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Studying for a test? Prepare with these 13 lessons on Integrating multivariable functions.
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Now that we hopefully have a conceptual understanding... ...of what a surface integral like this COULD represent, ...I want to think about how we can actually construct... ...a unit vector... ...a unit normal vector, at any point on the surface. And to do that, I will assume... ...that our surface can be parametrized... ...by the position vector function, r... ...and r is a function of two parameters. It's a function of u, and it is a function of v. You give me a u and a v and... ...that will essentially specify... ...a point on this two-dimensional surface right over here. It could be bent, so it kind of exists in three-dimensional space... But a u and a v will specify a given point on this surface. Now, let's think about what the directions of r... ...the partial of r with respect to... ...the partial of r with respect to u looks like... ...and what the direction of the partial of r... ...the partial of r with respect to v looks like. So let's say that we're at some... ...we're at some point. We're at some point, (u,v). So for some (u,v), if you'd find the position vector... ...it takes us to that point on the surface right over there. Now let's say that we increment u just a little bit. And as we increment u just a little bit, ...we're going to get to another point on our surface, ...and let's say that other point on the surface... ...is right over there. So what would r... What would this r_u vector look like? Well its magnitude is essentially going to be.. ...dependent on how fast it's happening, ...how fast we're moving towards that point, ...but its direction is going to be in that direction. It's going to be towards that point. It's going be along the surface. We're going from one point on the surface to another. It's essentially going to be tangent to the surface at that point. And I could draw a little bit bigger. It would look something like that. r... r_u. So I just zoomed in right over here. Now let's go back to this point. And now let's make v a little bit bigger. And let's say if we make v a little bit bigger, ...we go to this point right over here. So then our position vector, r, would point to this point. And so what would r_v look like? Well its magnitude, once again, would be dependent on... ...how fast we're going there, but the direction is what's interesting. The direction would also be tangential to the surface. We're going from one point on the surface to another... ...as we change v. So r_v might look something like that. And they're not necessarily... These two aren't necessarily perpendicular to each other. In fact, the way I drew them, they're not perpendicular. So r_v is like this, but they're both tangential to the plane. They're both essentially telling us, right at that point, ...what is the tangent? What is the slope in that... ...in the u direction, or in the v direction? Now, this is... When you have two... When you have two vectors that are... ...that are tangential to the plane, ...and they're not the same vector, these are actually... ...already specifying... ...these are already kind of determining a plane. And so you can imagine a plane that looks something like this. If you took linear combinations of these two things, ...you would get a plane that both of these would lie on. Now, we've done this before, but I'll re-visit it. What happens when I take the cross product... ...of r_u and r_v? What happens when I take the cross product? Well first, this is going to give us another vector. It's going to give us a vector... ...a vector that is perpendicular to both... ...to r_u AND r_v. Or another way to think about it is... ...this plane, that these... ...when you take the cross product... ...this plane is essentially a tangential plane... ...to the surface. And if something is going to be perpendicular... ...to both of these characters, ...it's going to have to be normal to them... ...or, it's definitely going to be perpendicular... ...to both of them, but it's going to be normal... ...to this plane. Which is essentially going to be... ...perpendicular to the surface itself. So this right over here... ...is going to be A normal vector. This is... I'll write it... Well, let me just write it this way. This is A normal vector. I'm not saying THE unit normal... I'm not saying THE normal vector, 'cause you have... ...you could have different normal vectors of... ...different magnitudes. This is A normal vector, when you take the cross product. And we can even think about what direction it's pointing in. And so when you have "something" cross "something else"... ...the easiest way I remember how to do it is... ...you point your left thumb... Oh, sorry. You point your RIGHT thumb... ...in the direction of the first vector... So, in this case, r_u. So let me see if I can... ...if I can draw this. I'm literally looking at my hand and trying to draw it. So you put your right thumb... -- so this is a right-hand rule, essentially -- ...in the direction of the first vector... ...and then you put your index finger in the direction of... ...the second vector... ...right over here. So this is the second vector. So that's the direction of my index finger. So my index finger is going to look something like... ...that. And then you bend... ...you bend your middle finger inward... ...and that will tell you the direction of the cross product. So if I bend my middle finger inward, ...it will look something... ...it will look something like this. And then of course, my other two fingers are just going to be... ...folded in like that, and they're not really relevant. But my other two fingers and my hand looks like that. And so that tells us the direction. The direction is going to be like that. It's going to be upward-facing. That's important, because you have normal vectors. One could... Or there's two directions of "normalcy," I guess you could say. One is going out like that... ...outwards... ...or I guess... ...in the upward direction... ...one would be going downwards, ...or going -- I guess you could say -- into the surface. But the way I have set it up right now, ...this would be going outwards. It would be A... It would be A normal vector... ...to the surface. Now, in order to go from A normal vector... ...to the UNIT normal vector, ...we just have to normalize it. We just have to divide this... ...by its magnitude. So now we have our drumroll. The unit vector... And it's going to essentially be... It's going to be a function of u... It's going to be... ...a function of u and v. You give me a u or a v... ...and I'll give you a... ...that unit normal vector. It is going to be equal to... ...the partial of r... ...the partial of r with respect to u... ...crossed with... ...the partial of r with respect to v... That just... Now that gives us A normal vector, but it hasn't been normalized. So we want to divide... ...by the magnitude... We want to divide by the magnitude... ...of the exact same thing. r_u crossed with r_v. And we're done! We have constructed a unit normal vector. And in future videos, we'll actually do this with concrete examples.