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- [Voiceover] Hello everyone.
So here, I'd like to talk about harmonic functions.
Now, harmonic functions
are a very special kind
of multivariable function,
and they're defined in terms of the Laplacian,
which I've been talking about
in the last few videos.
So the Laplacian, which we denote
with this upper right-side-up triangle,
is an operator that you might take
on a multivariable function.
So it might have two inputs,
it could have, you know, a hundred inputs,
just some kind of multivariable function
with a scalar output.
And I talked about it in the last few videos,
but as a reminder,
it's defined as the divergence
of the gradient of F,
and it's kind of like the second derivative.
It's sort of the way to extend the idea
of the second derivative
into multiple dimensions.
Now, what a harmonic function is,
is one where the Laplacian is equal to zero.
And it's equal to zero
at every possible input point.
And sometimes the way that people
write this to distinguish it,
they'll make a kind of triple equals sign,
maybe saying like, "Equivalent to zero."
And this is really just a way of emphasizing
that it's equal to zero
at all possible input points.
It's, you know, not an equation
that you're solving for the specific X and Y,
where it equals zero.
It's a statement about the function.
And to get our head around this,
because it's kind of a,
you know, as you're just starting
to learn about the Laplacian,
it's hard to just immediately
see the intuition for what this means,
let's think about what it means
for a single variable function.
If you just have some single variable function of X,
and you're looking at its second derivative,
which is kind of the analog of the Laplacian,
what does it mean if that's equal to zero?
Well, we can integrate it,
we can take the antiderivative,
and say, that means that the single derivative of F,
well let's see, what functions
have a derivative that's zero?
The only functions are the constant ones,
so C is just gonna mean some constant here.
And if you integrate that again,
say what function has,
as its derivative, a constant,
well, it's gonna be that constant times X,
plus some other constant,
some other constant K.
So basically, linear functions.
So if you're thinking of a graph,
it's just something that's got a line,
line passing through it like that.
And this should kind of make sense
if you think of the geometric interpretation
for the second derivative.
Because if you're just looking at a random,
you know, arbitrary function
that's kind of curving as it does,
the second derivative is negative
when this concavity is down.
So this right here would be a point
where the second derivative,
it's not zero, it's negative.
And over here, when the concavity is up,
and it's got a sort of bowl shape,
that's where the second derivative is positive.
So if we're saying that the second derivative
has to always be zero,
then it can't curve down and it can't curve up,
and it can't do that anywhere,
so basically there's no curving allowed,
so whatever direction it starts at,
it's not allowed to curve,
so it just sticks straight like that.
But once we extend this to the idea
of a multivariable function,
things can get a lot more interesting
than just a straight line.
So an example,
I've got the graph here
of a multivariable function
that happens to be harmonic.
So the graph that you're looking at,
this is of a two variable function,
and the function specifically is
F of XY,
XY,
is equal to E to the X
multiplied by
sine of Y.
And as we're looking at the graph here,
hopefully it makes a little bit of sense
why this is sort of an E to the X
sine of Y pattern.
'Cause as we're moving in the positive X direction,
this here is the positive X direction,
you have this exponential shape,
and this corresponds with the fact that over here
we've got an E to the X,
so as you move X,
it kinda looks like E to the X,
and it's being multiplied by something
that is a function of Y.
So if you're holding Y constant,
this just looks like a constant.
But notice, if that was a,
if that was a negative constant,
if sine of Y at some point
happens to be negative,
then your whole exponential function
actually kind of goes down.
It's sort of like a negative E to the X look.
But if you imagine moving in the Y direction,
so instead of the pure X direction like that,
if we imagine ourselves moving
with the input going along,
let's see what it would be,
it would be this way,
positive Y direction,
you have this sort of sinusoidal shape,
and that should make sense
because you've got this sine of Y.
And depending on what E to the X is,
the amplitude of that sine wave
is gonna get, you know,
really high at some points here.
It's going way up and way down.
But if E to the X was really small, you know,
it hardly, hardly even looks
like it's wiggling over here.
It pretty much looks flat.
So that's the graph that we're looking at.
And I'm telling you right now,
I claim that this is harmonic.
This is a function whose Laplacian
is equal to zero.
And what that would mean is that
as we go over here and we say,
we evaluate the Laplacian of F,
which, just to remind you,
there's a different formula
rather than thinking divergence of gradient,
that turns out to be completely the same
as saying, you take the second derivative
of that function with respect to X,
that's its first input,
and you add that,
let's see, second derivative with respect to X,
you add that to the second derivative
of your function
with respect to the next variable.
And you keep doing this
for all of the different variables that there are,
but this is just a two variable function,
so you do this twice.
The claim is that this
is always equal to zero.
So I might say, kind of equivalent,
at every possible input,
it's equal to zero.
And I think I'll leave that
as something for you to compute.
It might be kind of good practice
to kind of get a feel for
computing the Laplacian.
But what I wanna do is interpret
what does this actually mean, right?
'Cause you can plug it through,
and you can see, ah yes,
at all possible inputs, it will be zero.
But what does that mean?
Because in the single variable context,
once we started thinking about
the geometric interpretation
of a second derivative as this concavity,
it sort of made sense
that forcing it to be zero
will give us a straight line.
