# But what is a partial differential equation? | DE2

After seeing how we think about ordinary differential

equations in chapter 1, we turn now to an example of a partial differential equation,

the heat equation. To set things up, imagine you have some object

like a piece of metal, and you know how the heat is distributed across it at one moment;

what the temperature of every individual point is. You might think of that temperature here as

being graphed over the body. The question is, how will that distribution

change over time, as heat flows from the warmer spots to the cooler ones. The image on the left shows the temperature

of an example plate with color, with the graph of that temperature being shown on the right,

both changing with time. To take a concrete 1d example, say you have

two rods at different temperatures, where that temperature is uniform on each one. You know that when you bring them into contact,

the temperature will tend towards being equal throughout the rod, but how exactly? What will the temperature distribution be

at each point in time? As is typical with differential equations,

the idea is that it’s easier to describe how this setup changes from moment to moment

than it is to jump to a description of the full evolution. We write this rule of change in the language

of derivatives, though as you’ll see we’ll need to expand our vocabulary a bit beyond

ordinary derivatives. Don’t worry, we’ll learn how to read these

equations in a minute. Variations of the heat equation show up in

many other parts of math and physics, like Brownian motion, the Black-Scholes equations

from finance, and all sorts of diffusion, so there are many dividends to be had from

a deep understanding of this one setup. In the last video, we looked at ways of building

understanding while acknowledging the truth that most differential equations to difficult

to actually solve. And indeed, PDEs tend to be even harder than

ODEs, largely because they involve modeling infinitely many values changing in concert. But our main character now is an equation

we actually can solve. In fact, if you’ve ever heard of Fourier

series, you may be interested to know that this is the physical problem which baby face

Fourier over here was solving when he stumbled across the corner of math now so replete with

his name. We’ll dig into much more deeply into Fourier

series in the next chapter, but I would like to give at least a little hint of the beautiful

connection which is to come. This animation is showing how lots of little

rotating vectors, each rotating at some constant integer frequency, can trace out an arbitrary

shape. To be clear, what’s happening is that these

vectors are being added together, tip to tail, and you might imagine the last one as having

a pencil at its tip, tracing some path as it goes. This tracing usually won’t be a perfect

replica of the target shape, in this animation a lower case letter f, but the more circles

you include, the closer it gets. This animation uses only 100 circles, and

I think you’d agree the deviations from the real path are negligible. Tweaking the initial size and angle of each

vector gives enough control to approximate any curve you want. At first, this might just seem like an idle

curiosity; a neat art project but little more. In fact, the math underlying this is the same

as the math describing the physics of heat flow, as you’ll see in due time. But we’re getting ahead of ourselves. Step one is to build up to the heat equation,

and for that let’s be clear on what the function we’re analyzing is, exactly. The heat equation To be clear about what this graph represents,

we have a rod in one-dimension, and we’re thinking of it as sitting on an x-axis, so

each point of the rod is labeled with a unique number, x. The temperature is some function of that position

number, T(x), shown here as a graph above it. But really, since this value changes over

time, we should think of it this a function as having one more input, t for time. You could, if you wanted, think of the input

space as a two-dimensional plane, representing space and time, with the temperature being

graphed as a surface above it, each slice across time showing you what the distribution

looks like at a given moment. Or you could simply think of the graph of

the temperature changing over time. Both are equivalent. This surface is not to be confused with what

I was showing earlier, the temperature graph of a two-dimensional body. Be mindful of whether time is being represented

with its own axis, or if it’s being represented with an animation showing literal changes

over time. Last chapter, we looked at some systems where

just a handful of numbers changed over time, like the angle and angular velocity of a pendulum,

describing that change in the language of derivatives. But when we have an entire function changing

with time, the mathematical tools become slightly more intricate. Because we’re thinking of this temperature

as a function with multiple dimensions to its input space, in this case, one for space

and one for time, there are multiple different rates of change at play. There’s the derivative with respect to x;

how rapidly the temperature changes as you move along the rod. You might think of this as the slope of our

surface when you slice it parallel to the x-axis; given a tiny step in the x-direction,

and the tiny change to temperature caused by it, what’s the ratio. Then there’s the rate of change with time,

