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Studying for a test? Prepare with these 13 lessons on Integrating multivariable functions.
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In the last video, we saw that if we had some curve in the x-y plane, and we just parameterize it in a very general sense like this, we could generate another parameterization that essentially is the same curve, but goes in the opposite direction. It starts here and it goes here, as t goes from a to b, as opposed to the first parameterization, we started with t equals a over here, and it went up like that. And the question I want to answer in this video is how a line integral of a scalar field over this curve, so this is my scalar field, it's a function of x and y, how a line integral over a scalar field over this curve relates to, that's a line integral of that same scalar field over the reverse curve, over the curve going in the other direction. So the question is, does it even matter whether we move in this direction or that direction when we're taking the line integral of a scalar field? And in the next video, we'll talk about whether it matters on a vector field. And let's see if we can get a little intuition to our answer before we even prove our answer. So let me draw a little diagram, here. Actually, let me do it a little bit lower, because I think I'm going to need a little bit more real estate. So let me draw the y-axis, that is the x-axis, let me draw the vertical axis, just like that, that is z. Let me draw a scalar field, here. So I'll just draw it as some surface, I'll draw part of it. That is my scalar field, that is f of xy right there. For any point on the x-y plane we can associate a height that defines this surface, this scalar field. And let me put a curve down there. So let's say that this is the curve c, just like that. And the way we define it first, we start over here and we move in that direction. That was our curve c. And we know from several videos ago that the way to visualize what this line integral means, is we're essentially trying to figure out the area of a curtain that has this curve as its base, and its ceiling is defined by this surface, by the scalar field. So we're literally just trying to find the area of this curvy piece of paper, or wall, or whatever you want to view it. That's what this thing is. Now, if we take the same integral but we take it the reverse curve, instead of going in that direction, we're now going in the opposite direction. We're not taking a curve, we're going from the top to the bottom. But the idea is still the same. You know, I don't know which one is c, which one is minus c. I could have defined this path going from that way as c, and then the minus c path would have started here, and gone back up. So it seems in either case, no matter what I'm doing, I'm going to try to figure out the area of this curved piece of paper. So my intuition tells me that the either these are going to give me the area of this curved piece of paper, so maybe they should be equal to each other. I haven't proved anything very rigorously yet, but it seems that they should be equal to each other, right? In this case, let's say I'm taking a, let me just make it very clear. I'm taking a ds. a little change in distance, let me do it in a different color. A little change in distance, and I'm multiplying it by the height, to find kind of a differential of the area. And I'm going to add a bunch of these together to get the whole area. Here I'm doing the same thing. I'm taking a little ds, and remember, the ds is always going to be positive, the way we've parameterized it. So here, too, we're taking a ds, and we're going to multiply it by the height. So once again, we should take the area. And I want to actually differentiate that relative to, when you take a normal integral from a to b of, say, f of x dx, we know that when we switch the boundaries of the integration, that it makes the integral negative. That equals the negative of the integral from b to a of f of x dx. And the reason why this is the case, is if you imagine this is a, this is b, that is my f of x. When you do it this way, your dx's are always going to be positive. When you go in that direction, your dx's are always going to be positive, right? Each increment, the right boundary is going to be higher than the left boundary. So your dx's are positive. In this situation, your dx's are negative. The heights are always going to be the same, they're always going to be f of x, but here your change in x is a negative change in x, when you go from b to a. And that's why you get a negative integral. In either case here, our path changes, but our ds's are going to be positive. And the way I've drawn this surface, it's above the x-y plane, the f of xy is also going to be positive. So that also kind of gives the same intuition that this should be the exact same area. But let's prove it to ourselves. So let's start off with our first parameterization, just like we did in the last video. We have x is equal to x of t, y is equal to y of t, and we're dealing with this from, t goes from a to b. And we know we're going to need the derivatives of these, so let write that down right now. We can write dx dt is equal to x prime of t, and dy dt, let me write that a little bit neater, dy dt is equal to y prime of t. This is nothing groundbreaking I've done so far. But we know the integral over c of f of xy. f is a scalar field, not a vector field. ds is equal to the integral from t is equal to a, to t is equal to b of f of x of t y of t times the square root of dx dt squared, which is the same thing as x prime of t squared, plus dy dt squared, the same thing as y prime of t squared. All that under the radical, times dt. This integral is exactly that, given this parameterization. Now let's do the minus c version. I'll do that in this orange color. Actually, let me do the minus c version down here. The minus the c version, we have x is equal to, you remember this, actually, just from up here, this was from the last video. x is equal to x of a plus b minus t. y is equal to y of a plus b minus t. And then t goes from a to b, t goes from a to b, and this is just exactly what we did in that last video. x is equal to x of a plus b minus t, y is equal to y of a plus b minus t, same curve, just going in a different direction as t increases a to