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What I want to do in the next few videos is try to see what

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happens to a line integral, either a line integral over a

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scalar field or a vector field, but what happens that line

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integral when we change the direction of our path?

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So let's say, when I say change direction, let's say that

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I have some curve C that looks something like this.

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We draw the x- and y- axis.

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So that's my y-axis, that is my x-axis, and let's say my

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parameterization starts there, and then as t increases, ends

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up over there just like that.

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So it's moving in that direction.

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And when I say I reverse the path, we could

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define another curve.

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Let's call it minus C, that looks something like this.

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That is my y-axis, that is my x-axis.

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And it looks exactly the same, but it starts up here, and then

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as t increases, it goes down to the starting point

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of the other curve.

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So it's the exact same shape of a curve, but it goes in

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the opposite direction.

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So what I'm going to do in this video is just understand how we

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can construct a parameterization like this, and

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hopefully understand it pretty well.

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And then next two videos after this, we'll try to see what

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this actually does to the line integral, one for a scalar

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field, and then one for a vector field.

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So let's just say, this parameterization right here,

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let's just define it in the basic way that we've

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always defined them.

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Let's say that this is x is equal to x of t, y is equal to

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y of t, and let's say this is from t is equal, or t,

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let me write this way.

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t starts at a, so t is greater than or equal to

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a, and it goes up to b.

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So in this example, this was when t is equal to a, and the

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point right here is the coordinate x of a, y of a.

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And then when t is equal to b up here, this is really just a

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review of what we've seen before, really just a review of

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parameterization, when t is equal to b up here, this is

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the point x of b, y of b.

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Nothing new there.

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Now given these functions, how can we construct another

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parameterization here that has the same shape, but

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that starts here?

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So I want this to be, t is equal to a.

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Let me switch colors.

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Let me switch to, maybe, magenta.

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So I want this to be t is equal to a, and as t increases, I

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want this to be t equals b.

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So I want to move in the opposite direction.

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So when t is equal to a, I want my coordinate to

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still be x of b, y of b.

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When t is equal to a, I want a b in each of these functions,

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and when t is equal to b, I want the coordinate to

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be x of a, y of a.

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Right?

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Notice, they're opposites now.

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Here t is equal to a, x of a, y of a, here t is

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equal to b, our endpoint.

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Now I'm at this coordinate, x of a, y of a.

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So how do I construct that?

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Well, if you think about it, when t is equal to a, we want

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both of these functions to evaluate it at b.

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So what if we define our x, in this case, for our minus C

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curve, what if we say x is equal to x of, and when I say x

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of I'm talking about the same exact function.

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Actually, maybe I should write it in that same exact color.

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x of-- but instead of putting t in there, instead of putting a

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straight-up t in there, what if I put an a plus b

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minus t in there?

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What happens?

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Well, let me do it for the y as well.

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So then our y, y, is equal to y of a plus b minus t.

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a plus b minus is t.

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I'm using slightly different shades of yellow, might be

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a little disconcerting.

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Anyway, what happens when we define this?

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When t is equal to a, when t is equal to a, let's say that this

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parameterization is also for t starts at a and

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then goes up to b.

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So let's just experiment and confirm that this

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parameterization really is the same thing as this thing,

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but it goes in an opposite direction.

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Or at least, confirm in our minds intuitively.

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So when t is equal to a, when t is equal to a, x will be equal

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to x of a plus b minus a, right?

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This is when t is equal to a, so minus t, or minus a,

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which is equal to what?

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Well, a minus a, cancel out, that's equal to x of b.

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Similarly, when t is equal to a, y will be equal to

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y of a plus b minus a.

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The a's cancel out, so it's equal to y of b.

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So that worked.

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When t is equal to a, my parameterization evaluates to

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the coordinate x of b, y of b.

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When t is equal to a, x of b, y of b.

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Then we can do the exact same thing when t is equal to b.

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I'll do it over here, because I don't want to lose this.

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Let me just draw a line here.

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I'm still dealing with this parameterization over here.

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Actually, let me scroll over to the right, just so that

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I don't get confused.

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When t is equal to b, when t is equal to b, what does x

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equal? x is equal to x of a plus b minus b, right?

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a plus b minus b when t is equal to b.

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So that's equal to x of a.

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and then when she's able to be why is equal to lie of a plus b

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minus b, and of course, that's going to be equal to y of a.

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So the endpoints work, and if you think about it intuitively,

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as t increases, so when t is at a, this thing is going

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to be x of b, y of b.

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We saw that down here.

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Now as t increases, this value is going to decrease.

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We started x of b, y of b, and as t increases, this value is

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going to decrease to a, right?

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It starts from b, and it goes to a.

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This one obviously starts at a, and it goes to b.

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So hopefully, that should give you the intuition why this is

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the exact same curve as that.

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It just goes in a completely opposite direction.

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Now, with that out of the way, if you accept what I've told

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you, that these are really the same parameterizations,

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just opposite directions.

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I shouldn't say same parameterizations.

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Same curve going in an opposite direction, or same path going

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in the opposite direction.

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In the next video, I'm going to see what happens when we

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evaluate this line integral, f of x ds, versus this

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line integral.

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So this is a scalar field, a line integral of a scalar

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field, using this curve or this path, but what happens if we

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take a line integral over the same scalar field, but we do

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it over this reverse path?

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That's what we're going to do in the next video.

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And the video after that, we'll do it for vector fields.

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