All of the work we've been doing so far with line
integrals has been with scalar functions or
scalar-valued functions.
And when I say that, that just means you give me an x and a y
and you evaluate the function at that x and y, and you
get another scalar value.
You get just a number.
You don't get a vector.
So what I want to do in this video is start to get ourselves
warmed up with regards to vectors so that we can
understand what it means to take a line integral with
vector-valued functions.
So let me write down some vector-valued functions.
Actually, even a better place to start, let me draw a curve
or let me describe a curve.
So let's put that little f of x, y to the side.
We can ignore it for now.
Let's say I have some curve c and it's described, it can be
parameterized-- I can't say that word-- as let's say, x is
equal to x of t, y is equal to some function y of t.
And let's say that this is valid for t is between a and b.
So t is greater than or equal to a and then,
less than or equal to b.
So if I were to just draw this on-- let me see-- I
could draw it like this.
I'm staying very abstract right now.
This is not a very specific example.
This is the x-axis.
This is the y-axis.
My curve-- let's say this is when t is equal to a.
And then the curve might do something like this.
I don't know what it does.
Let's say it's over there.
This is t is equal to b.
This actual point right here will be x of b.
That would be the x-coordinate.
You evaluate this function at b and y of b.
And this is, of course, when t is equal to a.
The actual coordinate in r2 on the Cartesian coordinates will
be x of a, which is this right here.
And then, y of a, which is that right there.
And we've seen that before.
That's just a standard way of describing a parametric
equation or curve using 2 parametric equations.
What I want to do now is describe this same exact curve
using a vector-valued function.
So if I define a vector-valued function-- and if you don't
remember what those are, we'll have a little bit
of review here.
Let me say I have a vector-valued function, r,
and I'll put a little vector arrow on top of it.
And a lot of textbooks, they'll just bold it and they'll
leave scalar-valued functions unbolded.
But it's hard to draw bold, so I'll put a little
vector on top.
And let's say that r is a function of t.
And these are going to be position vectors.
And I'm specifying that because, in general, when
someone talks about a vector, this vector and this vector
are considered equivalent.
As long as they have the same magnitude and direction, no one
really cares about what their start and end points are as
long as their direction's the same and their
length is the same.
But when you talk about position vectors you're saying
no, these vectors are all going to start at 0, at the origin.
And when you say it's a position vector, you're
implicitly saying this is specifying a unique position.
In this case, it's going to be in two-dimensional
space, but it could be in three-dimensional space.
Or really, even four, five, whatever-- n dimensional space.
So when you say it's a position vector, you're literally
saying, OK, this vector literally specifies
that point in space.
So let's see if we can describe this curve as a position
vector-valued function.
So we could say r of t.
Let me switch back to that pink color.
This can stay in green.
Is equal to x of t times the unit vector in the x direction.
The unit vector gets a little caret on top-- a little hat.
That's like the arrow for it.
That just says it's a unit vector.
Plus y of t times j.
If I was dealing with a curve in three dimensions I would
have plus z of t times k.
But we're dealing with two dimensions right here.
And so the way this works is you're just taking your-- well,
for any t and still, we're going to have t is greater
than or equal to a and then, less than or equal to b.
And this is the exact same thing as that.
Let me just redraw it.
So let me draw our coordinates.
Our coordinates right here, our axes.
So that's the y-axis and this is the x-axis.
So when you evaluate r of a, that's our starting point.
So let me do that.
So r of a-- maybe I'll do it right over here.
Our position vector-valued function evaluated at t is
equal to a, is going to be equal to x of a times our unit
vector in the x direction.
Plus y of a times our unit vector in the vertical
direction, or in the y direction.
And what's that going to look like?
Well, x of a is this thing right here, so it's x of
a times a unit vector.
You know, maybe the unit vector is this long.
It has length 1, so now we're just going to have a length
of x of a in that direction.
And then, same thing in y of a.
It's going to be y of a length in that direction.
But the bottom line, this vector right here-- if you add
these scaled values of these two unit vectors, you're going
to get r of a looking something like this.
it's going to be a vector that looks something like that.
Just like that.
It's a position vector.
That's why we're nailing it at the origin, but drawing
it in standard position.
And that right there is r of a.
Now what happens if a increases a little bit?
What is r of a plus a little bit?
And I don't know, we could call that r of a plus
delta or r of a plus h.
I'll do it in a different color.
Let's say we increase a a little bit. r of a
plus some small h.
Well, that's just going to be x of a plus h times
a unit vector i.
Plus y times a plus h times the unit vector j.
And what's that going to look like?
Well, we're going to go a little bit further
down the curve.
That's like saying the coordinate x of a plus
h and y plus a plus h.
I might be that point right there.
So it'll be a new unit vector.
Sorry, it'll be a new vector-- position vector--
not a unit vector.
These don't necessarily have length 1.
That might be right here.
Let me do that same color as this.
So it might be just like that.
So that right here is r of a plus h.
So you see, as you keep increasing you value of t until
you get to b, these position vectors-- we're going to keep
specifying points along this curve.
So the curve-- let me draw the curve in a different color.
The curve looks something like this.
It's meant to look exactly like the curve that I have up here.
And for example, r of b is going to be a vector
that looks like this.
It's going to be a vector that looks like that.
I want to draw it relatively straight.
That vector right there is r of b.
So hopefully you realize that, look, these position vectors
really are specifying the same points on this curve as this
original, I guess, straight up parameterization that
we did for this curve.
And I just wanted to that as a little bit of review because
we're now going to break in into the idea of actually
taking a derivative of this vector-valued function.
And I'll do that in the next video.