- Computing the partial derivative of a vector-valued function
- Visual parametric surfaces
- Partial derivative of a parametric surface, part 1
- Partial derivative of a parametric surface, part 2
- Partial derivatives of vector valued functions
- Partial derivatives of vector fields
- Partial derivatives of vector fields, component by component
Partial derivative of a parametric surface, part 2
Taking the same example surface used in the last example, we now take a look at the partial derivative in the other direction. Created by Grant Sanderson.
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- We can see in this e.g that there are 2 vectors which are tangent on the point you have mentioned, one vector we get from the partial derivative with respect to "t" and the other vector we get from the partial derivative with respect to "s". If we now also draw a plane which is tangent on the point you have mentioned would those 2 vectors be lying on this plane? I am asking this because you were talking about a plane tangent to this point.(3 votes)
- Let's say we were to take 2 points on the yellow line (constant t line) on the TS plane, and then transform the plane into a surface, so then, the distance between the two point on the transformed surface divided by the distance between them on the TS plane is the value of (partial V/partial S)? Am I correct? Where distance is the "nudge" as grant calls it.(1 vote)
- [Voiceover] Hello hello again. So, in the last video, I started talking about how you interpret the partial derivative of a parametric surface function, of a function that has a two variable input, and a three variable vector-valued output. And we typically visualize those as a surface in three dimensional space, and the whole process, I was saying, you think about how a portion of the t-s plane moves to that corresponding output. And again, I'm kind of cheating with this animation, where, really, this isn't the t-s plane, right? This is on the x-y plane. t-s 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 x-y plane here. We're thinking about the squares being t and s ranging each from 0 to 3. And, what I said, for 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. And the partial derivative vector gives you a certain tangent vector to the curve representing that line, which corresponds to movement in the t direction. And the longer that is, the faster the movement, The more sensitive it is to nudges in the t direction. In the s direction, let's say we were to take the partial derivative with respect to s. So I'll kind of clear this up here. Also clear up this guy. And if you said instead, "what if we were doing it with respect to s," right? Partial derivative of v, the vector-valued function, with respect to s. Well, you do something very similar. You would say "OK, What is the line that corresponds "to movement in the s direction?" And the way I've drawn it, it's always going to be perpendicular, because we're in the t-s plane, the t axis is perpendicular to the s plane. And in this case, this line represents t = 1, right? You're saying t constantly equals 1, but you're letting s vary. And 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. It kind of starts curving this way, and then it curves very much up and kind of goes off into the distance there. And 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. And for partial derivatives, we think very similarly. You think of that partial s as representing- (voiced zooming sounds) Zoom on back here. That partial s you think of as representing a tiny movement in the s direction, just a little smidge and 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, and... and maybe that tiny nudge corresponded with one that's 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. And it's going to correspond to the rate at which 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 get some good practice computing things. And 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 -s², with respect to s, has a derivative of -2s. st: s looks like a variable, t looks like a constant, the derivative is just that constant, t. Down here, ts²: t looks like a constant, s looks like a variable, so... 2 times t times s. And then over here, we're subtracting off, s is the variable, t² looks like a constant, so that constant. And let's say we plug in the value (1,1). This red dot corresponds to (1,1), so what we would get here, s is equal to 1, so that's -2, t is equal to 1, so that's 1, then 2 times 1 times 1, I'll write it. 2*1*1 minus 1² is gonna correspond to 1, that's 2-1. 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. And if we go over, and we take a look at what the movement along the curve actually is, that lines up, right? Because, 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, and you can also kinda see that the leftward movement is kind of twice as fast as the upward motion. The slope favors the x direction. And then as far as the z component is concerned, you are, in fact, moving up. And maybe you could say, "Well, how do you know which way you're moving, "are you moving that way, or is everything "switched the other way around?" And the benefit of animation here is, we can say, "Ah, as s is ranging from zero "up to three, this is the increasing direction." And 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. And one kinda 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, kinda goes in one direction, and the other one kinda gives you a different notion of what a tangent vector on the surface could be. And you could have a notion of directional derivative too, that kind of combines these in various ways, and that will get you all the different ways that you can have a vector tangent to the surface. And 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. And the next couple videos, I'll talk about what partial derivatives of vector-valued functions can mean in other contexts, 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, "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.