DAVID JORDAN: Hello, and
welcome back to recitation.
In this problem, what
I'd like us to do
is I'd like us to sketch the
graphs, in three dimensions,
of these functions.
So z here is a
function of x and y.
On this second one, z is
also a function of x and y.
It just happens
not to depend on y.
When you graph these, I'd
suggest to consider slices,
so what happens if you
consider x equals 0
or if you consider z equals 0.
As you graph these, let's
see what you can do.
So why don't you
pause the video,
and I'll check back
with you in a moment,
and I can show you
how I solved these.
OK, welcome back.
So why don't we start by
looking at this function:
z is the square root of
x squared plus y squared.
OK, I'll try to always draw
my axes in the same way
as we do in lecture.
So x is pointing towards us,
y to the right, and z up.
So as I suggested,
I think a nice way
to get started
with these problems
is to just try setting the
variables x and y variously
equal to 0, and then
seeing-- instead
of a surface in that case,
then we'll get a curve,
and we'll see what curve we get.
So for instance, if
we set x equals to 0,
then we just get z is the
square root of y squared,
so we just get that z is
the absolute value of y.
So what that means is that
whatever this surface looks
like, we know what it
looks like if we slice it
in the blackboard, in the
plane of the blackboard.
We know that it just
looks like-- this
is just the graph of the
function absolute value of y, z
equals absolute value of y.
So now, if you think about it,
what I just said works just
as well for x instead of for y.
So if we were to graph
this in the xz-plane
where we set y equals to
0, then we would get-- OK,
I'm going to try to draw this.
So let me draw that
in blue, actually.
So the blue is in the
xz-plane and the white
is in the yz-plane.
OK, now I think what's going
to be really illustrative
is if we think about what
happens as we fix values of z.
Well, obviously, if
we set z equals to 0,
then there's just one solution,
which is this point here.
But what's going
to be interesting
is if we set z to be
some positive value.
So, for instance,
let's take z to be 2.
So, for instance,
we set z equals 2,
then we get 2 equals
the square root
of x squared plus y squared.
Solving this, this is the
same as saying x squared
plus y squared equals 4.
So what that tells us is that
at the height z equals 2,
we're just going to have
a circle of radius 2.
This is just the equation
for a circle of radius 2,
and so at height 2,
we just have a circle.
And actually, as
you can see, there's
nothing special about 2.
At every height, we're just
going to have a circle,
and so this is
what's called a cone.
OK.
Now for b, we can expect,
when we go over here
to b, that something
funny is going to happen
because it doesn't depend on y.
So let's see if we can
see how the fact that z
doesn't depend on y, how
this enters into our picture.
So I'll just walk over
here, and we'll consider z
equals x squared.
OK.
So again, we have our
x-axis, x, y, and z axes.
Now, let's consider
what this looks
like when we intersect
with the xz-plane,
so when we set y equals to 0.
Well, setting y
equal to 0 actually
doesn't change the equation,
and we get z equals x squared.
So we know what that looks like.
It's a parabola.
And this parabola, I want you
to think that it's, you know,
coming out at us, so
it's in the xz-plane,
going in and out of the board.
But now if you
think about it, what
it means to say that this
function doesn't depend on y,
what that means is that we
have the exact same picture
at every value of y.
So if we go out here, then we're
going to have the same picture.
And if we go over here, we're
going to have the same picture.
And, in fact, what
you're going to get
is you're going to get a prism.
Oh, that's really hard to read.
Let's see if we
can-- so let me--
since that's a bit hard
to read on the axes,
let's draw this again.
What we'll get is
we're going to have
a prism, which looks
like a parabola,
looks like a sheet
that's just stretched out
in the shape of a parabola.
And so, this we could call
a prism of a parabola.
Now let's see if we can get
any more insight from these two
pictures.
So look what happened
in this instance.
So here, the function z, it
obviously didn't depend on y.
And we can see that by
looking at the graph,
because, you know,
as you vary y,
the picture had to be unchanged.
So the fact that this was
a prism in y and the fact
that the function didn't depend
on y are one and the same fact.
Now if we go over
to the cone-- OK,
so here, our function z very
much depended on both x and y.
But you notice that it depended
on x and y only in the sense--
so z is actually equal
just to the radius r, which
is x squared plus y squared.
So the fact that this cone--
thank you-- the square root
of x squared plus y squared.
So the fact that
this only depends
on the radius and not the
relative angle of x and y
is why we got what--
this is an example
of a surface of revolution.
So we can always expect
that if the dependence
of z on the variables x
and y, if you can actually
just rewrite that as a
dependence on r, then
you'll get this nice
radial symmetry,
just like we had translational
symmetry for the prism.
And I think I'll
leave it at that.
