Hello!

So far, we have learned quite a few things, basic

We have established the infrastructure that vectors We have learned the lesson

We made the introduction, the first of vectors application

In multivariable functions as a We looked species.

They gave us space curves.

If you take this year again

In multivariate functions, which is equal to fx

only one independent variable in the first kind

t also said her name, dependent variable

We said at x as a vector, but the vector functions

We have defined this vector of functions Coming opposed to the curve in space.

Already at this stage the other end

an argument that will go before We took the number,

We bought one argument, now If one will take the dependent variable.

However, two independent variable

We will work as an independent variable that vector, the dependent variable

space curves, whereas a number of independent whereas a variable number of

was a vector for its dependent variable We said vector functions.

Such a surface in space functions show.

The next program, we will examine as follows;

these surfaces before

We need a little recognition, what with We need to understand that we take care of this rather

simple, partly because of what you know, but regularly

that will be presented here some basic

We will examine the functions, will recognize.

How a plane surface of the simplest, but

there is a first-order z, x and y were from the first degree.

However, in terms of x and y are no longer functions

it is not limited functions of all kinds I might.

Once you understand this, 3 and n variables

functions conceptually 'll see.

After that just two variables function starting from the most basic

We'll see two processes, their derivatives and integral concepts.

Here's the basic derivatives and integrals We will see the concepts.

In subsequent sections, various We will see applications.

Here again, the basic functions

I'm going to do the promotion emphasizes.

Easy job.

Our argument was x and y,

arguments say that a plane point.

For example, as defined by the coordinates x1 y1 get a a point.

We function f at this point a to z We carry the number.

This is a reference, where the term given.

This concept still univariate as included in functions

independent and dependent variables

In a combining axes compile.

Here, of course, three-variable three-component we need to work in a space.

Yet, as we did in the third dimension

a type 2 dimension, just a painter his surveyors

We will try to move objects, such as, that here

Is the x-axis, y-axis is the z-axis have

Although the two-dimensional plane drawn that

Three-dimensional object, you will understand.

If we change the points where x1 y1 certain plane

him these numbers change within the region z opposing the change.

This is our place in the geometry of a surface.

Still surface from a different perspective to emphasize that

x y z vary for all three of the free plane all

because we fill the space anywhere We could go free, but not z, x, and

How can be calculated from the function y staying there, do not go anywhere.

Therefore, we can not fill all the space.

Here this two degree of freedom Because the surface is formed.

Yet if he shows up on my hand a this surface

remain on the surface in two orthogonal directions I can move, but it

When I'm away from the surface in the direction perpendicular surface interval, that the second of freedom

a geometric solid with a degree of We emphasize that surface.

Now we are the same, as I said, this surface

As a painter makes a surveyor even

when taking pictures as we did in the third dimensional

In two-dimensional space, from space show We will seek ways.

Among the various methods for that pronounced as three main ways

perspective, perspective from a point When we look at the same photo

from the camera to the plane of the three-dimensional objects

to show it as a shot I will try the second equal

The methods he introduced value lines.

Here, the same cartographers lakes or anything and mountains

To mark at equal height points a

As combined with the curve in which lines of equal value method is a

this way, for example an object such as watermelon Think in this direction intersected

When we get a look at open

Do her in a different direction you cut

still obtain a different way when Do you.

This is a vertical cross-sectional pick, where the horizontal process

reflecting sections are able to receive, wherein the take vertical sections.

Now a little more detail to understand

The main surfaces of the surfaces Let's think.

Of course infinitely many surface You can define.

But one point of their most basic ones

a depression around, around 1 point 1 peak.

Now a pit, that's like a lake Consider, as a container

Think about where you've reversed this container Or as

Think about it like a mountain peak, already Univariate

the most basic of functions directly from the curve immediately

was the parabola, which is the parabola is happening to them already

paraboloid is called, in more detail We'll see.

Now here is the direction in which if you cut a

parabola, but it turns up here ministers

involved a parabola, where the downward pointing

involved a parabola, which is the If you cut certain.

So this surface just above the parabola ministers

from the downward facing parabola that only consists of parabolas.

Without equivalent in the plane with a surface more encounter.

Upwardly in a direction parabolas a in the direction of the downward facing parabola.

