 2011-2012, as luck would have it, I chanced upon two women who would change my life forever. One held a master's degree in math education from UNC, and the other held a PhD in mathematics from Brown. These ladies were diamonds in a thrift store, my friend. There were questions they simply wouldn't answer because they *wanted* us to wrestle with the concept. My brain ached and bled but in solving those widow-maker take-home quizzes, I made connections with the natural world I wouldn't have conceived in a thousand years. Me, low-born trailer trash, solving partial differential equations at a fucking community college in my mid-30s a mile away from a crime nucleus. Anyway, before I get too maudlin, I'd like to thank you for your work. It's beautiful stuff.

What software u use to create your animations

*TIL:* 2 very important terms 1:18 14:45 Laplacian Operator: ∇² (nabla squared) : it simply means sum of 2nd order PDs wrt all spatial dimensions - i.e. multivariable version of second derivative 10:54 ΔΔ or ∂² : as (diff b/w T and average of its neighbours)

15:14 ahhhh i see why ur videos were mentioned in khan academy. Your experience there must have been the foundation stone for this channel rightttt??

Another sketchy part at 11:36. I would not just skip over how your finite difference becomes a derivative, because they have different physical dimensionality. Such “intuitions” can confuse students a lot.

Mistake at 6:46. A partial derivative is not a ratio, unlike a full derivative.

Has "Infinite Powers" been released as an audio book? I'd love to listen while I'm commuting.

teach me senapi.

Showing the graph with both time and distance was a great decision

Usually maths gives me The Fear but I feel so tranquil when I watch 3b1b

What the function?!

17:38 Well I did not love math until I saw your channel!

If there was a Patreon, I would love to help in making such awesome videos!

And, I just actually think that this is the best channel to gain rock solid intuition in math. Thank you ☺️

What I don't understand is... on the "difference of the difference" part you've explained (10:43), you showed that the middle point will tend to move towards the "mean value" of the neighbouring points. What if all of the points create a perfect 45 degree line for example? They wont move since the points are pefrectly "in the middle" of their neighbouring points, but that shouldnt happen if a rod has evenly increasing temperature (from 0-90c for example, distributed exactly just like the points)

And there are Partial Integrals.

Man this is just so fucking cool. Thank you very much for putting out for free. I love you.

12:25 first time that my mind is slowly starting to grasp that the rate of change of temperature at a middle point can be described the same way as acceleration of a particle. Need to think about it a little more still. Each word in this video contains a lot of thought.

This is how science and engineering needs to be taught in college. Not back to the students copying notes onto the chalkboard as quickly as possible.

salute to you sir for making such a difficult topic interesting by your explanation

very juicy video. My biggest takeaway is on the interpretation of the second partial derivatives: it measures how the value compares with the value of its neighbors

wonderful!, as always!!

whosoever you are....you are a genius....its been 5 years since i have studied the PDEs for the first time and today i have understood them. The fact that no teacher could ever tell me the statement 'rate of change of rate of change' tells us how messed up our education system is....this channel should be made compulsory for first year engineering undergrads

Baby-faced Fourier, poodle-haired Fourier. Okay!

Super visualization! Appreciate the work done by the 3b1b

You are helping me to feed my love for physics and maths and you really dont know how much this helps me understand this beautiful subjects better. Thank You Sir

this channel is the reason why I am still alive

omg, i seem to have learnt something.

Old Babyface Fourier, the Math Gangster, hey?

Wonderful explanation!

Only if they could teach maths like this at school, everyone would be digging it!

i am no wiser about what a PDE is. Can someone help me?

I'm going to try to answer both of your comments here: 1. What is a PDE? Do you understand what what a differential equation is in general? A *partial* differential equation then is just a DE that includes partial derivatives. Meaning that it relates changes in one thing (e.g. time) with changes in another thing (e.g. space). 2. How do we come up with DEs? That's a good question. It's essentially asking "how do we come up with scientific models?" DEs are mathematical formulations of the rules that govern a system's behavior. For example we know that heat spreads out over time and we know how it does so from experimental observations. The heat equation encapsulates that in mathematical form. Another example: When we have a population of some organisms we know that the more there are, the faster they can reproduce until they hit the limit of what their enviroment can support. That can then be expressed in a DE which has the logistic growth curve as its solution. Often these rules come from more fundamental models such as with the DE of the pendulum he talked about in the first episode.

i dont understand. I am looking for the steps to create the PDE in the first place. what is the first step? What are the next steps frm there, so I understand please? From what I gather, is the first step (s1) that he observes the function (curvy line in the x, T plain) in real life? Does he then manually observe it changing over time in the T-t plain? Is he then able to calculate dT/dx and dT/dt by observation? Is that how he is able to come up with the PDE? can someone please confirm or correct?

thank you!

what a great explanation

You made us hit peak of imagination

In My Mind I Always Feel Like Maybe 3blue1brown Would Be Famous..... He Should Be On Jeopardy

love you

Amazing work....the plots that follow the explanation makes everything much more easier...keep educating us!

