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.