Stokes example part 2

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.