But clearly, that's not the case.
This is much more complicated
than a straight line.
And for that,
I want to give a kind of a different
way that you can think about
the single variable second derivative.
On the one hand, you can think of,
let's say, negative second derivative,
as being this concavity
where it's kind of frowning down.
But another way you can maybe think about this
is saying that all of the neighbors of your point,
if you go a little bit to the left, right,
you've got an input point here,
and if you go a little bit to the left,
the neighbor is less than it,
and if you go a little bit to the right,
that other neighbor is also less than it.
So it's kind of a way of saying,
hey, if you look at the neighbors of your input,
so if you, if you happen to be making the claim
that F double prime, at some particular input,
like X sub O,
is less than zero,
it's saying that all of the neighbors
of X sub O,
all of the neighbors of that point,
are less than it.
And if you do a similar thing
over at a positive concavity point
where it's kind of smiling up,
you say, well its neighbor to the right
has a greater value,
and its neighbor to the left has a greater value.
So at some point, where the second derivative,
instead of being less than zero,
happens to be,
happens to be greater than zero,
that means that the neighbors
tend to be greater than the point itself.
And even if you're looking at a circumstance
that isn't this idealized, you know,
it happens to be a local minimum,
but let's say you're looking at a graph.
Let's say you're looking at a function
at a point where it's concave up, right?
It's concave up, but it's not this idealized,
local minimum kind of circumstance.
So instead, you might be looking
at a point like this,
and if you look at, you know,
its neighbor to the left,
that'll have some value that's actually less
than your original guy,
so the neighbor looks like
it's less than it on the left,
but if you move that same distance to the right,
its neighbor is greater.
But you would say, on average,
if you took the average value of the neighbors,
the neighbor on the right kind of outbalances
the neighbor on the left,
and you would say, on average,
its neighbors are greater than the point itself.
So let's say that input point there
was like X sub O,
that would mean that the second derivative
of your function at that point,
you know, is greater than zero.
So this positive concavity,
you can also think of it
as a measure of, on average,
are the neighbors greater than your original point
or less than it?
And the reason I'm saying this
is because this idea,
of kind of comparing your neighbors
to the original point
is a much better way to contemplate
the Laplacian in the multivariable world.
So if we look at a function like this,
and let's say we're looking at it
kind of from a bird's eye view.
So we've got our XY plane, right?
This over here is the X-axis,
and this up here is the Y-axis.
And let's say that we're looking
at some specific input point.
With the Laplacian,
you wanna start thinking about
a circle of points around it,
all of its neighbors,
and in fact, think of a perfect circle,
so all of the points
that are a specified distance away.
The question the Laplacian is asking is,
"Hey, are those neighbor points, on average,
"greater than or less than your original point?"
And this is actually,
this is how I introduced the Laplacian
in the original video where I was giving
kind of the intuition for the Laplacian,
you're asking, "Do the points around a given input
"happen to be greater than it or less than it?"
And if you're looking at a point
where the Laplacian of your function
happens to be greater than zero at some point,
that would mean all of the neighbors
tend to be, on average, greater than your point.
Whereas if you're looking at a point
where the Laplacian of your function
is less than zero,
then all of those neighbors, on average,
would be less than your point.
So in particular, you know,
if the Laplacian was less than zero,
your point might look like a local maximum.
Or if the Laplacian was greater than zero,
it might look like a local minimum,
'cause all of its neighbors
would be greater than where it is.
But for harmonic functions,
what makes them so special
is that you're saying
the value of the function itself,
or the value of the Laplacian of the function
at every possible point
is equal to zero.
So no matter what point you choose,
those neighbors are gonna be,
on average, the same value as this guy.
So the height of the graph above those neighbors
will, on average, be the same.
So,
so if we kind of look around the graph,
what that should mean is,
let's say you're looking at an input point,
you know, the output of this guy.
And if we looked at all of its,
the circle of its neighbors,
and kind of projected them up onto the graph,
what it should mean is that the height
of all the points on this circle,
on average, are the same as that.
And no matter where you look,
that should kind of average out.
And again, I encourage you to
take a look at,
take a look at this function
and actually evaluate the Laplacian
to see that it's zero.
But what's interesting is it's not at all clear,
just looking at this E to the X
times sine of Y formula,
that the average value of a circle of input points
is always gonna kind of equal the value
of the point at the center.
That's not something you can easily tell
just looking at that formula.
But with what's not that hard a computation,
you can make this conclusion,
which is pretty far-reaching.
And this comes up all the time in physics.
You know, for example, heat is one
where maybe you wanna describe
how the heat at a certain point in a room
is related to the average value of the heat
of all of the points kind of around it.
And in fact, it comes up in all sorts of circumstances
where you have some point in physical space
and something about that point,
maybe like the rate at which
some property of it is changing
corresponds to the average value
at points around it.
So whenever you're sort of relating neighbors
to your original point, the Laplacian comes in,
and harmonic functions have this tendency
to correspond to some notion of stability.
And I won't go deeper into that now.
This is really,
that really starts to get into the topic
of partial differential equations.
But at least in the context of
just multivariable calculus,
I wanted to shed some light
on interpreting this operator,
and kind of interpreting
the physical and geometric properties
that that implies about a function.
And with that,
I will see you next video.