In a simulation of this kind to them surfaces,

See here a parabola facing up, a parabola in this side facing downwards.

Think of a horse's saddle and even a horse Consider the body of the horse is the backbone of

such an upward-facing for a parabola, but also the sides of the body

at least rearward parabolas subject at the beginning and then turns off

but around this point that interests us behavior.

In the morning we drink from our Turkish culture If you careen down his cup of tea

In this direction a parabola up a similar

a similar line up facing the parabola is there.

If a washer on the side surface, although the up and down

If you consider that more than half cut creates a similar surface.

You get more colorful as well as a

MATLAB is a software program that called fine varieties can draw shapes.

From there, a picture taken.

As you can see a cap again.

Ministers from the bottom up, on the contrary it

in a similar modern architecture shapes is done, it's like a hammock

It looks like a swing like this one It looks like a modern dome.

We also see this in the work of the saddle surface in this way

facing upward, downward facing in this direction comprising parabolas.

surface.

These three very basic surfaces and their common We will frequently encounter.

Surface of the minimum maximum points We'll see.

Here numerical them very early in the course but to make it as geometrical

Do you see this as the point where a small points, wherein

The highest peak of the peak point, it

a hill with a lake, the deepest and most such as the high point.

The saddle also has a slightly different situation.

Now this surface equal

If we cut with a horizontal plane

that all our points on the horizontal plane identical points

We have combined them such a varieties such as elliptical curves creates.

Upward-facing horizontal gene in a container When you cut with a plane here again

such as elliptic curves occurs, of course, A circle is a special case of the ellipse.

If you cut this kind of a curve in the saddle When going on a hyperbolic surface.

We will see them in numbers, but now here

albeit somewhat qualitative Our understanding is useful.

Here, let us consider an island in the middle of the sea, a little

Looks like even Heybeli candidate has two peaks.

This candidate we cut with the horizontal plane As you can see when

where there is a hill, where another hill There, he had a valley in between the two.

Here we see that's just basic way here from the hill

a valley in between the second peak a surface

occurs, the geography of opposing it

books, maps, atlases in the As we have seen

equal to or equal to the height points depth points is achieved.

The third method is the first to say perspective

a photographer's camera's 3 the two-dimensional plane dimensional solid

such as reduction, in the second method the horizontal by plane

We estimate this surface, sometimes many Once you cut through the vertical plane

We can understand the surface, for example, this kind Or a half door

Think watermelon, taken from the top, vertical by plane

When you cut lengthwise parabolas you will find.

This is still the opportunity for us to understand this surface brings.

I can show three types of functions I've mentioned before.

Indication of the first kind.

Where x and y arguments open it seems to be.

As our dependent variable is also clear that it seems.

Off the dependent notation independent We do not make a distinction between variables.

Provide all have an equation.

See, here are two of freedom.

x and y free.

3 of freedom here, while x y z between 1

With this function we restrict to 2 degrees of freedom still remains.

In another form of the same curve, space curves

As we have seen through a parameter defining surface.

Because there are two degrees of freedom of the surface Two parameters are needed so 3

variables x y z variables u and v as Gösterebilliy two variables are denominated.

They sometimes such that sometimes this kind, sometimes such further

but all that could be plant-available basis only two remain notation of freedom.

To give an example, a sphere, a sphere Let's examine the representation of the surface.

We are most familiar structure, x squared plus y squared plus z is equal to the square is a square.

if you receive a frame to the left, that of x y z a

be incorporated into relationship is zero, that implicit.

But a little while on this account can be difficult because we

usually a single variable functions We have become accustomed to fx y equals.

Here z separating easily, this

A squared minus x squared minus y squared z We return to the structure.

The square root of it, as you know the pros and minus the square

the northern half of the globe, and because the roots will be spheres themselves

Because a separate function are displayed in the first If you remember what I said in the beginning only

be valid in variable functions here for a simple generalization.

Independent variables dependent on a single variables must come opposite.

If these two do not separate hemispheres 1 argument 1

1 in the Northern Hemisphere Southern Hemisphere opposing points of revenue

and it is uncertain what will not function You do not know how to party,

equivalent to two functions for it here it seems implicit.

We have seen that it is already in the circle.

