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Studying for a test? Prepare with these 13 lessons on Integrating multivariable functions.
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What we will attempt to start to do in this video is take the surface integral of the function x squared over our surface, where the surface in question, the surface we're going to care about is going to be the unit sphere. So it could be defined by x squared plus y squared plus z squared is equal to 1. And what I'm going to focus on in this first video, because it will take us several videos to do it, is just the parameterization of this surface right over here. And as you'll see, this is often the hardest part because it takes a little bit of visualization. And then after that, it's kind of mechanical, but it can be kind of hairy at the same time. So it's worth going through. So first, let's think about how we can parameterize-- and I have trouble even saying the word. How we can parameterize this unit sphere as a function of two parameters. So let's think about it. Let's think about it a little bit. So first, let's just think about the unit sphere. I'm going to take a side view of the unit sphere. So let's take the unit sphere. So this right over here is our z-axis. That's our z-axis. And then over here, I'm going to draw-- this is going to be not just the x or the y-axis. This is going to be entire xy-plane viewed from the side. That is the xy-plane. Now, our sphere, our unit sphere, might look something like this. The unit sphere itself is not too hard to visualize. It might look something like that. The radius-- let me make it very clear. The radius at any point is 1. So this length right over here is 1. That length right over there is 1. And this is a sphere, not just a circle. So I could even shade it in a little bit, just to make it clear that this thing has some dimensionality to it. So that's shading it in. It kind of makes it look a little bit more spherical. Now, let's attempt to parameterize this. And as a first step, let's just think. If we didn't have to think above and below the xy-plane, if we just thought about where this unit sphere intersected the xy-plane, how we could parameterize that. So let's just think about it. So where it intersects the xy-plane. It intersects it there, and there, and actually everywhere. So it intersects it right over there. So let's just draw the xy-plane and think about that intersection, and then we could think about what happens as we go above and below the xy-plane. So on the xy-plane, this little region where we just shaded in. So let me draw. So now you could view this as almost a top view. The z-axis is now going to be pointing straight out at you, straight out of the screen. So that's x. So let me draw it. So that's x, and then this right over there is y. So this thing that we were viewing sideways, now we're viewing it from the top. And so now our unit sphere is going to look something like this viewed from above. What I just drew, this dotted circle right over here, this is going to be where our unit sphere intersects-- I labeled that y. That should be x. Don't want to confuse you already. Let me clear that. So this is our x-axis. This is our x-axis. So this little dotted blue circle, this is where our unit sphere intersects the xy-plane. And so using this, we can start to think about how to parameterize at least our x- and y-values, our x- and y-coordinates, as a function of a first parameter. So the first parameter, we can think of something that is-- so this is the z-axis popping straight out at us. So we're essentially, if we're rotating around that z-axis viewed from above, we could imagine an angle. I will call that angle s, which is essentially saying how much we're rotating from the x-axis towards the y-axis. You could think about it in the xy-plane or in a plane that is parallel to the xy-plane. Or you could say, going around the z-axis. The z-axis popping straight up at us. And the radius here is always 1. It's a unit sphere. So given this parameter s, what would be your x- and y-coordinates? And now we're thinking about it right if we're sitting in the xy-plane. Well, the x-coordinate-- this goes back to the unit circle definition of our trig functions. The x-coordinate is going to be cosine of s. It would be the radius, which is 1, times the cosine of s. And the y-coordinate would be 1 times the sine of s. That's actually where we get our definitions for cosine and sine from. So that's pretty straightforward. And in this case, z is obviously equal to 0. So if we wanted to add our z-coordinate here, z is 0. We are sitting in the xy-plane. But now, let's think about what happens if we go above and below the xy-plane. Remember, this is in any plane that is parallel to the xy-plane. This is saying how we are rotated around the z-axis. Now, let's think about if we go above and below it. And to figure out how far above or below it, I'm going to introduce another parameter. And this new parameter I'm going to introduce is t. t is how much we've rotated above and below the xy-plane. Now, what's interesting about that is if we take any other cross section that is parallel to the xy-plane now, we are going to have a smaller radius. Let me make that clear. So if we're right over there, now where this plane intersects our unit sphere, the radius is smaller. The radius is smaller than it was before. Well, what would be this new radius? Well, a little bit of trigonometry. It's the same as this length right over here, which is going to be cosine of t. So the radius is going to be cosine of t. And it still works over here because if t goes all the way to 0, cosine of 0 is 1. And then that works right over there when we're in the xy-plane. So the radius over here is going to be-- so that right over there is cosine of 0. So this is when t is equal to 0. And we haven't rotated above or below the xy-plane. But if we have rotated above the xy-plane, the radius has changed. It is now cosine of t. And now we can use that to truly parameterize x and y anywhere. So now, let's look at this cross section. So we're not necessarily in the xy-plane, we're in something that's parallel to the xy-plane. And so if we're up here, now all of a sudden, the cross section-- if we view it from above, might look something like this. It might look something like this. We're viewing it from above, this cross section right over here. Our radius right over here is cosine of t. And so given that-- I guess altitude that we're at, what would now be the parameterization using s of x and y? Well, it's the exact same thing, except now our radius isn't a fixed 1. It is now a function of t. So we're now a little bit higher. So now, our x-coordinate is going to be our radius, which is cosine of t. That's just our radius. Times cosine of s. Times cosine of s, how much we've angled around. And in this case, s has gone all the way around here. S has gone all the way around there, so it's going to be cosine of t times cosine of s. And then, our y-coordinate is going to be our radius, which is cosine of t times sine of s. Same exact logic here, except now we have a different radius. Our radius is no longer 1. Times sine of s. I'm running out of space, let me scroll to the right a little bit. And I know this looks very confusing, but you just have to say, at any given level we are, we're parallel to the x-axis. We're kind of tracing out another circle where another plane intersects our unit sphere. We're now then rotating around with s. And so our radius will change. It's a function of how much above and below we've rotated-- how much above or below the xy-plane we've rotated. So this is just our radius instead of 1. And then, s is how much we've rotated around the z-axis. Same there for the y-coordinate. And then the z-coordinate is pretty straightforward. It's going to be completely a function of t. It's not dependent on how much we've rotated around here at any given altitude. It is what our altitude actually is. Now, we can go straight to this diagram right over here. Our z-coordinate is just going to be the sine of t. So our z is equal to sine of t. So let me write that down. So the z is going to be equal to sine of t. So now, every point on this sphere can be described as a function of t and s. Now, we have to think about over what range will they be defined. Well, s is going to go-- at any given level, you could think. For any given t, s is going to go all the way around. We see that right over here. At any given level viewed from above, s is going to go all the way around. So thinking about in radians, s is going to be between 0 and 2 pi. And t is essentially our altitude in the z-direction. So t can go all the way down here, which would be negative pi over 2. So t can be between negative pi over 2. And it can go all the way up to pi over 2. It doesn't need to go all the way back down again. And so it goes all the way back-- it goes only up to pi over 2. And then we have our parameterization. Let me write this down in a form that you might recognize even more. If we wanted to write our surface as a position vector function, we could write it like this. We could write it r is a function of s and t, and it is equal to our x-component. Our i-component is going to be cosine of t cosine of s i. And then plus our y-component is cosine of t sine of s plus our z-component, which is the sine-- which is just-- oh, I forgot our j vector. j plus the z-component, which is just sine of t sine k. And we're done. And these are the ranges that those parameters will take on. So that's just the first step. We've parameterized this surface. Now we're going to have to actually set up the surface integral. It's going to involve a little bit of taking a cross product, which can get hairy, and then we can actually evaluate the integral itself.