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All of the work we've been doing so far with line

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integrals has been with scalar functions or

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scalar-valued functions.

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And when I say that, that just means you give me an x and a y

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and you evaluate the function at that x and y, and you

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get another scalar value.

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You get just a number.

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You don't get a vector.

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So what I want to do in this video is start to get ourselves

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warmed up with regards to vectors so that we can

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understand what it means to take a line integral with

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vector-valued functions.

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So let me write down some vector-valued functions.

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Actually, even a better place to start, let me draw a curve

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or let me describe a curve.

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So let's put that little f of x, y to the side.

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We can ignore it for now.

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Let's say I have some curve c and it's described, it can be

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parameterized-- I can't say that word-- as let's say, x is

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equal to x of t, y is equal to some function y of t.

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And let's say that this is valid for t is between a and b.

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So t is greater than or equal to a and then,

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less than or equal to b.

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So if I were to just draw this on-- let me see-- I

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could draw it like this.

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I'm staying very abstract right now.

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This is not a very specific example.

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This is the x-axis.

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This is the y-axis.

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My curve-- let's say this is when t is equal to a.

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And then the curve might do something like this.

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I don't know what it does.

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Let's say it's over there.

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This is t is equal to b.

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This actual point right here will be x of b.

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That would be the x-coordinate.

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You evaluate this function at b and y of b.

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And this is, of course, when t is equal to a.

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The actual coordinate in r2 on the Cartesian coordinates will

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be x of a, which is this right here.

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And then, y of a, which is that right there.

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And we've seen that before.

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That's just a standard way of describing a parametric

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equation or curve using 2 parametric equations.

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What I want to do now is describe this same exact curve

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using a vector-valued function.

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So if I define a vector-valued function-- and if you don't

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remember what those are, we'll have a little bit

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of review here.

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Let me say I have a vector-valued function, r,

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and I'll put a little vector arrow on top of it.

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And a lot of textbooks, they'll just bold it and they'll

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leave scalar-valued functions unbolded.

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But it's hard to draw bold, so I'll put a little

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vector on top.

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And let's say that r is a function of t.

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And these are going to be position vectors.

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And I'm specifying that because, in general, when

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someone talks about a vector, this vector and this vector

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are considered equivalent.

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As long as they have the same magnitude and direction, no one

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really cares about what their start and end points are as

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long as their direction's the same and their

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length is the same.

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But when you talk about position vectors you're saying

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no, these vectors are all going to start at 0, at the origin.

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And when you say it's a position vector, you're

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implicitly saying this is specifying a unique position.

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In this case, it's going to be in two-dimensional

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space, but it could be in three-dimensional space.

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Or really, even four, five, whatever-- n dimensional space.

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So when you say it's a position vector, you're literally

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saying, OK, this vector literally specifies

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that point in space.

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So let's see if we can describe this curve as a position

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vector-valued function.

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So we could say r of t.

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Let me switch back to that pink color.

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This can stay in green.

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Is equal to x of t times the unit vector in the x direction.

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The unit vector gets a little caret on top-- a little hat.

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That's like the arrow for it.

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That just says it's a unit vector.

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Plus y of t times j.

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If I was dealing with a curve in three dimensions I would

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have plus z of t times k.

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But we're dealing with two dimensions right here.

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And so the way this works is you're just taking your-- well,

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for any t and still, we're going to have t is greater

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than or equal to a and then, less than or equal to b.

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And this is the exact same thing as that.

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Let me just redraw it.

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So let me draw our coordinates.

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Our coordinates right here, our axes.

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So that's the y-axis and this is the x-axis.

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So when you evaluate r of a, that's our starting point.

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So let me do that.

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So r of a-- maybe I'll do it right over here.

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Our position vector-valued function evaluated at t is

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equal to a, is going to be equal to x of a times our unit

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vector in the x direction.

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Plus y of a times our unit vector in the vertical

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direction, or in the y direction.

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And what's that going to look like?

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Well, x of a is this thing right here, so it's x of

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a times a unit vector.

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You know, maybe the unit vector is this long.

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It has length 1, so now we're just going to have a length

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of x of a in that direction.

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And then, same thing in y of a.

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It's going to be y of a length in that direction.

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But the bottom line, this vector right here-- if you add

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these scaled values of these two unit vectors, you're going

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to get r of a looking something like this.

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it's going to be a vector that looks something like that.

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Just like that.

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It's a position vector.

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That's why we're nailing it at the origin, but drawing

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it in standard position.

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And that right there is r of a.

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Now what happens if a increases a little bit?

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What is r of a plus a little bit?

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And I don't know, we could call that r of a plus

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delta or r of a plus h.

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I'll do it in a different color.

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Let's say we increase a a little bit. r of a

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plus some small h.

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Well, that's just going to be x of a plus h times

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a unit vector i.

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Plus y times a plus h times the unit vector j.

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And what's that going to look like?

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Well, we're going to go a little bit further

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down the curve.

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That's like saying the coordinate x of a plus

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h and y plus a plus h.

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I might be that point right there.

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So it'll be a new unit vector.

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Sorry, it'll be a new vector-- position vector--

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not a unit vector.

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These don't necessarily have length 1.

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That might be right here.

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Let me do that same color as this.

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So it might be just like that.

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So that right here is r of a plus h.

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So you see, as you keep increasing you value of t until

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you get to b, these position vectors-- we're going to keep

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specifying points along this curve.

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So the curve-- let me draw the curve in a different color.

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The curve looks something like this.

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It's meant to look exactly like the curve that I have up here.

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And for example, r of b is going to be a vector

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that looks like this.

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It's going to be a vector that looks like that.

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I want to draw it relatively straight.

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That vector right there is r of b.

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So hopefully you realize that, look, these position vectors

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really are specifying the same points on this curve as this

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original, I guess, straight up parameterization that

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we did for this curve.

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And I just wanted to that as a little bit of review because

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we're now going to break in into the idea of actually

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taking a derivative of this vector-valued function.

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And I'll do that in the next video.

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