which you might think of as the slope of this surface when we slice it in a direction parallel

to the time axis. Each one of these derivatives only tells part

of the story for how the temperature function changes, so we call them “partial derivatives”. To emphasize this point, the notation changes

a little, replacing the letter d with this special curly d, sometimes called “del”. Personally, I think it’s a little silly

to change the notation for this since it’s essentially the same operation. I’d rather see notation which emphasizes

the del T terms in these numerators refer to different changes. One refers to a small change to temperature

after a small change in time, the other refers to the change in temperature after a small

step in space. To reiterate a point I made in the calculus

series, I do think it’s healthy to initially read derivatives like this as a literal ratio

between a small change to a function’s output, and the small change to the input that caused

it. Just keep in mind that what this notation

is meant to convey is the limit of that ratio for smaller and smaller nudges to the input,

rather than for some specific finitely small nudge. This goes for partial derivatives just as

it does for ordinary derivatives, and I believe can make partial derivatives easier to reason

about. What you and I will build to is that the way

this temperature change with respect to time depends on the second derivative with respect

to space. At a high level, the intuition is that at

points where the temperature distribution curves, it tends to change in the direction

of that curvature. Since a rule like this is written with partial

derivatives, we call it a partial differential equation. This has the funny result that to an outsider,

the name sounds like a tamer version of ordinary differential equations when to the contrary

partial differential equations tend to tell a much richer story than ODEs. The general heat equation applies to bodies

in any number of dimensions, which would mean more inputs to our temperature function, but

it’ll be easiest for us to stay focused on the one-dimensional case of a rod. As it is, graphing this in a way which gives

time its own axis already pushes the visuals into three-dimensions. But where does an equation like this come

from? How could you have thought this up yourself? Well, for that, let’s simplify things by

describing a discrete version of this setup, where you have only finitely many points x

in a row. This is sort of like working in a pixelated

universe, where instead of having a continuum of temperatures, we have a finite set of separate

values. The intuition here is simple: For a particular

point, if its two neighbors on either side are, on average, hotter than it is, it will

heat up. If they are cooler on average, it will cool

down. Focus on three neighboring points, x1, x2,

and x3, with corresponding temperatures T1, T2, and T3. What we want to compare is the average of

T1 and T3 with the value of T2. When this difference is greater than 0, T2

will tend to heat up. And the bigger the difference, the faster

it heats up. Likewise, if it’s negative, T2 will cool

down, at a rate proportional to the difference. More formally, the derivative of T2, with

respect to time, is proportional to this difference between the average value of its neighbors

and its own value. Alpha, here, is simply a proportionality constant. To write this in a way that will ultimately

explain the second derivative in the heat equation, let me rearrange this right-hand

side in terms of the difference between T3 and T2 and the difference between T2 and T1. You can quickly check that these two are the

same. The top has half of T1, and in the bottom,

there are two minuses in front of the T1, so it’s positive, and that half has been

factored out. Likewise, both have half of T3. Then on the bottom, we have a negative T2

effectively written twice, so when you take half, it’s the same as the single -T2 up

top. As I said, the reason to rewrite it is that

it takes a step closer to the language of derivatives. Let’s write these as delta-T1 and delta-T2. It’s the same number, but we’re adding

a new perspective. Instead of comparing the average of the neighbors

to T2, we’re thinking of the difference of the differences. Here, take a moment to gut-check that this

makes sense. If those two differences are the same, then

the average of T1 and T3 is the same as T2, so T2 will not tend to change. If delta-T2 is bigger than delta-T1, meaning

the difference of the differences is positive, notice how the average of T1 and T3 is bigger

than T2, so T2 tends to increase. Likewise, if the difference of the differences

is negative, meaning delta-T2 is smaller than delta-T1, it corresponds to the average of

these neighbors being less than T2. This is known in the lingo as a “second

difference”. If it feels a little weird to think about,

keep in mind that it’s essentially a compact way of writing this idea of how much T2 differs

from the average of its neighbors, just with an extra factor of 1/2 is all. That factor doesn’t really matter, because

either way we’re writing our equation in terms of some proportionality constant. The upshot is that the rate of change for

