That in our previous section, you

basically starting with curves on plane curvature

the concept of the unit and the unit tangent vector other

how the concepts of vectors and their I showed the account can be made.

Now we learn that here with this We will now space curves.

This transition is very easy.

A curve equation with one variable a vector function specified.

This two-component vector in the plane.

In space has x and y as well as a de z.

Therefore, a straightforward generalization.

Now how in the plane when i and j

space i, j k also had an addition, Gene in the curve

t and n is the unit tangent in the plane and perpendicular If vectors

a third vector as well as the would naturally emerge.

Similarly, curvature of a container in space had.

Sorry, the plane had a close curvature.

In space curvature of the window, as well as than a second curvature is required.

Consider a curve because it is long, space, plane

curled in one direction, but in the space of two direction may be curled.

Its curvature than a second is the future.

Gene accounts arc length or t we can do in terms of parameters.

With the function no longer able to show clear not in space

The third way that it is possible for curves will not.

Let's start with just remembered it again.

A beam in this plane on the curve had received.

We showed that the delta with s.

P is zero p is a length of said beam delta h said.

This is also a vector p is zero p delta

We are also components of the vector x delta x and The delta y.

Burr, it showed between the two lines We define the length.

This is in accordance with the Pythagorean theorem, this beam throughout the delta delta x squared plus y was of square.

From here we move to the infinitely small

d to achieve this by changing deltas We are.

A straightforward generalization of the space thing.

These x and y as well as z will also b.

Delta z ka, and will come here, but mainly We will try to end d'Progress.

d z frame the future.

S derivative of x by the unit tangent vector.

In two dimensions in this geometrically 've seen.

Is it really as a long Let's see.

Let T's size.

T is the length dx is the size of the DS.

The denominator is a numerical value for share

dx dx is the future value, but also the definition of has DS.

Therefore, the future will be a DSi DSi divide.

Therefore, the derivative of x is based on s, we received

When the unit tangent vector directly we find.

Unit E, find the perpendicular vector and curvature again for two

T as we did in size Let the product of their own.

This will be the square of the length of T.

M, T, we know that the length of a.

According to Gene s derivative of the nose divided by dt we get

the inner product of T with ds is zero we find.

So divide dt ds can be written as follows: He

a vector perpendicular, of course, a length thereof will not.

This, it says you're off.

So it's own length N dt Slashes DS is divided.

Because in the direction perpendicular to T dt ds divide.

Close the absolute value of the closing of this dt divide this,

Get the length of both sides of the equation by dt divide ds

Where n is the length of a cover for the dt DS divided by the absolute

income equal to the value e, trap.

Here you also find the sign off If we want to take this dt divided

If we multiply by N N N DS with the multiplication since there is a decrease.

Therefore, to cover the N dt divided ds'yl be multiplied.

See how easy it works with vectors.

Here is a little bi note.

Equivalent to T in terms of the geometry in the plane, perpendicular two

There are many vectors, but this one We seçiyo, which is the direction of curvature.

But when you consider that in the space of a e, towards this

right tangent vector T and infinitely many other has.

Here we take as dt ds divide this

only one of an infinite vectors As we have chosen.

This is a detail, but important.

Accounts crucial.

Now we have the T and N as well as a There is a third vector.

The third vector that is perpendicular to both bi

The simplest way of producing these vectors vector get multiplied.

How the vector multiplication of i and j Get k

If the vector product involved in T and N, B interests.

There are a special case in the plane.

Because T is in the plane, for the N

multiplying a vector perpendicular to the plane of the two will be.

M, the plane perpendicular to the vector k and only hard.

Thus have D but trivial in the plane.

Because not only Siya k and variable b.

How the i, j, k vectors perpendicular to each other a

oluşturuyos triple T, N, B also orthogonal ternary forms.

See it on as though a curve If these aspects of T, N perpendicular to them if so

again using the right hand rule to N T Remove a unit vector in the direction of B gone.

It's easy to keep in mind the T, N, B. so we can write.

M, N, multiplication of T'yl B.

N, B's product T.

This trigonometric we travel in the positive direction When the spouse, we get the essence thresholds.

Because if we go in the opposite direction from time to time ih reverse multiplication, are needed.

Or to T'yl T'yl N gives B's product.

