Hello.

In the last session of curvature factors account We have seen ways.

With a clear representation of the curvature function y is equal to that fx

x in the function of As we can show

we can show as a parametric 've seen.

So we found two different ways.

The curves in this plane but for in the plane

I repeat again, the best curves we will be able to understand the concepts in space.

It is possible to understand, but also directly in space E, I thoroughly

to internalize this useful reminder.

Now we'll see a third way and in space this third

the road will be valid generalization.

Now we have a tangent to the curve the truth

When we draw it fx y is equal to the slope of the was given.

y is the base for the tangent.

We also know that since middle school.

Derived slope's.

Here we can obtain a vector.

When we get a unit horizontal vertical the tangent

As for when we go here u feature vector.

But this is not one length of the vector.

Thus from the vector e, units

vector to go to this length, We need to partition.

As you can see, E, root comes below the length.

If we look at it as a geometric e, and y The following base year equal to the tangent of the base.

BI in the denominator is the length.

One plus tangent frame fi.

Because the base year equal to the tangent of the plug.

When we look at it this denominator Let us remember again.

One plus frames for the tangent there.

This is a split cosine square fi.

So up, share a cosine fi income.

For when we hit the cosine of the cosine fi fi here and sinus occurs.

This geometrically we can find bi Şiyar.

Because we move the x direction cosine fi time

we will proceed in the y direction for sinus you saying this.

As long as it still Pythagorean theorem sine cosine square frame.

Where a is the length of a we see.

These two vectors, it is also a unit vector u T We show through.

This geometric and algebraic representations like this.

These two vector, the vector perpendicular to the second There vector.

We call this the orthogonal vectors.

This is a vector perpendicular to T in the perpendicular vector.

We define it like this:This is where

As shown in this way the normal vector You can also choose direction.

E, we can choose it in the opposite direction.

Among them the adoption of this decision got to give.

Both could not but indecision.

This content is bowing in the direction of N were selected.

Tion of describing it, the way k T get vector multiplication.

This is a vector perpendicular to T as we know it will give.

N as we want.

We can do this kind of account.

Since T is for sine cosine for this vector

old when he calculated by multiplying the sine fu cosine fi income.

Or when we look in terms of algebraic

With this one year minus the base year income base Get multiplication of vectors.

We are now able to provide bi.

We claim that N together T'yl steep.

Really be multiplied by the cosine of minus fi times

sinus to the first two, the first component the product comes from.

The second component of the product of the same size but

with a plus sign, this time really zero giving.

Again, this is similar algebraic representation to me, when we get N T times

base year plus to minus y, y, again as a base gives zero.

You can ask the question:Why ourselves Besides yoral at bi N I T Let?

This has two meanings.

How is it with unit vectors i and j point

We tanımlıyabiliyo position wherein T and N acts are the same.

And moreover, this curve better ta, They included the possibility to define.

Because if you know the point T and N T curve E,

E, N also what gives the tangential direction toward the side that gives curl.

Because such a right tangent of this

toward the side opposite to a bendable can be curled.

N is also for us which side curls shows.

This bi them from the application of a along the trajectory to calculate acceleration.

Momentum in the direction of a tangent to the tangent at b

that the centrifugal acceleration in the direction perpendicular to it we say.

E, there are components.

E, they are also in real life things encountered.

For example, in amusement park death train called rails

b out of the train when suddenly comes down.

Meanwhile, our house would be a little bad.

But that's going a little fun in the have no interest in them.

This is what makes our inner bad, modeled either car in a curve

Since the time to enter, itilme Us

As the centrifugal force, gives impetus.

They say that what works in practice things.

Now another way

the relationship between this tangent and perpendicular vector We can examine.

Sine cosine fi fi t we said this.

Let s based on this derivative.

Look, when you get the derivative with respect to s N is happening.

We can not directly.

Because T as a function of s

not given as a function of f given.

Thus, indirectly, ie chained derivatives We take the derivative rules.

After the first plug derivative of f at According to s derivative.

As the first term but is kalıyor fi We know that split off from the DS.

By definition, we know.

Here, too, we find that close.

Therefore derivative by T s We can find just taking off.

Two encores, the second bi thing you know, we find.

The following derivative with respect to t we get See here

mA, minus sine cosine fi fi here is coming.

E, is the unit vector that we, We know that the perpendicular vector.

I once long one.

A sine squared plus cosine squared is the.

In addition to this, because they see that other T. both

When multiplied by e, we find zero.

So it's one, easily perpendicular vector we find.

T is the fie derivatives.

This geometric.

With a more algebraic way it functions As we arrived there a way second.

Let t multiplied with itself.

This is due to the inner product of T's neck be square.

T is a length.

A frame.

Thus T is T is a point.

T and bury it, e, h, according to the derivative of T dT Taking the first one

E derivative, dt ds split times T plus

e first, the second of the first times derivative.

But this may change as the inner product ds dt T will divide twice.

One derivative of the derivative at the right side is zero.

Therefore, the result is very interesting bi we find.

Two of course does not matter.

Here is simplified.

DS with T dt divided inner product is zero.

E, I've learned a lot already.

If the inner product of two vectors is zero is the angle between ninety degrees.

Because the length of the inner product means that it intermediate length cosine fi.

Means that the cosine fi is zero.

F is zero, the cosine of the two vectors be other shows.

So here we divided dt T DS We see that it is perpendicular.

But of these units is sized to E b, There's no guarantee.

For this reason, there will be general, dt divided ds in the direction of N

but in order to do unit divided by dt we divide the length of the DS.

When we divide dt ds inasmuch as it splits The size of N is that it closes.

Because here we saw that divide dt ds N, these dT I have seen for what it is divided by d.

Once we find immediately to cover that.

Here we see that the curvature to our unit We find the tangent vector.

Also, the unit vector perpendicular to the tangent We find from the vector.

That's us in space, space curves in will provide the opportunity to expand.

Now here's an example I'm doing.

This example perfectly clear in detail functionally

y is equal to the circle when given as f x We give here in this circle.

This circle also parametric representation We know.

Locate the unit tangent vector of this circle here, the derivatives

we want to take the curvature we can find.

Open because it will pass quickly function, la fine detail

We did and the curvature of the circle and a slash

a and the radius of curvature of the circle We have found that the radius.

This is consistent with our intuition as a result stated.

DS already, so the curvature of the arc divided by df

That length coming from derivative results of our own

Find intuitively consistent in what we are is chosen to be.

Parametric t of this circle

a point on the center combining

When we get the job right angle We found all of these formulas.

The denominator of the second derivative of the curvature of the first with derivatives

wherein it is obtained from the supplier We know.

That means that all the work to be done first derivatives account

to the second derivative to account for this from the x and y.

They also put in place.

Here, as the gene should be divided by the radius of curvature

so that the radius of curvature the

now is the radius of the circle we find.

The third way