the temperature of a point is proportional to the second difference around it. As we go from this finite context to the infinite

continuous case, the analog of a second difference is the second derivative. Instead of looking at the difference between

temperature values at points some fixed distance apart, you consider what happens as you shrink

this size of that step towards 0. And in calculus, instead of asking about absolute

differences, which would approach 0, you think in terms of the rate of change, in this case,

what’s the rate of change in temperature per unit distance. Remember, there are two separate rates of

change at play: How does the temperature as time progresses, and how does the temperature

change as you move along the rod. The core intuition remains the same as what

we just looked at for the discrete case: To know how a point differs from its neighbors,

look not just at how the function changes from one point to the next, but at how that

rate of change changes. This is written as del^2 T / del-x^2, the

second partial derivative of our function with respect to x. Notice how this slope increases at points

where the graph curves upwards, meaning the rate of change of the rate of change is positive. Similarly, that slope decreases at points

where the graph curves downward, where the rate of change of the rate of change is negative. Tuck that away as a meaningful intuition for

problems well beyond the heat equation: Second derivatives give a measure of how a value

compares to the average of its neighbors. Hopefully, that gives some satisfying added

color to this equation. It’s pretty intuitive when reading it as

saying curved points tend to flatten out, but I think there’s something even more

satisfying seeing a partial differential equation arise, almost mechanistically, from thinking

of each point as tending towards the average of its neighbors. Take a moment to compare what this feels like

to the case of ordinary differential equations. For example, if we have multiple bodies in

space, tugging on each other with gravity, we have a handful of changing numbers: The

coordinates for the position and velocity of each body. The rate of change for any one of these values

depends on the values of the other numbers, which we write down as a system of equations. On the left, we have the derivatives of these

values with respect to time, and the right is some combination of all these values. In our partial differential equation, we have

infinitely many values from a continuum, all changing. And again, the way any one of these values

changes depends on the other values. But helpfully, each one only depends on its

immediate neighbors, in some limiting sense of the word neighbor. So here, the relation on the right-hand side

is not some sum or product of the other numbers, it’s also a kind of derivative, just a derivative

with respect to space instead of time. In a sense, this one partial differential

equation is like a system of infinitely many equations, one for each point on the rod. When your object is spread out in more than

one dimension, the equation looks quite similar, but you include the second derivative with

respect to the other spatial directions as well. Adding all the second spatial second derivatives

like this is a common enough operation that it has its own special name, the “Laplacian”,

often written as an upside triangle squared. It’s essentially a multivariable version

of the second derivative, and the intuition for this equation is no different from the

1d case: This Laplacian still can be thought of as measuring how different a point is from

the average of its neighbors, but now these neighbors aren’t just to the left and right,

they’re all around. I did a couple of simple videos during my

time at Khan Academy on this operator, if you want to check them out. For our purposes, let’s stay focused on

one dimension. If you feel like you understand all this,

pat yourself on the back. Being able to read a PDE is no joke, and it’s

a powerful addition to your vocabulary for describing the world around you. But after all this time spent interpreting

the equations, I say it’s high time we start solving them, don’t you? And trust me, there are few pieces of math

quite as satisfying as what poodle-haired Fourier over here developed to solve this

problem. All this and more in the next chapter. I was originally inspired to cover this particular

topic when I got an early view of Steve Strogatz’s new book “Infinite Powers”. This isn’t a sponsored message or anything

like that, but all cards on the table, I do have two selfish ulterior motives for mentioning

it. The first is that Steve has been a really

strong, perhaps even pivotal, advocate for the channel since its beginnings, and I’ve

had the itch to repay the kindness for quite a while. The second is to make more people love math. That might not sound selfish, but think about

it: When more people love math, the potential audience base for these videos gets bigger. And frankly, there are few better ways to

get people loving the subject than to expose them to Strogatz’s writing. If you have friends who you know would enjoy

the ideas of calculus, but maybe have been intimidated by math in the past, this book

really does an outstanding job communicating the heart of the subject both substantively

and accessibly. Its core theme is the idea of constructing

solutions to complex real-world problems from simple idealized building blocks, which as

you’ll see is exactly what Fourier did here. And for those who already know and love the

subject, you will still find no shortage of fresh insights and enlightening stories. Again, I know that sounds like an ad, but

it’s not. I actually think you’ll enjoy the book.

