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Studying for a test? Prepare with these 11 lessons on Green's, Stokes', and the divergence theorems.

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# Stokes example part 2

Video transcript

Now that we've set up
our surface integral, we can attempt to
parametrise the surface. And one way to think about
is we want our x and y values to take on all of the
values inside of the unit circle, what I'm shading
in right over here. And that our z values can be
a function of the y values. We can express this
equation right here, z is equal to 2 minus y. And then we could figure
out how high above to go to get our z value. And by doing that, we'll
be able to essentially get to every point that
sits on our surface. And so first let's think about
how we can get every x in y value inside of the unit circle. So let's just focus
on the xy plane. We're kind of rotated
around a little bit, so it looks a little
bit more traditional. So this is my x-axis and then
my y-axis would look something like that. Let me draw it a
little bit different. This is my y-axis. And then if I were
to draw the unit circle, some kind of the base
of this thing, or at least where it intersects
the xy plane-- actually this thing would
keep going down, if I wanted to
draw the x squared plus y squared equals 1. But if I draw where it
intersects the xy plane, we get the unit circle. So let me just draw it. That's my best attempt
at drawing a unit circle. We get the unit
circle and we need to think of using
parameters so that we can get every x and
y-coordinate that's inside of the unit circle. And to think about that,
I'll introduce one parameter that's essentially the
angle with the x-axis. And I'll call that
parameter theta. So theta is the angle
with the x-axis. And so theta will
essentially sweep things all the way around. So theta can go
between 0 and 2 pi. So theta will take on
values between 0 and 2 pi. And if we just fix the
radius at some point, say radius 1, that
would only give us all of the points
on the unit circle. But we want all the
points inside of it too. So we need to vary
the radius as well. So let's introduce
another parameter, let's call it r,
that is the radius. So for any given r, if
we keep changing theta, we would essentially sweep
out a circle of that radius. And if you change radius
a little bit more, you'll sweep out another circle. And if you vary radius
between 0 and 1, you'll get all of
the circles that will fill out this entire area. So the radius is going
to go between 0 and 1. Another way of thinking about
it is for any given theta, if you keep varying
the radius, you'll sweep out all of the
points on this line. And then as you
change theta, it'll sweep out the entire circle. So either way you
think about it. So with that, let's actually
define x and y in those terms. So we could say that x is equal
to-- so the x value whatever r is, the x value is going
to be r cosine theta. It's going to be
that component, it's going to be r cosine theta. And then the y component-- this
is just basic trigonometry-- is going to be r sine theta. And then the z
component, we already said z can be expressed
as a function of y. Right over here we can
rewrite this as z is equal to 2 minus y. That'll tell us how high to
go so we end up on that plane. So if z is equal to 2 minus
y and if y is r sine theta, we can rewrite z as being
equal to 2 minus r sine theta. So there, we're done. That's our
parametrization, if we wanted to write this
as a position vector with two parameters--
I'll call it lowercase s, it's already used r. Lowercase s, this
is our surface, and it's going to be
parametrized with r and theta. We can write it as r cosine
theta i plus r sine of theta j plus r plus 2 minus
r sine of theta k.