JOEL LEWIS: Hi.
Welcome back to recitation.
In lecture, you've been learning about the divergence theorem,
also known as Gauss's theorem, and flux,
and all that good stuff.
So I have a nice exercise on it for you here.
So what I want-- so I want you to take F,
and I want it to be the field whose components are
x over rho cubed, y over rho cubed, and z over rho cubed.
So here, rho is your usual rho from spherical coordinates.
Rho is equal to the square root of x squared
plus y squared plus z squared.
And I want S to be the surface of the box whose
vertices are plus or minus 2, plus or minus 2,
plus or minus 2.
So it's a cubical box.
So what I'd like you to do is, first in part a,
I'd like you to show that the divergence of F
is 0, wherever the field F is defined.
In part b, what I'd like you to think about
is whether we can conclude from that,
that the flux through the surface of S is equal to 0.
All right.
And in part c, what I'd like you to do
is to use the extended version of Gauss's theorem--
or the extended version of the divergence theorem--
in order to actually compute the flux through S
by computing an integral.
So why don't you pause the video for a couple of minutes,
work out this problem, come back,
and we can work it out together.
Hopefully you had some luck with this problem.
Let's get started.
Part a asks you to compute the divergence of F.
So in order to compute that, we're
going to need to take the partial derivatives
of the components of F. And in order to do that, at some point
I'm going to need to take a partial derivative of rho.
So let me first compute the partial derivatives of rho,
and that will save me a tiny bit of work later.
So rho is equal to the square root of x squared
plus y squared plus z squared.
So partial rho partial x-- well, you just
apply your usual chain rule here.
And I guess we get a half, but then we
get a 2 that cancels it, so I think
this works out to x divided by the square root of x squared
plus y squared plus z squared, so that's x divided by rho.
All right.
And I'm just going to keep rho around here,
because otherwise I have to write out the square root of x
squared plus y squared plus z squared over and over again,
and this is going to save me some effort
and would save you some effort as well.
So OK.
So this is rho.
So this is d rho dx.
So we want to take the x partial of the first component of F. So
that's the x partial of x over rho cubed.
OK.
And you just apply your usual quotient rule,
so what do we get?
We get the derivative of the top.
So that's rho cubed minus-- OK, so the top
is x times the derivative of the bottom, which
is going to be 3 rho squared times x over rho--
so that's 3-- so we have an x-- so it's
3 x squared rho, divided by the bottom squared,
which is rho to the sixth.
And I guess there's a common factor of rho
everywhere that we can cancel out.
So this is equal to rho squared minus 3 x squared divided
by rho to the fifth.
OK.
So that's the x-partial derivative
of the first component of F.
Now we need the y-partial derivative
of the second component of F, and the z-partial derivative
of the third component of F. But if you go and look
back at what the formula for F was,
you see that this is a very, very symmetric formula.
So in order to get from the first component
to the second component, we just change
x to y, and to get from the second component to the third,
we just change y to z, because of course rho
treats x, y, and z the same.
So what does that mean?
Well, that means that the partial derivatives
are easy to compute.
Having computed this x-partial derivative,
we also get that partial over partial y
of the second component-- which is y over rho cubed--
is equal to rho squared minus 3 y squared, over rho
to the fifth.
And the last one we get, partial over partial z of z over rho
cubed is equal to rho squared minus-- I'm
getting a little cramped here-- 3 z
squared, over rho to the fifth.
And so adding these up, we get that div F
is equal to the sum of those three things.
So let's see what we've got.
We've got a 3-- so the denominators
are all rho to the fifth.
And we've got 3 rho squared minus 3
x squared minus 3 y squared minus 3 z squared.
So this is equal to 3 rho squared
minus 3 x squared minus 3 y squared minus 3 z squared, all
over rho to the fifth.
But of course, rho squared is x squared
plus y squared plus z squared, so this numerator is just 0.
So this is equal to 0.
OK.
Which is what we thought it should be.
All right.
Good.
So that's part a.
We just computed the partial derivatives of F,
and then added them together to get the divergence.
And we found that, in fact, yes, the divergence was equal to 0.
Great.
So that's part a.
So let's go look at what part b was.
Part b asks, can we conclude that the flux
through the surface S is 0?
All right.