I guess it will Hard for anyone else To Make Better then these videos 🙂 Thanks Mate! (y)

Is the white line at @12:40, the first, or second derivative?

發出讚嘆的聲音Wow .. you are truly gifted in utilizing modern visualization for complex topics!!

Love this! Such a wonderful presentation! The graphics is awesome and I understood something I had never been able to grasp. Thank you!

I am reading Strogatz's "The Joy of x". Honestly, a lot of things go over my head, but it's a wonderful journey on math. I'll be looking up the book you mentioned here. Thank you again!

I hardly ever comment but this.is.A.W.E.S.O.MAHH

Really nice video, thanks

baby face Fourier 😂😂😂

Brilliant

sir your lecture is so much helpful and concept cearing but you speak very fast (speed) so please next video upload with slow speed -thank you

3:37 what the fu….nction 😂

Great animations! Thank you very much for putting the effort to make all these videos.

WOW!!! I'm an electrical engineer I had no idea about heat equations, I was the best way to teach a heat problem. Thank you so much!

kinda neat

This stuff is blowing my mind, but it feels like things that would be common sense to a rocket engineer.

Your comment about thinking of derivatives as actual ratios. Read this, which demonstrates that you can do this if you use corrected notation: https://arxiv.org/pdf/1811.03459.pdf

Omg, babyface Fourier!!!

This guy is terrific! I'm subscribing to his channel!

I like how the pi creature always captures our emotions so perfectly like at 1:21 haha

Wow, can tell ur videos r getting even better!

i think the notation with 'del' and 'd' are really usefull and important since they signal the difference between a partial differential and a total differential. Specially with things like 'del'f/'del't' != df/dt since x could depend on t! so they not just differ in notation

Portuguese pleease

You are a valuable human being.

youre awesome and your channel.

dude i love you

I am curious. Do any college professors use videos like these for students to appreciate the beauty of the subject?

Can any one suggest a book to learn PDE

Oh god my look on heat transition will not same any more.

11:25 so why we do not have a function T'_ x = const T' '_ tt when we use the same explanation?

I watched your intro to calculus video before I went into calc 1 and now I'm watching this as I'm going through calc 3, I must say your videos do indeed make math fun. thanks for the great videos!

Man, do you offer any degree or certification? I'd like to get an approvement latter from you rather than from my college.

BTW, just curious about the time cost that you to film a video like this?

Why can’t my classes be this fun🙃

no words can describe how much thankful I am

thank you sir thank you a lot

Why didn't you derive the equation on the basis of heat flux gradients ?!

This is so freakin beautiful

Amazing graphics, perfectly narrated – as many others have commented, this is what the teaching and learning of mathematics should be like. This is a brilliant example of how a powerful visualisation (this video) can make understanding so much greater for what would otherwise be a difficult concept for many students.

Thanks a lot for your videos. Is it possible for you to explain hyperbolic PDE, characteristics etc.

I Love how this is taught, I wish my classes were like this.

I am high school student. Am I eligible to watch this?

3:37 "WTF we're analysing is" 🙂

quan o，but u hard

z ？ a n

2+e d m

你的車是4s，r/w

Apart from the deep understanding for math, your ability to use python for generating such these graphical tutorials is incredible!

I love you.

I can feel my brain pulsating

dude, i love you

Watching this type of videos makes me proud studying math and being able to understand them lol

Incredible work! Thanks for this great explanation! Greetings from Mexico. 🙂

NICE FUCKING ANIMATION 4:13

Some people will say i'm doing black magic 😂

"for finitley many vectors this representation won't be usually a perfect representation, but in this case it is a lower f" hahaha lshidmtamafo

Do a video on how you make the graphics in the video!!!

My brain is a a differencial equasion constantly solving your every question. What would you do?

Time without observers would be based on neighbors closer=faster

Gravity attracts itself.

from active projects.one.part1.shared constructs import * ModuleNotFoundError: No module named 'active_projects'

Math has never ever been as beautiful until 3Blue1Brown. Like Mozart he makes you feel and see it. As opposed to compute and confuse which happens all the time

Finally. I've been wondering about this stuff.

