In the last couple of videos we saw that we can describe

a curves by a position vector-valued function.

And in very general terms, it would be the x position as a

function of time times the unit vector in the

horizontal direction.

Plus the y position as a function of time times

the unit victor in the vertical direction.

And this will essentially describe this-- though, if you

can imagine a particle and let's say the parameter

t represents time.

It'll describe where the particle is at any given time.

And if we wanted a particular curve we can say, well, this

only applies for some curve-- we're dealing, it's r of t.

And it's only applicable between t being greater

than a and less than b.

And you know, that would describe some curve

in two dimensions.

Just me just draw it here.

This is all a review of really, the last two videos.

So this curve, it might look something like that where this

is where t is equal to a.

That's where t is equal to b.

And so r of a will be this vector right here that

ends at that point.

And then as t or if you can imagine the parameter being

time, it doesn't have to be time, but that's a convenient

one to visualize.

Each corresponding as t gets larger and larger, we're just

going to different-- we're specifying different

points on the path.

We saw that two videos ago.

And in the last video we thought about, well, what does

it mean to take the derivative of a vector-valued function?

And we came up with this idea that-- and it wasn't an idea,

we actually showed it to be true.

We came up with a definition really.

That the derivative-- I could call it r prime of t-- and

it's going to be a vector.

The derivative of a vector-valued function is once

again going to be a derivative.

But it was equal to-- the way we defined it-- x prime of t

times i plus y prime of t times j.

Or another way to write that and I'll just write all

the different ways just so you get familiar with--

dr/dt is equal to dx/dt.

This is just a standard derivative.

x of t is a scalar function.

So this is a standard derivative times i

plus dy/dt times j.

And if we wanted to think about the differential, one thing

that we can think about-- and whenever I do the math for

the differential it's a little bit hand wavy.

I'm not being very rigorous.

But if you imagine multiplying both sides of the equation by a

very small dt or this exact dt, you would get dr is equal to--

I'll just leave it like this.

dx/dt times dt.

I could make these cancel out, but I'll just write

it like this first.

Times the unit vector i plus dy/dt times dt.

Times the unit vector j.

Or we could rewrite this.

And I'm just rewriting it in all of the different ways

that one can rewrite it.

You could also write this as dr is equal to x prime of t dt

times the unit vector i.

So this was x prime of t dt.

This is x prime of t right there times the unit vector i.

Plus y prime of t.

That's just that right there.

Times dt.

Times the unit vector j.

And just to, I guess, complete the trifecta, the other way

that we could write this is that dr is equal to-- if we

just allowed these to cancel out, then we get is equal

to dx times i plus dy times dy y times j.

And that actually makes a lot of intuitive sense.

That if I look at any dr, so let's say I look at

the change between this vector and this vector.

Let's say the super small change right there, that is our

dr, and it's made up of-- it's our dx, our change in x

is that right there.

You can imagine it's that right there times-- but we're

vectorizing it by multiplying it by the unit vector in

the horizontal direction.

Plus dy times the unit vector in the vertical direction.

So when you multiply this distance times the unit

vector, you're essentially getting this vector.

And when you multiply this guy-- and actually our change

in y here is negative-- you're going to get

this vector right here.

So when you add those together you'll get your change in

your actual position vector.

So that was all a little bit of background.

And this might be somewhat useful-- a future

video from now.

Actually, I'm going to leave it there because really I just

wanted to introduce this notation and get you

familiar with it.

In the next video, what I'm going to do is give you a

little bit more intuition for what exactly does

this thing mean?

And how does it change depending on different

parameterizations.

And I'll do it with two different parameterizations

for the same curve.