Hello.

The simplest vector B in our previous I have seen the function.

This is the most basic right of a vector function Was equation.

The equation of a straight line in space.

Does y equals x plus the plane he We know since middle school.

The easiest way to show the vector in space.

I have seen it.

Furthermore, in space at the simplest surfaces 've seen.

These planes.

In this first part, these vectors Inc., following the

In the first part of the non-linear vector functions we'll see.

They will give us the space curves.

In this space in the next section surfaces, curve,

which is different from the plane surfaces of league We'll see.

Of course, as in everything a little we also need preparation.

Here I just say that x t a

but now it is a linear function not function.

In terms of components, we can do it.

When there is only one variable only independent

geometrically variable latitude degree means.

It can proceed on a curve t

but this is due to the outward curve do not go.

Because there is a single degree of freedom.

He however you ride along the curve gives the possibility.

Here it again in three dimensions in space bi

in perspective form x, y, z coordinates a

as did the painter had seen three in space size bi

shape such as plane mirror We show figure.

This curve b.

Curves A, also the collection of vectors, bi

by number multiplication, inner product, vector product I have seen such things even triple product.

This x, y, y, y is constant thereof You do not have to deal with derivatives.

Derivatives will be zero because of the constant.

Therefore, this work with algebraic We have carried out.

But here there are derivatives with respect to t.

T because x and y only the first force does not consist of.

But this is a generalization so easy.

We are a function of how the derivative If we take a

vectors derived from the individual components is derived.

x component, y component, z component We take the derivative.

Likewise integrally vector x x, y, z of the components of this vector to oluşuyos

integration of these components also means that individual is integral.

There is also a short demonstration on here.

When we say that y is equal to how f x

b simple, simple, quick views, a short bi representation

we provide, where x is the vector x of According to t

the point on the derivative of the vector x We're putting sets.

Similarly the components derivatives of the components in the gene x

and y and z components on those points We're putting sets.

We are also naming as vectors:

The first component of the name of the vector We give.

I.e. components x, y, z in the vector but

We are writing in bold this time vector x We are naming as.

Yet if we consider the inner product of this process derivative thereof

u have the same function of the product of bi , a first derivative of the

Multiply the first time derivative of the second plus the second derivative.

Bura, the only difference from functions numerical functions

values for that row is not important.

This also does not mean that this point b.

But the point here is to show the inner product continues.

Similarly, if the vector product

vector multiplication with a cross in We have shown.

The receipt of the first gene of its derivatives derivative

second plus the second times the first time derivative.

We need to protect the order of these processes and

We recall that the vector product need.

Because if we change the order, this time minus vector multiplication sign in revenue.

We can change the order of the inner product.

Value does not change.

If you remember the initial part of this Go to the kosinüsl

Go to the product of these vectors to the sinus

When you change the order of minus sine-sine

the minus sign means that the vector multiplied.

Therefore, there is nothing new here.

I just need to keep the same order.

E, curve

we usually y equals x in the plane for as we show.

But also, among other representations is there.

For example, in parametric we can do.

t is a function of x b, y t is the bi We can show as a function.

In physics, for example, you use a lot of this bi thing.

If x and y at time t of a point oblique The trajectory of the shot

indicating the coordinates of this, the When location

With this show in terms of parametric We determined.

This parametric representation, for example, the t solving with X,

here take the place of x of y as a function'll find it.

So you can go from one to another.

These two functions as a parametric The x and y, an

When we take as components of the vector we get a vector function.

This is exactly equivalent.

Carries the same information.

But we'll see three work with these vectors size will be much easier.

BI also have representation with the function off.

Off function equal to x and y We take the weight and

a function, the function of x and y is equal to zero would indicate.

Yet there is one, from one of these impressions always possible to switch to another.

Let's see that in both examples.

And as t a t square vector representation Whether a given vector.

Vek, vector function.

Here separating components of the x component T, y

We also found that T square of the component.

To eliminate t between these two very easy.

