Partial derivative of a parametric surface, part 2

Hello, hello again!

So in the last video, I started talking about how you interpret the partial derivative of a parametric surface function, right? Of a function that has a two-variable input and a three-variable vector-valued output. We typically visualize those as a surface in three-dimensional space. The whole process, I was saying, you think about how a portion of the ts plane moves to that corresponding output.

And again, I'm kind of cheating with this animation, where really this isn't the ts plane; right? This is on the xy plane. The ts plane should just be some separate space over here, and we're imagining moving that separate space over into three dimensions. But that's harder to animate, so I'm just not gonna do it, and I'm gonna instead keep things inside the xy plane here.

And you know, we're thinking about the squares being t and s ranging each from 0 to 3. What I said for the partial derivative with respect to t is you imagine the line that represents movement in the t direction. You see how that line gets mapped as all of the points move to their corresponding output. The partial derivative vector gives you a certain tangent vector to the curve representing that line, which corresponds to movement in the t direction.

The longer that is, the faster the movement, the more sensitive it is to nudges in the t direction.

So in the s direction, let's say we were to take the partial derivative with respect to s. I’ll kind of clear this up here, also clear up this guy. If you said instead, "What if we were doing it with respect to s?" Right? The partial derivative of v, the vector-valued function, with respect to s is something very similar. You would say, "Okay, what is the line that corresponds to movement in the s direction?"

The way I've drawn it, it's always going to be perpendicular because we're in the ts plane. The t axis is perpendicular to the s plane. In this case, this line represents t = 1, right? You're saying t constantly equals 1, but you're letting s vary. If you see how that line maps as you move everything from the input space over to the corresponding points in the output space, that line tells you what happens as you're varying the s value of the input.

And I guess it kind of starts curving this way, and then it curves very much up and kind of goes off into the distance there. Again, the grid lines here really help because every time that you see the grid lines intersect, one of the lines represents movement in the t direction, and the other represents movement in the s direction.

For partial derivatives, we think very similarly. You think of that partial s as zooming back here; that partial s you think of as representing a tiny movement in the s direction, just a little smidge. Nudge somehow nudging that guy along, and then the corresponding nudge, you look for in the output space. You say, "Okay, if we nudge the input that much, and we go over to the output end..."

And you know, maybe that tiny nudge corresponds with one that’s like three times bigger. I don’t know! But it looked like it stretched things out so that tiny nudge might turn into something that’s still quite small but maybe three times bigger. But it's a vector!

What you do is you think of that vector as being your partial v, and you scale it by whatever the size of that partial s was. Right? So the result that you get is a tangent vector that's not puny, not a tiny nudge, but is actually a sizable tangent vector. It's going to correspond to the rate at which tiny changes, not just tiny changes, but the rate at which changes in s cause movement in the output space.

So let's actually compute it for this case, just to get some good practice computing things. If we look up here, the t value, which used to be considered a variable when we were doing it with respect to t, but now that t value looks like a constant, so its derivative is zero. Then negative s squared with respect to s has a derivative of negative 2s.

st looks like a variable; t looks like a constant. The derivative is just that constant t. Down here, ts squared p looks like a constant; s looks like a variable, so it’s 2 times t times s. And then over here, we’re subtracting off s as the variable; t squared looks like a constant, so that constant.

Let’s say we plug in the value 1, 1. Right? This red dot corresponds to 1, 1. So what we would get here: s is equal to 1, so that’s negative 2. t is equal to 1, so that’s 1. Then 2 times 1 times 1; oh, let’s see, I’ll write it 2 times 1 times 1 minus 1 squared minus 1 squared is going to correspond to 1. That’s 2 minus 1.

So what we would expect for the tangent vector, the partial derivative vector is the x-component should be negative, and then the y and z components should each be positive. If we go over and we take a look at what the movement in the, you know, along the curve actually is, that lines up right, because you're moving, as you kind of zip along this curve, you're moving to the left.

So the x component of the partial derivative should be negative, but you're moving upwards as far as y is concerned. You can also kind of see that the leftward movement is kind of twice as fast as the upward motion; the slope favors the x direction.

As far as the z component is concerned, you are, in fact, moving up. Maybe you could say, "Well, how do you know what you're moving? Are you moving, you know, that way or is everything switched the other way around?"

The benefit of animation here is we can say, "Ah, as s is ranging from zero up to three, you know, this is the increasing direction." You just keep your eye on what that direction is as we move things about, and that increasing direction does kind of correspond with moving along the curve this way.

So you get a tangent vector in the other way. One kind of nice thing about this then is the two different partial derivative vectors that we found, each one of them you could say is a tangent vector to the surface, right? So the one that was a partial derivative with respect to t over here kind of goes in one direction, and the other one gives you a different notion of what a tangent vector on the surface could be.

Various different... you could have a notion of directional derivative too that kind of combines these in various ways, and that'll get you all the different ways that you can have a vector tangent to the surface.

Later on, I’ll talk about things like tangent planes if you want to express what a tangent plane is, and you kind of think of that as being defined in terms of two different vectors. But for now, that’s really all you need to know about partial derivatives of parametric surfaces.

In the next couple of videos, I’ll talk about what partial derivatives of vector-valued functions can mean in another context because it's not always a parametric surface, and maybe you're not always thinking about a curve that could be moved along, but you still want to think, you know, how does this input nudge correspond to an output nudge, and what's the ratio between them?

So with that, I’ll see you next video!