Defining and differentiating vector-valued functions
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Vector-valued functions differentiation
In the last video, we hopefully got ourselves a respectable understanding of how a vector-valued function works, or even better, a position vector-valued function, that is, in some ways, a replacement for traditional parameterization to describe a curve. And what I want to do in this video is just get a little bit of gut sense of what it means to take of a derivative of a vector-valued function. In this case, it'll be with respect to our parameter t. So let me draw some new stuff right here. So let's say I have the vector-valued function r of t, and this is no different than what I did in last video. x of t times unit vector i plus y of t times the unit vector j. If we were doing it in 3 dimensions, we'd add a z of t times k, but let's keep things relatively simple, and let's say that this describes a curve, and let's say the curve we're dealing with, t is between a and b, and this curve will look something like, let me do my best effort to draw the curve, I'll just draw some random curve here, so let's say the curve looks something like that. This is when t is equal to a, so it's going to go in this direction. This is when t is equal to b right here, this is t is equal to a, so this right here would be x of a, this right here is y of a, and similarly, this up here, this is x of b, and this over here is y of b. Now, we saw in the last video that the endpoints of these position vectors are what's describing this curve. So r of a we saw in the last video, it describes that point right there. I don't want to review that too much. But what I want to do is think about, what is the difference between 2 points? So let's say that we take some random point here. Let's say, some random t here. Let's call that r of t. Well actually, I'm going to do a different point, just because I want to make it a little bit clearer. So let's say-- I'm going to switch colors-- let's say that that right there is r of some t. Some particular t, right there. That is r of t. It's going to be, you know, a plus something. So that's some a particular t. And let's say that we want to figure out, and let's say we increase t by a little bit. By h. So let's say that r of t plus h, well, if we view the parameter t as time, we've moved in forward in time by some amount, so our little particle has moved a little bit. And let's say that we're over here. So that is that right there, in yellow, is r of t plus h. Just a slightly larger value for h. Now, one question we might ask ourselves, is how quickly is r changing with respect to t? So the first thing we might want to say, well, what's the difference between these two? If I were to take, and I want to visualize it. If I were to take r, the position vector, that we get by evaluating r at t plus h, and from that, I would subtract r of t. What do we get? Well, you might want to review some of your vector algebra but we're essentially just going to get this vector. Let me do it in a nice, vibrant color. We're going to get this vector right there, that I'm doing in magenta. So that magenta vector right there is, let me do it, that magenta one right there, is the vector r of t plus h minus r of t. And it should make sense, because when you add vectors, you go heads to tails. You could alternatively write this as r of t plus this character right here, plus r of t plus h minus r of t. When you add two vectors, you're adding, let me make it very clear, I'm adding this vector to this vector right here. You put the tail of the second vector at the head of the first. So this is the first vector, and I put the tail of the second there, and then the sum of those two, as we predicted, should be equal to this last one. It should be equal to r of t plus h. And we see that is the case, and algebraically, you would see that obviously this guy and that guy are going to cancel out. So hopefully that satisfies you. And I want to be clear. This, all of a sudden, this isn't a position vector. We're not saying that hey, let's nail this guy's tail at the origin and use this guy to describe a unique position. Now all of a sudden he's, it's just kind of a pure vector. It's describing just a change between two other position vectors. So this guy is right out here. But this vector literally describes the change. But say we care, and how would this look algebraically if we were to expand it like that? So this is going to be equal to, what's r of t plus h? That's the same thing as x of, let me do it over here. This is the same thing as x of t plus h times the unit vector i plus y of t plus h times the unit vector j, that's just that piece, that piece right there is that piece, minus this piece, so minus, I'll do it in the second line, I could have done it out here, but I'm running out of space. Minus x of t, right r of t is just x of t times i, plus, but I'll just distribute the minus sign, so it's minus y of t times j. Actually let me write it, this would be minus, let me write this way, plus this. So you realize that this is really just this guy right here. I'm just evaluating at t. So you have x of t and y of t, and then later we can distribute, right? If you distribute this minus sign, you get a minus x of t and a minus y of t. And in vector addition, you might need a little review on this if you haven't seen it in a while, you know that you can just add the corresponding components. You can add the x-components and you can add the y-components. So this is going to be equal to, let me rewrite it over here, because I think I'm going to need some space later on. So let me rewrite it over here. So I have r of t plus h minus r of t is equal to, and I'm just going to group the x- and the y-components, this is equal to the x-components added together, but this is a negative, so we're going to subtract this guy from that guy. So x of t plus h minus x of t, and then all of that times our unit vector in the x-direction, and then we'll have plus y of t plus h minus y of t times a unit vector the j-direction, I'm just rearranging things right now, and this will tell us what is our change between any 2 r's for given change in distance. And our change in distance here is h between any 2 position vectors. Now, what I set out at the beginning of this video, I said, well, I wanted to figure out the change, and we're going to think about the instantaneous change with respect to t. So I want to see, well, how much did this change over a period of h? Instead of writing h