Because t equal to x.

So if a t t x A x squared in square interests.

So get the representation of the function open we have had.

We have come to the following structure.

X of y on the same side here and a Summing y equals x squared minus then becomes zero.

This representation of the function off is going on.

Of course we all here, most of the time is equal to y We would prefer to frame the curve x

but that equivalent structures to define There are also specific benefits.

Because it will always be the most appropriate.

For example, when we think of a circle the best known of the circle

representation of x squared plus y squared equals a is square.

Get a square on the left side of this f x y is equal to zero, the structure has arrived.

This is a closed representation.

From the definition of functions each of x to y b y We know that should come.

If you remember the first part of the function In order to be

x must not opposed to a single year.

Indeed, when we solve here, the y y square equals

We take the square root of x squared minus a squared need.

In the square root of the pros and cons of this for the future function of representation

is equal to double the functions come, one of you, half way up

rim of the other semicircle down see that.

Of a circle, as well as their E, theta terms of the angle

theta, wherein the parameter t

we choose, the representation of a cosine of theta x component

y also see that a sine of theta, We know.

Therefore, since these two parametric we can get representation.

By combining these two circle we can obtain the vector representation.

Circles vector was easier to work with.

Because there is nothing like the square root.

Here, when the implicit function derivatives blah

in taking the team to b, e difficulties, may occur.

In addition to verbatim here, an email, a single one year does not mean the opposite of x b.

Coming two opposing.

Here we see that the parametric sometimes be useful is shown.

For example, as in the circle.

Now again on the curve in the plane We are continuing to curve.

The significant size of the bow curves b is the length.

Means that arc length.

What kind of moves along the curve length is happening.

In order to do this infinitesimal

As that differential calculus We are approaching.

Length of the curve directly We do not know.

His bi-beamed before we start.

These beams.

A pin with a pin datum curve Let the point.

We combine these two with bi true.

We call this beam.

A zero for p.

As you can see here, p is zero for merger vertical

When we get a vertical and horizontal lines We create a triangle.

This is zero for p in a right triangle.

Es have shown as Delta.

We have shown in the Delta water.

This is the hypotenuse of the triangle.

Therefore, the hypotenuse of the triangle is zero for p One delta p

wherein p is zero but also the length f a vector

Let's just say it because of the length of the delta x components of the delta x and delta y.

This is the delta Pythagorean theorem we know

ie s is the length of your hypotermia expression.

As derivatives have you know.

Point x in the delta function

value, the value of the function x plus with the value of x is the delta are account.

That the value of the function at x our interests

When you have found the delta y are.

Delta delta x to y when we divide this We find the slope of the beam.

Delta x p a zero mean to take

point on this curve gradually p is closer to zero.

see also that when p approaches zero continue to draw a continuous beam

Let us, at the end of çizdik becomes tangent.

This is also the slope of the tangent, say f.

Here we see the importance of limits.

Delta xi directly take zero I put you see

where f x f x is zero denominator minus.

Put a zero in the denominator of x in the delta uncertainty would be divided by zero is zero.

Here is the account of this uncertainty limits In order to have occurred.

In this univariate function Did you see.

This brings us to the derivative.

Equivalent to each other in a variety of derivatives There are impressions.

d y d x may be called split.

is divided by d * may be called for.

There are minor differences between the two meanings, but equivalent to their use.

In less than a year base software f prime displayed or is displayed.

The reason for using derivatives of di The difference comes from the fact that the ratio.

Of these issues for the first time in Western languages

which examined the differential in Western languages is the word comes from.

It to Leibniz, found.

Newton also slightly different in this way I have used forms.

D was used but international a universal

as a form of representation of the differential the word comes from.

Showing that way.

Now we have found here s the delta.

This right triangle from hipotenüz.

Here you hit the delta X, delta y and Let's split.

Delta X, A, si leave the outside.

Therefore, we will put inside the delta x.

While inside the square root is under entering the delta x squared.

Then from the first term future.

