Partial derivatives of vector fields, component by component

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 calculus, and it's also just good to sit down and take a complicated thing and kind of break it down piece by piece.

A vector field like the one that I just showed is represented by a vector-valued function. 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 we'll just write P of XY for that X component and Q of XY for that Y component. Each of these is 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. 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 * Y and Q is equal to y² - x².

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. I think it's a good way to understand vector-valued functions in general, but here that's not what I'm going to do. It's actually another useful skill 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. You can think about the partial derivative of P with respect to X or the partial derivative of P with respect to Y. Then similarly for Q, you could think about the partial derivative of Q with respect to X 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. In this specific example, let's actually compute these.

The derivative of P with respect to X; P is this first component we're taking the partial of 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 rules are reversed, and its partial derivative is X because X looks like that constant.

For Q, its partial derivative with respect to X; Y looks like a constant, x² goes to -2X, but then when you're taking it with respect to Y, y² now looks like a function whose derivative is 2Y, and x² 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 does it mean in terms of the picture that we're looking at up here? In particular, let's focus on a point—a specific point. Let's do this one here; it's something that's sitting on the X-axis, so this is where Y equals 0, and X is something positive. This is probably when X is around 2-ish, let's say.

So the value we want to look at is XY when X is 2 and Y is 0. If we start plugging that in here, what that would mean, this guy goes to 0, this guy goes to 2; this guy, -2 * X is going to be -4, and then -2 * Y is going to be 0.

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 want to look at the two neighboring vectors and consider what's going on in 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 0. The partial derivative of P with respect to X is 0, so we wouldn't expect a change.

But on the other hand, if we're looking at the partial derivative of P with respect to Y, this should be positive. 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 a change in the X direction, but instead we're wondering what happens as we move generally upwards.

We're going to 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 is a little bit to the left, then we get to our main guy here, and it's 0. The X component is 0 because 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 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 because 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. 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.

Again, we look at these neighboring guys, and now the Y component starts off small but negative, then gets a little more negative, then gets even more negative. 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 -2.

So that's telling us, given that it started negative, it's getting more negative. If it started positive, they would have 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 the 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 to the left, so down it's a little bit negative here. The Y component is also a little bit negative over here; it remains a little bit negative. From our analysis, 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 0. The 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 0.

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. You'll get plenty of chances 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. You'll see what I mean by that in just a couple of videos.