The presentation, and the concepts it's illustrating, is beautiful.

Never begin a sentence with "but"

How about simultaneous non liner partial differential equations?.

9:20 Just came to point out that if we had a universe consisting of nothing but an infinitely long rod with a uniformly ascending temperature value across its length, it would be in equilibrium even though one end would have an infinitely greater temperature than the other. You're welcome.

8:55 " the bigger the difference the faster T2 heats up". But doesn't the two neighbour temperatures for being near a relatively low temperature loses heat and settles at some point? If it does, doesn't the nearby neighbor temperature is also affected and so on so forth?

Omg i hope this is exactly what i think it is Thank you

Huh, I could have sworn I studied that the rate of flow of heat was proportional to dT/dX, wonder why its a function of d2T/dX2 here I must be missing something, but its been bugging me for a while now, any help?

After watching this video, I realized that it gave me another way of thinking style which changed my style of looking to things and thanks to you Grant, I am thinking of the world much more versatile right now. I am putting a mark in history here, claiming that this video had increased my attentions to calculus so much that I feel like I owe you... So, Thank you.

Please give this man a Nobel Prize. This was absolutely good.

8:03 holy hell dude I wish you were my thermodynamics professor back in undergrad days. This is such a beautiful way to teach. The animations are brilliant too. Kudos to you!

Wow.

baby face fourier lmfao

wow, that is the best description of partial derivatives I have ever seen!!

De2 is a brutal nu metal band from Colombia...

In the example of the flat bar, would the curve representing temperature temporarily invert as it approaches a flat line? Could a particular point switch signs for it's relative temperature in regards to its neighbors? Also great video, thanks for giving me an opportunity to finally try to understand Diff-E-Q.

so you're saying that the partial derivative means that you are relating the change in function in one dimension to the change in second one (dimension) provided that function depends upon two variables ???

Learning every day.Thank's!

very tough simply wet over nmy head

Oh no. I was recommended this one. Guess I'll have to pick up the lesson as I go, just like real school

Steven is wrong...not star ... a GOD is born...AMAZING!!!

Amazing explanation and video quality!

I wish my math class is like this.

Your videos are like arts. Thanks you so much.

@7:43 now we know where all the blender deleted cubes went !

I'd heard that heat equation tells that we can't move backward in time. If that's true, please make a video on that.

Baby face poodle head fourier. Haha.

I have already ordered the book. This video made me fall in love with mathematics again.

this is due to pré hilbertian space, wich Fourrier transformation is jus one approximation of anyfunction, others exist also

Make video on wavelet transformation

Always he has such a beautiful and intuitive perspective that reduces the complexity of intimidating mathematical objects to something simple and approachable. This is probably how Fourier himself built it up.

what is your math software tool you use?

This video makes me want to binge watch all of your other videos and ignore my phone all day!!!!

oh my god ..... this is what i am trying to understand for a very long time... thank you so much

This, and the previous video, cracked open so many relationships between how ancients saw planets - "wanderers" - to how Illustrator's path handles pull curves along, to Fourier transforms, that I'm tempted to find the second derivative to how quickly my mind has been blown tonight.

Amazing video

"first, let's be clear about WHAT THE FU-nction we are analyzing is" lol

I recommend that anyone watching this video goes away and solves the 1D case in Excel.

PDEs are interesting because of how they describe the world so simply. Everything is only affected by its immediate neighbours. This video is really amazingly eye opening.

wow

Amount of thought put in putting the things in best way in this video is amazing and it has truly made things intuitive. This is the place where I learned how to read PDE.

Never before have I been as hooked by any video.

What an amazing video! Thanks a lot! You're doing a great job.

I didn’t realize how helpful this would be until my engineering calculus class where we started 3 dimension calculus. This helps with just understanding it so much

I was looking for smth like this for months. Thank you!

I hope this video can be dubbing with Indonesian language, and many people like me can more understand. Thank you very much

I came to understand math , now I understand the meaning of life

I seriously have so many other things to be doing, I'm a blood neuroscience major, but GOD IM DRAWN IN - it's sooooooooo well done; it makes me smile just watching it.

Can you imagine if there were such a thing as "blood neuroscience", how metal that would be?

@Etienne Sellar Oh cool 😅 That makes sense

@Izzy kinda like a blood mage. I meant bloody; as in "I have no bloody business watching a partial differential video"

What's "blood neuroscience"?

How to solve an equation uxx+(x^2)uyy=(1/x)ux

These videos are so good, I'm getting emotional watching them and idk why

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