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 by piece so a vector field like the one that 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 going to 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 X this should be a partial or the partial derivative of Q with respect to Y so for 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 let's sexually 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 the con constant times X derivative is just that constant if we took the derivative with respect to Y the roles are reversed and it's partial derivative is X because X looks like that constant but Q it's 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 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 want to look at is XY when X is 2 and Y is 0 so if we start plugging that in here what that would mean this guy goes to 0 this guy goes to 2 this guy negative 2 times X is going to be negative 2 and then negative 2 times y is going to be 0 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 so 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 with 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 0 the partial derivative of P with respect to X is 0 so we wouldn't expect to 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 going to 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 zero because it's pointing purely down and up here is 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 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 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 think it's a little 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 because the partial derivative here was negative 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 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 into 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 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 and you'll see what I mean by that in just a couple videos