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- [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.