- [Voiceover] In the last video I started to talk about

the formula for curvature.

Just to remind everyone of where we are

you imagine that you have some kind of curve in let's say

two dimensional space just for the sake of being simple.

Let's say this curve is parameterized by a function S of T.

So every number T corresponds to some point on the curve.

For the curvature you start thinking about

unit tangent vectors.

At every given point what does the unit

tangent vector look like?

The curvature itself which is denoted by this sort of

Greek letter Kappa

is gonna be the rate of change of those unit vectors

kind of how quick they're turning in direction

not with respect to the parameter t

but with respect to arc length d s.

What I mean by arc length here is

just a tiny step you can think

the size of a tiny step along the curve would be d s.

You're wondering, as you take a tiny step like that

does the unit tangent vector turn a a lot or

does it turn a little bit?

The little schematic that I said you might have in mind

is just a completely separate space

where for each one of these unit tangent vectors

you go ahead and put them in that space

saying okay so this one would look something like this

this one is pointed down and to the right so it would

look something like this.

This one is pointed very much down.

You're wondering basically as you take tiny little steps

of size d s what is this

change to the unit tangent vector

and that change is gonna be some kind of vector.

Because the curvature's really just a value

a number that we want all we care about is the size

of that vector.

The size of the change to the tangent vector

as you take a tiny step in d s.

Now this is pretty abstract right?

I've got these two completely separate things that

are not the original functions that you have to think about.

You have to think about this unit tangent vector function

and then you also have to think about this

notion of arc length.

The reason, by the way, that I'm using an s here

as well as here for the parameterized of the curve

is because they're actually quite related.

I'll get to that a little bit below.

To make it clear what this means I'm gonna go ahead

and go through an example here where let's say our

parameterized with respect to t is

a cosine sine pair.

So we've got cosine of t as the x component

and then sine of t as the y component

sine of t.

Just to make it so that it's not completely boring

let's multiply both of these components by constant r.

What this means, you might recognize this,

cosine sine pair

what this means

is that in the x y plane

you're actually drawing a circle with radius r.

This would be some kind of circle with the radius r.

While I go through this example I also want to make a note

of what things would look like a little bit more abstractly.

If we just had s of t equals not specific functions

that I laid down but just any general function

for the x component and for the y component.

The reason I want to do this is because the concrete

version is gonna be helpful and simple and something

we can deal with but almost so simple as to

not be indicative of just how complicated the normal

circumstances but the more general circumstances

so complicated I think it will actually confuse

things a little bit too much.

It'll be good to kind of go through

both of them in parallel.

The first step is to figure out what is this

unit tangent vector?

What is that function that at every given point

gives you a unit tangent vector to the curve?

The first thing for that

is to realize that

we already have a notion of what should give the

tangent vector

the derivative of our vector valued function

as a function of t

the direction in which in points is

in the tangent direction.

If I go over here and if I compute this derivative

and I say s prime of t

which involves just taking the

derivative of both components so the

derivative of cosine is negative sine

of t multiplied by r and the derivative

of sine is cosine

of t multiplied by r.

More abstractly, this is just gonna be

anytime you have two different component functions

you just take the derivative of each one.

Hopefully you've seen this, if not

maybe take a look of videos on

taking the derivative of a position vector valued function.

This we can interpret as that tangent vector

but it might not be a unit vector right?

We want a unit tangent vector and this

only promises us the direction.

What we do to normalize it

and get a unit tangent vector function

which I'll call capital t of lowercase t

that's kind of confusing right?

Capital t is for tangent vector

lowercase t is the parameter.

I'll try to keep that straight.

It's sort of standard notation but

there is the potential to confuse with this.

What that's gonna be is your vector value derivative

but normalized. So we have to divide by

whatever it's magnitude is

as a function of t.

In this case, in a specific example

that magnitude, if we take the magnitude of

negative sine of t r

multiply by r

and then cosine of t

multiplied by r

so we're taking the magnitude of this whole vector

what we get

make myself even more room here

is the square root

of sine squared

negative sine squared is just gonna be sine squared

so sine squared of t multiplied by r squared

and then over here cosine square times r squared

cosine squared of t times r squared

we can bring that r squared outside of the radical

to sort of factor it out

turning it into an r

and on the inside we have sine squared plus cosine squared

I'm being too lazy to write down the t's right now

because no matter what the t is that whole value

just equals one.

This entire thing is just gonna equal r.

What that means is that our unit tangent vector up here

is gonna be the original function but divided by r.

It happens to be a constant usually it's not

but it happens to be a constant in this case.

What that looks like

given that our original function is

negative sine of t times r and cosine of t times r

we're dividing out by an r

the ultimate function that we get

is just negative sine of t

and then cosine of t.

For fear of running a little bit long

I think I'll call it an end to this video and

continue on with the same argument in the next video.