b. But let's get the derivative. I'll do it in the derivative color, maybe. So dx dt. For this path, it's going to be a little different. We have to do the chain rule now. Derivative of the inside with respect to t. Well, these are constants. Derivative of minus t with respect to t is minus 1. So it's minus 1 times the derivative of the outside with respect to the inside. Well, that's just x prime of a plus b minus t. Or, we could rewrite this as, this is just minus x prime of a plus b minus t. dy dt, same logic. Derivative of the inside is minus 1 with respect to t, right? Derivative minus t is just minus 1. Times the derivative of the outside with respect to the inside. So y prime of a plus b minus t, same thing as minus y prime a plus b minus t. So given all of that, what is this integral going to be equal to, the integral of minus c of the scalar field f of xy ds? What is this going to be equal to? Well, it's going to be the integral from, you could almost pattern match it. t is equal to a to t is equal to be of f of x. But now x is no longer x of t. x now equals x of a plus b minus t. It's a little bit hairy, but I don't think anything here is groundbreaking. Hopefully it's not too confusing. And once again, y is no longer y of t. y is y of a plus b minus t. And then times a square root, I'll just switch colors, times the square root of dx dt squared. What is dx dt squared? dx dt squared is just this thing squared, or this thing squared. This thing, if I have minus anything squared, that's the same thing as anything squared, right? This is equal to minus x prime of a plus b minus t squared, which is the same thing is just x prime of a plus b minus t squared, right? You lose that minus information when you square it. So that's going to be equal to x prime of a plus b minus t squared, the whole result function squared, plus dy dt squared. By the same logic, that's going to be, you lose the negative when you square it. y prime of a plus b minus t squared. Let me extend the radical. And then all of that dt. So that's the line integral over the curve c, this is the line integral over the curve minus c. They don't look equal just yet. This looks a lot more convoluted than that one does. So let's see if we can simplify it little bit. And we can simplify it, perhaps, by making a substitution. Let's let, let me get a nice substitution color, let's let u equal to a plus b minus t. So first we're going to have to figure out the boundaries of our integral, well actually, let's just figure out, what's du? so du dt, the derivative of u with respect to t is just going to be equal to minus 1, or we could say that du, if we multiply both sides by the differential dt, is equal to minus dt. And let's figure out our boundaries of integration. When t is equal to a, what is u equal to? u is equal to a plus b minus a, which is equal to b. And then when t is equal to b, u is equal to a plus b minus b, which is equal to a. So if we do the substitution on this crazy, hairy-looking interval, let's simplify a little bit, and it changes our-- so this integral is going to be the same thing as the integral from u, when t is a, u is b. When t is b, u is a. And f of, x of, this thing right here is just u. x of u. So it simplified it a good bit. And y of, this thing right here, is just u. y of u. Times the square root-- let me do it in the same color. Times the square root of x prime of u squared plus y prime of u squared. Instead of a dt, we have to write a, or could write, if we multiply both sides of this by minus, we have dt is equal to minus du. So instead of a dt, we have to put a minus du here. So this is times minus du, or, just so we don't think this is a subtraction, let's just put that negative sign out here in the front, just like that. So we're going from b to a of this thing, right, like that. And just to make the boundaries of integration make a little bit more sense, because we know that a is less than b, let's swap them. And I said at the beginning of this video, for just a standard, regular, run of the mill integral, if you swap, if you have something going from b to a of f of x dx, or du, maybe I should write it this way. f of u du. This is equal to the minus of the integral from a to b of f of u du. And we did that by the logic that I had graphed up here. That here, when you switch the order, your du's will become the negatives of each other, when you actually visualize it, when you're actually finding the area under the curve. So let's do that. Let's swap the boundaries of integration right here. And if we do that, that will negate this negative, or make it a positive. So this is going to be equal to the integral from a to b. I'm dropping the negative sign, because I swapped these two things. So I'm going to take the negative of a negative, which is a positive. Of f of x of u y of u times the square root of x prime of u squared plus y prime of u squared du. Now remember, everything we just did was a substitution. This was all equal to, just to remember what we're doing, this was the integral of the minus curve of our scalar field, f of xy ds. Now how does this compare to when we take the regular curve? How does this compare to that? Let me copy and paste it to see. You know, I'm using the wrong tool. Let me copy and paste it to see how they compare. Copy, then let me pick to down here, edit, paste. So how do these two things compare? Let's take a close look. Well, they actually look pretty similar, right? Over here, for the minus curve, we have a bunch of u's. Over here, for the positive curve, we have a bunch of t's, but they're in the exact same places. These integrals are the exact same integrals. If you make a u-substitution here, if you just make the substitution u is equal to t, this thing is going to be integral from a to b of, it's going to be the exact same thing. Of f of x of u, y of u times the square root of of x prime of u squared plus y prime of u squared du. These two things are identical. So we did all the substitution, everything, but we got the exact same integrals. So hopefully that satisfies you that it doesn't matter what direction we go on the curve, as long as the shape of the curve is the same. Doesn't matter if we go forward or backward on the curve, we're going to get the same answer. And I think that meets our intuition, because in either case, we're finding the area of this curtain.