Parametrization of a Reverse Path Understanding how to parametrize a reverse path for the same curve.
Parametrization of a Reverse Path
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- What I want to do in the next few videos is try to see what
- happens to a line integral, either a line integral over a
- scalar field or a vector field, but what happens that line
- integral when we change the direction of our path?
- So let's say, when I say change direction, let's say that
- I have some curve C that looks something like this.
- We draw the x- and y- axis.
- So that's my y-axis, that is my x-axis, and let's say my
- parameterization starts there, and then as t increases, ends
- up over there just like that.
- So it's moving in that direction.
- And when I say I reverse the path, we could
- define another curve.
- Let's call it minus C, that looks something like this.
- That is my y-axis, that is my x-axis.
- And it looks exactly the same, but it starts up here, and then
- as t increases, it goes down to the starting point
- of the other curve.
- So it's the exact same shape of a curve, but it goes in
- the opposite direction.
- So what I'm going to do in this video is just understand how we
- can construct a parameterization like this, and
- hopefully understand it pretty well.
- And then next two videos after this, we'll try to see what
- this actually does to the line integral, one for a scalar
- field, and then one for a vector field.
- So let's just say, this parameterization right here,
- let's just define it in the basic way that we've
- always defined them.
- Let's say that this is x is equal to x of t, y is equal to
- y of t, and let's say this is from t is equal, or t,
- let me write this way.
- t starts at a, so t is greater than or equal to
- a, and it goes up to b.
- So in this example, this was when t is equal to a, and the
- point right here is the coordinate x of a, y of a.
- And then when t is equal to b up here, this is really just a
- review of what we've seen before, really just a review of
- parameterization, when t is equal to b up here, this is
- the point x of b, y of b.
- Nothing new there.
- Now given these functions, how can we construct another
- parameterization here that has the same shape, but
- that starts here?
- So I want this to be, t is equal to a.
- Let me switch colors.
- Let me switch to, maybe, magenta.
- So I want this to be t is equal to a, and as t increases, I
- want this to be t equals b.
- So I want to move in the opposite direction.
- So when t is equal to a, I want my coordinate to
- still be x of b, y of b.
- When t is equal to a, I want a b in each of these functions,
- and when t is equal to b, I want the coordinate to
- be x of a, y of a.
- Notice, they're opposites now.
- Here t is equal to a, x of a, y of a, here t is
- equal to b, our endpoint.
- Now I'm at this coordinate, x of a, y of a.
- So how do I construct that?
- Well, if you think about it, when t is equal to a, we want
- both of these functions to evaluate it at b.
- So what if we define our x, in this case, for our minus C
- curve, what if we say x is equal to x of, and when I say x
- of I'm talking about the same exact function.
- Actually, maybe I should write it in that same exact color.
- x of-- but instead of putting t in there, instead of putting a
- straight-up t in there, what if I put an a plus b
- minus t in there?
- What happens?
- Well, let me do it for the y as well.
- So then our y, y, is equal to y of a plus b minus t.
- a plus b minus is t.
- I'm using slightly different shades of yellow, might be
- a little disconcerting.
- Anyway, what happens when we define this?
- When t is equal to a, when t is equal to a, let's say that this
- parameterization is also for t starts at a and
- then goes up to b.
- So let's just experiment and confirm that this
- parameterization really is the same thing as this thing,
- but it goes in an opposite direction.
- Or at least, confirm in our minds intuitively.
- So when t is equal to a, when t is equal to a, x will be equal
- to x of a plus b minus a, right?
- This is when t is equal to a, so minus t, or minus a,
- which is equal to what?
- Well, a minus a, cancel out, that's equal to x of b.
- Similarly, when t is equal to a, y will be equal to
- y of a plus b minus a.
- The a's cancel out, so it's equal to y of b.
- So that worked.
- When t is equal to a, my parameterization evaluates to
- the coordinate x of b, y of b.
- When t is equal to a, x of b, y of b.
- Then we can do the exact same thing when t is equal to b.
- I'll do it over here, because I don't want to lose this.
- Let me just draw a line here.
- I'm still dealing with this parameterization over here.
- Actually, let me scroll over to the right, just so that
- I don't get confused.
- When t is equal to b, when t is equal to b, what does x
- equal? x is equal to x of a plus b minus b, right?
- a plus b minus b when t is equal to b.
- So that's equal to x of a.
- and then when she's able to be why is equal to lie of a plus b
- minus b, and of course, that's going to be equal to y of a.
- So the endpoints work, and if you think about it intuitively,
- as t increases, so when t is at a, this thing is going
- to be x of b, y of b.
- We saw that down here.
- Now as t increases, this value is going to decrease.
- We started x of b, y of b, and as t increases, this value is
- going to decrease to a, right?
- It starts from b, and it goes to a.
- This one obviously starts at a, and it goes to b.
- So hopefully, that should give you the intuition why this is
- the exact same curve as that.
- It just goes in a completely opposite direction.
- Now, with that out of the way, if you accept what I've told
- you, that these are really the same parameterizations,
- just opposite directions.
- I shouldn't say same parameterizations.
- Same curve going in an opposite direction, or same path going
- in the opposite direction.
- In the next video, I'm going to see what happens when we
- evaluate this line integral, f of x ds, versus this
- line integral.
- So this is a scalar field, a line integral of a scalar
- field, using this curve or this path, but what happens if we
- take a line integral over the same scalar field, but we do
- it over this reverse path?
- That's what we're going to do in the next video.
- And the video after that, we'll do it for vector fields.
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