Now remember what the divergence theorem says.
The divergence theorem says that the flux
through a surface of a field is equal to the triple integral
of the divergence of that field over the interior,
provided the field is defined and differentiable and nice,
or whatever, Everywhere inside.
OK?
But this field has a problem.
Almost everywhere, this field is nicely behaved,
but at 0, we have a real problem.
We're dividing by 0.
Right?
So this field is not defined at 0.
So there's a single point in the middle of this cube where
this field behaves badly.
And that means we can't apply the divergence
theorem inside this cube.
So since we can't apply the divergence theorem,
we aren't allowed to conclude immediately
that the flux through this surface is 0.
OK.
So the answer is no.
We can't conclude that the flux through S
is 0, because one of the hypotheses of the divergence
theorem isn't satisfied.
Namely, the field isn't defined everywhere inside the surface.
OK.
So the answer to b is no.
OK, I'm just going to write that.
But it's no because the hypotheses aren't satisfied.
OK, so now let's look at part c.
So part c suggests, we can't conclude that the flux is 0.
So we still want to know what the flux is.
That's still an interesting question,
so part c suggests, maybe you can still
use the divergence theorem-- well,
now we're calling it extended Gauss's theorem--
to compute what this flux is.
So let's think about how we could do that.
So remember what extended Gauss's theorem says?
Or extended divergence theorem.
I'm going to try and just say Gauss's theorem from now on,
so I stop having to say both.
But I mean both.
I mean, they're the same theorem, right?
OK.
So Gauss's theorem says, when you have a surface bounding
a region, the flux through the surface
is equal to the triple integral of divergence over the region,
provided everything is well-defined and nice.
Extended Gauss's theorem says, this
is still true if your region has more than one boundary.
So for example, if your region is a hollow something--
so if it's a spherical shell that
has an outside sphere and an inside sphere-- then extended
Gauss' theorem says, OK, so you do the same thing.
You take the triple integral of the divergence
over the solid region.
And then you take the flux, but you add up
the flux over all of the boundary pieces.
So you add up the flux over the outside boundary surface,
and also, if there is one, through any other boundary
surface.
OK?
And those two things are equal.
So the total flux through all of the boundary surface
is equal to the integral of divergence
over the whole region bounded by those surfaces.
So how are we going to use this?
We're trying to compute the flux through a surface.
OK, but we don't want to compute a double integral
if we can avoid it.
We don't want to compute the surface integral.
So what we'd like to do is we'd like
to find a convenient region over which
to compute this integral, to put us in a situation
where we can apply extended Gauss's theorem.
We can't use just the inside of the cube,
so we want some other region.
So what we're going to do is we're going to walk over here,
and we're going to do-- there are
many possible things you could do, but this is a nice one.
All right.
One thing you could do is you could take a big sphere.
Take a big sphere.
So we've got our cube here.
This is the point (2, 2, 2), and this is the point
2, 2, minus 2, and so on.
So we've taken a big sphere of radius R--
for some big R-- that contains our surface
S that we're interested in, that completely contains the cube.
OK?
So why have we done that?
Well, extended Gauss's theorem--
OK, so what does extended Gauss' theorem
say for the region between the sphere and this cube.
All right.
So our cube is named S. Let's call our sphere
S_2, because why not?
OK.
And let's call the solid region between them, between the cube
and sphere-- just for convenience,
let's give it a name-- so, I don't know,
we often call solid regions D, so let's call it D.
So it's this spherical region, but it has a cubical hole
in the middle of it.
OK.
So what does extended Gauss's theorem say?
So extended Gauss's theorem says that the triple integral
over D of the divergence of F dV is
equal to the sum of the fluxes through each of the surfaces.
But for this, we want the flux out of the solid region.
So for the sphere, the flux out of the inside of the sphere
is the flux out of the sphere.
So that's integral over S_2 of F dot n, d surface area.
But for the cube, the flux out of this region
is the flux into the cube.
Right?
Out here, you're living in a region outside the cube,
so when you leave that region, you're going into the cube.
So this is the negative of the flux that we really want.
So this is minus the flux through the cube of F dot n,
with respect to surface area.
So remember, the signs here are different,
because I'm taking this normal to be the outward pointing
normal to both surfaces.