From the second term y squared divided by the delta delta will be x squared.

Lim it, when we go to the limit

the differential d to the delta will return.

See here had delta es delta p had.

It will be d s.

Delta had the length of vector x.

He is going to be the length of the vector x.

Here delta divided by delta x of y rate base year would be at the limit.

Here d x d to divide, but we could also write

for the sake of easy writing the square of the base year we say.

d s d divided by x plus y prime seems to be square.

Where Bi is also going to need x divided by d s.

This time we bring you take d x d s time

this is a plus y squared denominator base future.

We find them easily.

Why are dealing with s, this delta?

Or d with s, the arc length.

Because this is a useful measure.

Now we will see the two application We'll see.

Bi accounts of curvature to them.

BI them in the definition of the integral be used.

Because a good measure.

On a curve on the way

distance you have covered, the length of a major measure.

We know the slope of the curve.

We show y as a base.

you, and also as the tangent f We show.

That is to mean, a derivative in the x direction we go

When we go in the y direction tangent fi g, where

We create e-angled triangle, the slope f becomes tangent.

Y is the base of this tangent for We know.

Or how we define.

Sense means in the x direction of the derivative in exchange

The slope of the tangent of the curve at that point.

It might be asked:I wonder how this slope are they changing?

How it progresses along the curve slope are they changing?

We came here to this point.

So here we are.

See, for example in such a way we moved on.

Here was a curvature.

So be a little more open or worsen, closed I might.

May be a little more open.

Therefore, this is an important change for is a measure.

With this, we define the derivative with respect to s.

How the slope of the derivative of y with respect to x found.

We're taking the derivative of f s.

F s and that of the change, curling One measure.

This is called the curvature.

Why we call it a little more curvature We will see in detail.

We have the following.

The following year equal to the tangent of the base.

Let s based on this derivative.

Why?

Because here, there, and we h by fi We would like derivatives.

According to the s derivative of both sides Let's take.

But because it can not calculate fi x the given function.

However, we take the derivative with respect to s in this we want.

We can not take a direct derivative of the plug by s.

In case we did not receive a direct derivative all e, we get a chain derived.

Therefore, before the tangent for their own We take the derivative with respect to variable plug.

Then we take the derivative with respect to s of f.

See them but this is a fraction The following symbolic

If you think the simplified d fi d s really a tangent remains.

Similarly, on the right y s is not function.

The function of x.

Thus, derivatives with respect to x, then According to the derivative of x s need to take.

Now this account can no longer be We came.

Because according to tangential derivative of the plug for it We know.

One plus tangent frame fi.

If you look at the information that preparations 'll see.

In addition, the plug tangent sine cosine divided by fi According to v divided by the writing,

If the account as part of its derivatives you will find.

So bring a plus tangent here frames are writing for.

Here d f d p.

Here again the derivative of the first derivative take the second derivative is.

We are writing to bring it on.

Here too there is x divided by d s.

Now we do replace them.

Because in the end we all are equal for y We want to find in terms of x.

Now shut up once d fi d s is we find.

We said this curvature.

This is fine.

Here, have this.

In addition, for the slope of the tangent, say derivatives

Base the tangent of the y instead of f we put y be the square of the base.

Told off.

Come here.

There are two base years.

BI also had made on the previous page.

Repeat it here.

This beam Pythagorean, your utterance of Pythagoras e by,

hibote your account, the account is divided by d s We found xi.

He is also a square root of the squared plus y base was trippin.

Here we have two pages of more interest before.

We take them places

When editing your work simple bi trap x's

y in terms of x, because certain terms we obtain the expression.

This is an important formula.

That in the course of a single variable functions You need to be seen.

If y'all forget to remember seeing bi opportunity.

And because of this wider concept We will see in space curves.

What if your hatırlıyosa nice.

Now that we have achieved curvature.

We will continue to do so.

Where b is a natural cut-off point alone.

Now you thinking and learning opportunities to take a break.

Bi goodbye until more opinions.