I just found this channel and I am amazed at the graphics, the explanations, everything. All I can say is, thank you.

Sir , please similary explain about wave equation ,,,we want to see in another video

4:05

haha 3 years of math degree and I always wondered how to think about the second order derivative and after enough of scratching and itching this video gave me something to think about. And those graphics on point

From baby faced Fourier to poodle haired fourier xD

what a legend

What tools do you use to animate?

Lmao I'm a highschool student watching this and just thinking how I have to do this in 3 years😂

Thanks for such communicative videos, they're really helping opening my mind to the beauty of math.

It's really being difficult to understand double and triple integration in graphical view

How can i view it n solve ??

By the end of this video I FINALLY UNDERSTOOD intuitively how the Laplacian is the divergence of the gradient! 🙂 So for example, regarding heat, if a point is surrounded by points hotter than it, then there is a flux of heat gradient flowing into that point from a small neighbouring circle around it and that heats it up.

Lol this is why professor Strogatz made us watch this video

man how you gunna go and do fourier like that?

The animation and explanation is cutting edge technique of teaching. It should be implemented in all educational institute all across the world.

Can you please explain the kinematics of fluid mechanics.

At 3:35 I have a feeling, that the script goes: "…what the f. is going on in here" and you somehow saved the situation at the very right moment

This is a dumb question, but I really wonder what your IQ is? I mean I don't fundamentally believe that IQ is a proper way to measure one's abilities, but is still the closest we've got into that area.

Life is funny. Where are we off to next? I seriously want to be a truck driver. That is not a pity party. I am looking forward to taking naps in them when they become self driving.

Twisted visions of one of my friends in trouble when it wasn't them is denying flares not me denying a close friend ranked is in trouble or not.

As a CS major when I heard the sentence at 13:57, "The way that any one of these values changes depends on the other values, but quite helpfully, each one only depends on its immediate neighbors (…)" my brain started screaming dynamic programming real hard. That was freaking awesome.

I have studied mechanical engineering for almost 5 years now, and this is the best explanation of PDE´s and fourie series I have ever heard. Thank you very much for fantastic illustrations and intuitive explanations.

I love you

Public

Displays of

Effection

I came to learn how to spell the new character but stayed for the whole video ,:)

Incredible

Your animation skills and ability to explain complex ideas is stellar.

3:18 🎼- nice touch

Hey! Loved this video! Absolute stunning how beautifully you explained these concepts!

But I have a note/question… In 13:39, when you show the 2-body problem equations: Why are the masses in the denominator? Shouldn't the mass be in the numerator (together with G)? So that way the units be correct?

Thank you!

Ur volume is too low

I have being studying numerical methods to solve PDEs for the last 4 years at university. I was feeling pretty confident on the topic. Now you come around and turn my world upside down – each one of your videos gives me a new, powerful way of looking at the same old problems! Truly eye-opening.

Case and point: your explanation of the second derivative as moving towards the average of its neighbors is nothing other than a central difference scheme. I've been using it for years and you just gave me a totally new insight into how it works.

Just… Wow.

PS: your visualisations are freaking amazing as well. I wish professors used such good material as well. If I'm ever teaching any of these topics, I'll be sure to make watching your videos compulsory 😉

So impressed by 3Blue1Brown! Thanks for these sweet bits of insight!

Your videos are very neat. but why complicate simple matters ?

Grant Sanderson is from Area 51

Sir, how to explain why planet moves in an elliptical path around another planet(heavier). While they must have revolved around the axis passing through their(system's) center of mass in circular path? 13:32

Love it

The One and Only

For those who are venturing in to world of solving PDEs..

as Grant nicely puts it from 3:04 to 3:11..

this approximation is the heart of solving for pdes…

impressive explanation, thank you

Leave Fourier alone :<

Seriously… 3B1B's animations always BLOWS MY FRICKING MIIIND HOOOLY SHIIIIT

Damn it! This video is fucking beautiful!

Great material!

You are the best math teacher that I have had.

thank you.