B'yl multiplied by N to T, but all the old data marked.

Wherein the negative trigonometric direction we go.

We know this because this vector

When you change the order of the product minus mark the future.

Refer to the N T'yl B veriyo but gets hit here

change the order of n, T, gets hit We know the future of a B-minus.

These plus the triangle and trigonometric Or turning in the direction of negative trigonometric

easy when faced with this product a simple mechanism to remember.

Now the vector product N of B T'yl We know that.

Let's take the derivative of B by s.

See it here first derivative divided by dt

second plus the second times the first time derivative.

Once we close the DS N is divided by dt We know.

Now see here dt divided by N ds'yl there.

Let us divide N dt ds off instead.

N will be N times.

I will be the radius vector.

So the angle is zero.

The sequence of the vectors of a vector with it We have seen that the product is zero.

So the first fall term.

Because dt ds N divide the direction vector b.

He gets hit by a vector parallel to the it is zero for the angle,

In the intervening sine theta and vector multiplication this term is reduced because it comes open.

Now these remains.

Therefore dB divided by d, h T times dn ds divide happens.

N is divided by dN DS do not know, but a We know that.

Now here we have done in the same T.

If we make the N dn ds divide itself is steep.

Now as a result, the DS divides dn We do not know what it is.

But just know that it is perpendicular to N..

There is no component perpendicular to the direction N to n.

There are only T and B direction components.

Sigma unknown to them now as and Tom let's say.

See if we take it replaces The first term in this place fell.

There are times where T DN DS.

DSA DN T times.

E it has a T component of a component We know that.

But here's something very interesting right again is going on.

T and T have multiplied.

Vector multiplication.

This means that the first term is reduced to zero.

Back to the times T times B stays.

W e T times N.

If you remember T is equal to N times B located.

When using this feature B T times

Tour times N is negative, we find here.

T times B is negative trigonometric direction We're going.

As you can see here the old N B T times.

Tour times N minus times minus those involved.

Refer to divide it with ds dt Let's compare.

divided by dt ds, T found at the beginning.

Ie turning off N times, warping show.

dB divide DSi has a negative sign, but so that it does not matter now

opinion, this means that something in the direction of N this shows a curvature.

This is one of these, showing rotation.

This close to the curvature in the DS offspring dt he said.

I.e., the slope of the variation of T show.

DB Ds divide the change of the slope of the B show.

B T from a vector finally too not different in nature.

Places, but it's a different direction vector, a vector that.

This is the unit vector, this is the unit vector.

This shows a change in slope.

This is called the torsion.

Deliberately chosen to name something anyway.

Tortion in Western languages, in English, tortion this

is universally recognized as a term we have used the same term.

So the absolute value of the DS dB Split If we are going from here Tom.

This torsional we are also found.

Now it is possible to improve a little more.

I'm going to look at it in this interim I would recommend the latter, but it is very pleasant.

Mind, now we have found dt DS with ease.

We also found dB DS easily.

But we have to find dn DS.

DS dn details on this brew I take calculated

Tour times are T and B minus time off.

Three of them in a matrix, which we la time dt

ds, ds dN, dB DS as a column Let's write.

Let's write a column B. In TN.

See also close to zero in this matrix is zero it seems.

Matrix crashing What are we doing?

You're getting hit by rotating multiply, with the number of wives in front.

Here goes close with N.

According to this first term are sowing.

T minus off times see here first

T minus the first term with the term off times produces.

To'yl it comes to product B's produces.

However, the structure of this matrix is very nice bi construction.

There are fine symmetry.

Full antisymmetric.

See zeros on the diagonal there.

Here you have a close, here are close minus.

Here are Tom, here are minus Tom.

This is quite a symmetrical structure.

Easy to keep in mind.

In addition, often the case in mathematics is beautiful, elegant, simple

in a very deep sense of the expression carries.

See T and N, and B is at elections

how beautiful that is selected We see here.

Other things you choose to so elegant impasse.

So far it s our account

, namely the arc length of the 've done.

This account via the parameter t possible to perform.

Our next session of this account will do.

You can also look at this last session notes it

Try to figure out in this t'yl See the difference calculation and their

superior in the sense in which what's what My kavrayabilel in what sense is superior.

Until next time, goodbye.