Partial derivatives of vector fields, component by component

- [Voiceover] Let's continue thinking about partial derivatives of vector fields. This is one of those things that's pretty good practice for some important concepts coming up in multivariable calc, and it's also just good to sit down and take a complicated thing and kind of break it down piece by piece. So a vector field like the one I just showed is represented by a vector-valued function, and since it's two-dimensional, it'll have some kind of two-dimensional input. And the output will be a vector, each of whose components is some kind of function of x and y, right? So I'll just write P of x, y for that x-component and Q of x, y for that y-component. And each of these are just scalar-valued functions. It's actually quite common to use P and Q for these values. It's one of those things where sometimes you'll even see a theorem about vector calculus in terms of just P and Q, kind of leaving it understood to the reader that, yeah, P and Q always refer to the x and y components of the output of a vector field. And in this specific case, the function that I chose, it's actually the one that I used in the last video. P is equal to x times y, and Q is equal to y squared minus x squared. And in the last video, I was talking about interpreting the partial derivative of v, the vector-valued function with respect to one of the variables, which has its merits, and I think it's a good way to understand vector-valued functions in general. But here that's not what I'm gonna do. It's actually, another useful skill is to just think in terms of each specific component. So if we just think of P and Q, we have four possible partial derivatives at our disposal here, two of them with respect to P. So you can think about the partial derivative of P with respect to x, or the partial derivative of P with respect to y. And then similarly, Q, you could think about partial derivative of Q with respect to x. This should be a partial. Or the partial derivative of Q with respect to y. So four different values that you could be looking at and considering and understanding how they influence the change of the vector field as a whole. And in this specific example, let's actually compute these. So derivative of P with respect to x. P is this first component. We're taking the partial of this with respect to x. y looks like a constant. Constant times x. Derivative is just that constant. If we took the derivative with respect to y, the roles have reversed, and its partial derivative is x, 'cause x looks like that constant. But Q, its partial derivative with respect to x, y looks like a constant, negative x squared goes to negative 2x. But then when you're taking it with respect to y, y squared now looks like a function whose derivative is 2y, and negative x squared looks like the constant. So these are the four possible partial derivatives. But let's actually see if we can understand how they influence the function as a whole. What it means in terms of the picture that we're looking at up here. And, in particular, let's focus on a point, a specific point, and let's do this one here. So it's something that's sitting on the x-axis. So this is where y equals zero and x is something positive. So this is probably when x is around two-ish, let's say. So the value we wanna look at is x, y when x is two and y is zero. So if we start plugging that in here, what that would mean, this guy goes to zero, this guy goes to two, this guy, negative two times x, is gonna be negative two. And then negative two times y is gonna be zero. And let's start by just looking at the partial derivative of P with respect to x. So what that means is that we're looking for how the x-component of these vectors change as you move in the x direction. For example, around this point, we're kind of thinking of moving in the x direction, vaguely. So we wanna look at the two neighboring vectors and consider what's going on with the x direction. But these vectors, this one points purely down. This one also points purely down, and so does this one. So no change is happening when it comes to the x-component of these vectors, which makes sense because the value at that point is zero. The partial derivative of P with respect to x is zero, so we wouldn't expect a change. But on the other hand, on the other hand, if we're looking at partial derivative of P with respect to y, this should be positive. So this should suggest that the change in the x-component as you move in the y direction is positive. So we go up here and now we're not looking at change in the x-direction. Not looking at change in the x-direction. But instead we're wondering what happens as we move generally upwards. So we're gonna kind of compare it to these two guys. And in that case, the x-component of this one is a little bit to the left. The one below it, it's a little bit to the left. Then we get to our main guy here and it's zero. The x-component is zero 'cause it's pointing purely down. And up here it's pointing a little bit to the right. So as y increases, the x-component of these vectors also increases. And again, that makes sense because this partial derivative is positive. This two suggests that as you're changing y, the value of p, the x-component of our function, should probably keep that on screen. You know, the x-component of our vector-valued function is increasing 'cause that's positive. For contrast, let's say we look at the Q component over here. So what this is doing, we're looking at changes in the x, and we're wondering what the y value of the vector does. So we go up here. And now, we're not looking at changes in the y direction. But instead, we're going back to considering what happens as we change x, as we're kind of moving in the horizontal direction here. So again, we look at these neighboring guys. And now, the y-component starts off small, but negative, then gets a little bit more negative, then gets even more negative. And if we kind of keep looking at these, the y-component is getting more, and more, and more negative so it's decreasing. The value of Q, the y-component of these vectors should be decreasing. And that lines up 'cause the partial derivative here was negative two, so that's telling us, given that it started negative, it's getting more negative. If it started positive, they would've been kind of getting shorter as vectors as their y-component got smaller. And then finally, just to close things off nice and simply here, if we look at partial of Q with respect to y, so now if we start looking at changes in the y direction, and we start considering how, as you move below, and then starting to go up, what happens to the y-component, here, the y-component is a little bit negative, right? It's pointing down and to the left. So down, it's a little bit negative. Here, the y-component is also a little bit negative. Over here, well, it remains a little bit negative. And you know, from our heuristic look, there's no discernible change. Maybe it's changing a little bit, and we don't have a fine enough vision of these vectors to see that, but if we actually go back to the analysis, and see what we computed, in fact, it is zero. That fact that it looked like there was not too much change in the y-component of each one of these vectors corresponds with the fact that the partial derivative of that y-component with respect to y, with respect to vertical movements, is zero. So this kind of analysis should give a better feel for how we understand the four different possible partial derivatives and what they indicate about the vector field, and you'll get plenty of chance to practice that understanding as we learn about divergence and curl, and try to understand why each one of those represents the thing that it's supposed to. And you'll see what I mean by that in just a couple videos.