we could have written delta t, it would've been the same thing. So I want to divide this by h. So I want to say, look. My vectors changed this much, but I want to say it's over a period of h. And this is analogous to when we do slope. We say rise over run, over delta y, or change in y, over change in x. This is kind of the change in our function per change in x. Let's just divide everything, or I shouldn't say change x, per change in t. So here, our change in t is h, right? The difference between t plus h and t is just going to be h. And so we're going to divide everything by h. When you multiply a vector by some scalar, or divide it by some scalar, or you're just the taking each of its components and multiplying or dividing by that scalar, and we get that right there. So this, for any finite difference right here, h, this'll tell us how much our vector changes per h. But if we want to find the instantaneous change, right, just like what we did when we first learned differential calculus, we said, ok. This is kind of analogous to a slope. This would be good, this would work out well for us, if the path under question looked something like this. If it was a linear path. If our path looked something like this. We could just calculate this, and we'll essentially have the average change in our position vectors, so you could imagine, 2 position vectors, that's one of them. Well, actually, they'd all be parallel. Well, the position vectors, they don't have to be parallel. They could be like that. And then, this would just describe the change between these 2 per h, or how quickly are the position vectors changing per our change in our parameter, right? This is, the h, you could also consider, is kind of a delta t. Sometimes people find the h simpler, or sometimes they find the delta t. But anyway, I'm concerned with the instantaneous. We're dealing with curves, we're dealing with calculus. This would have been OK if we were just in an algebraic, linear world. So what do we do? Well maybe, we can just take the limit as h approaches 0. Let me scroll this over. So let's just take the limit, let me do this in a nice vibrant color, let's take, I'm running out of colors, the limit as h approaches 0 of both sides of this. So here, too, I'm going to take the limit as h approaches 0, and here, too, I'm going to take the limit as h approaches 0. So I just want to say, well, what happens, how much do I change per a change in my parameter t, but what's kind of the instantaneous change, as the difference gets smaller and smaller and smaller? This is exactly what we first learned when we learned about instantaneous slope, or instantaneous velocity, or slope of a tangent line. Well, this thing looks a little bit undefined to me, right now. We haven't defined limits for vector-valued functions, we haven't defined derivatives for vector-valued functions. But lucky for us, all of this stuff here looks pretty familiar. This is actually the definition of our derivative. And these are scalar-valued functions right here. They're multiplied by vectors, in order for us to get vector-valued functions. But this right here, by definition, this is the derivative, this is x prime of t. Or this is dx dt. This right here is y prime of t, or we could write that as dy dt. So all of a sudden we can define, we can say, and I'm being a little hand-wavy here, but I want to give you the intuition, more than anything. We can say that the derivative, we can call this expression right here, as the derivative of my vector-valued function r with respect to t, or we could call it dr dt, notice I keep the vector signs there. This is its derivative, and all it's going to be equal to, r prime of t, is going to be equal to, well, this is just the derivative of x with respect to t, is equal to x prime of t times the x-unit vector, the horizontal unit vector, plus y prime of t, times the y-unit vector, times j, the unit vector in the horizontal direction. That's a pretty nice and simple outcome. But the hard thing may be to a kind of visualize what it represents. So if we think about what happens, let me draw a big graph, just to get the visualization in a healthy way. So let's say my curve looks something like this. That's my curve. And let's say that this is, we want to figure out the instantaneous change at this point right here. So that is r of t. And then if we take r of t plus h, we saw this already, you know, t plus h might be something like right there. So this is r of t plus h. Right now, the difference between these two, and this is just the numerator when you take the difference, or how fast we're changing from this vector to that vector in terms of t, and it's hard to visualize here. And I'm going to do a whole video so we can think about the magnitudes here. That might be some vector. Well, the difference between these two is just going to be that. But then when you divide it by h, it's going to be a larger vector, right, if we assume that h is a small number. Let's say h is less than one. We're going to get a larger vector, right? But this is kind of the average change over this time. But as h gets smaller and smaller and smaller, this r prime of t is going to, its direction is going to be tangential to the curve. And I think you can visualize that, right? As these two guys get closer and closer and closer, the dr's get smaller, so the change, the dr, the difference between the two, the delta r's, get smaller and smaller, you can imagine if h was even smaller, if it was right here. Then all of a sudden, the difference between those two vectors is getting smaller. And it's getting more and more tangential to the curve. But then we're also dividing by a smaller h, so the actual derivative, as the limit of h approaches 0, it might be you know, maybe it's even a bigger number there. And actually, the magnitude of this vector, it's a little hard to visualize. It's going to be dependent our parameterization for the curve. it's not dependent on the shape of the curve. The direction of this vector is dependent on the shape of this curve, and the direction, so the direction, this will be tangent to the curve. Or you could imagine that this vector is on the tangent line to the curve. The magnitude of it is a little bit hard to understand. I'll try to give you a little bit of intuition on that in the next video. But this is what I want you to understand right now, because we're going to be able to use this in the future, when we do the line integral over vector-valued functions.
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