The normal that points away from the origin.
But the normal pointing away from the origin on the cube
is the normal that points into the solid region instead
of the normal that points out of the solid region.
So that's why this minus is here.
OK.
Whew.
All right, so what does this mean?
Well, we've already computed, in part
a, that the divergence-- so first of all,
F is well-defined everywhere in this region
D. The only place F was badly behaved was the origin.
And this region doesn't contain it,
which is why this trick works.
So we've already computed that the divergence of F
is 0 everywhere.
It's defined, so it's 0 on all of D,
and so this triple integral is just 0.
So if this triple integral is 0, that
means we can just add the thing that we're
interested in to both sides, and we
get that the surface integral over the cube of F
dot n, with respect to surface area,
is equal to the surface integral over the sphere of F dot n,
with respect to surface area.
OK.
So we've converted this original integral-- our flux integral
that we're interested in-- and we
found that it's equal to this separate flux
integral over a different surface.
This time over a big sphere.
OK, so that's nice.
Why do we want to do that?
Well, we want to do that because F
is a really nicely behaved field with respect to a sphere.
F is a radial field.
So F dot n is really easy to understand.
F dot
n is just-- well, n is a unit normal and F is a radial field.
So on a sphere, the normal is radial, right?
It's parallel to the position vector.
And F is radial.
So they're both pointing in exactly the same direction.
So when you take that dot product,
n is the unit vector in the same direction as F,
so when you dot that with F, you just get the length of F. OK,
so what does that mean?
That means over here, this integrand
is really easy to understand.
OK?
This integrand F dot n on the sphere
is just equal to the length of the vector F.
Now what is the length of the vector F?
Well, we know what F is.
It's x over rho cubed i hat, plus y over rho cubed j hat,
plus z over rho cubed k hat.
So OK, so you compute the length of that vector,
and what do you get?
Well, it's exactly 1 over rho squared.
OK.
But we said that this is a sphere.
I guess I didn't write it down.
Let me write it down right here.
This is a sphere whose radius is big R. It doesn't really
matter very much what R we choose,
we just want it to be big enough so that it
contains the whole cube.
If you said this a sphere of radius 10,
that would completely do the trick.
That would be totally fine.
OK, so the radius of the sphere is big R,
so the length of the field, we said, back over here,
is 1 over R squared.
The length of the vector F. So this flux integral then,
is the integral over the sphere S_2 of a constant.
So it's the integral over the sphere of 1 over R squared dS.
But when you integrate a constant
over a surface, what you get is just that constant times
the surface area.
Well, what's the surface area?
This is a sphere.
It's easy to understand its surface area.
Its surface area is 4 pi R squared.
Right?
So this is equal to the surface area,
so that's 4 pi R squared, times whatever that constant was.
So the constant was 1 over R squared.
And so the R squareds cancel.
Right?
This is why it didn't matter what
R we chose, because they're just going
to cancel at the end, anyhow.
OK, so those cancel, and we're left with 4*pi.
So let's just quickly recap what we did in this part c.
We're looking to compute the flux over the cube.
But it's a kind of unpleasant integral we'd have to compute,
to total up the fluxes over these various different faces
and so on.
So instead, we had this clever idea
that we'll apply the divergence theorem to replace the cube
with a more congenial surface.
So the surface we choose, because this
is a nice radial vector field-- that's our main hint.
Because there was a rho involved in the problem, if you will.
So the surface that we choose is some big sphere.
And then we apply the extended Gauss's theorem
to the solid region between the cube and the sphere.
Outside the cube, but inside the sphere.
So because the divergence of the field is 0,
the extended Gauss's theorem tells us
that the two fluxes-- the flux out of the cube
and the flux out of the sphere-- are actually
equal to each other.
But since the fluxes are actually equal to each other,
in order to compute the flux out of the cube,
it's enough to compute the flux out of the sphere.
OK.
But computing the flux out of the sphere
is relatively easy, because on the sphere,
the integrand F dot n is just a constant.
And so then we're integrating a constant over the surface
of a sphere, and that just gives us the surface area
of the sphere times that constant,
which is 4 pi R squared times 1 over R squared, which is 4*pi.
So the flux out of the cube then is also equal to 4*pi.
I'll stop there.