You have less experience, but geography is also fine

There is a method you know of a sphere To determine the point on.

For example, Istanbul's location on the globe How we set.

The angle from Greenwich, latitude We measure the angle on.

On the meridian from the equator, longitude

We measure the angle on and we find.

That's the same approach they

angle point in a space for the z

with the axis including an angle which point p in space

When we get to the xy plane projection subject from the x axis, and

We can measure the angle from the y-axis, of course, The first angle is enough.

As for x-x from the axis angle

And when we say that we measure theta.

If we show it here on the globe for the plug, holding globe

If this endpoint returns on a 're getting latitude.

Similarly at this point constant theta If we change keeping you see

I would like to give a longitude and the second This first line of intersection of the line with

it on to Istanbul from Greenwich Or any until our province

until up to any point We measure the angle theta angle is happening.

Those of us on the way the angle of latitude giving.

In Geography from the equator angles but measured

sailors from the North Pole angle were measured throughout history.

His reason because the open northern North star,

north star does not change the location.

But of course it is more practical to measure each underfloor

you see the equator for sailors no practical value for.

But the sum of these two angles 90 degrees, its not in terms of mathematics for lights.

Here are two variables here, as you can see u and v in terms of the

Putting and theta angles for a on the surface of the sphere

the coordinates of a point to easily we find.

This pin point based on the vertical axis Get the projection, the

found wherein a distance z, wherein A other

There are triangles, right around the corner angle is 90 degrees.

Open it for the length of the side of his Because cosine is coming.

Same distance, in washer-sphere

radius a, the radius of the sphere using

with the corresponding terms for the length of this If we measure,

here again, the zoom angle near 90 degrees,

Using the right triangle are in the creek at 90

f becomes a sine times.

But our job is currently not on the x and y

but after finding the x and y r very easy to find.

Yet here there is a right triangle, the x at the point of 90 degrees

r is the next turn house.We

hence the projection is close to kosinüsl

are times when we get the cosine of theta r were also found in the acini fi.

Buddha cosine theta.

y is the same as the sine of theta be obtained by multiplying.

Here the sine of theta times sine fi as we find.

When we bring them in place x vector components

As you can see in an x function we obtain a vector function.

This vector function f and theta function.

For two degrees of freedom is

confirmed here again is a surface We're going.

It often spherical coordinates to be encountered.

For him it thoroughly from the beginning would be useful to learn.

There are two special surfaces.

Y of X, as we know it

f, x, y is zero function as bi Although the x, y plane shows a curve.

But it is this function in three dimensions, we but considering where z does not appear.

He 's not difficult.

We understand the very simple bi.

The most simple e, from the start line.

Please assume that this is the equation of a circle get.

x, y plane of the circle b.

But all of this circle on Have you received at points where the point.

On top of this exit.

of these, all having the same x and y points z are different.

But at this point, this equation offers.

Therefore, this kind of three-dimensional that is, a component may function

a cylinder surface without function shows.

Cylinder surface diyince practice the most The cylinder used

b is a circular cylinder stove pipe, water

such as pipes etc. E, heating system, such as pipes, machinery

parts such as pipes in electricity such as the system of coils wound.

Diyince circular cylinder immediately to mind but here, the definition of the cylinder is coming

Although the parabola, for example in accordance with base b and to the top of the parabola,

If it's a roll of the point because it would be in this building.

So this is also a parabolic cylinder happens.

A line with an even simpler if you receive the x, y and this is true in the plane of the

When you exit struts on it a You get the plane.

So, in this context a plane cylinder.

Because the E in the equation of the plane, z does not appear

When we think in three dimensions, but also geometrically all have our points.

Here, the following circle of concrete given x, y

but that in the plane in space of this circular cylinder is happening.

Given these kind of parabola.

y equals x squared plus a he.

He is a parabola.

x, y plane in space in the parabolic surface.

The parabolic cylinder surface.

Gene where x, y plane a correct

usually a vertical plane in three dimensions is going on.

So a special surface of the first kind without variable

face of the cylinder surfaces is happening.

The second type of the function z r and theta can be expressed as.

How is variable on the x and y in r and theta may be variable.

But if it does not find theta z, r plane the

b z is the vertical plane, as it is at r

the x, y plane connecting the centers

including the length of this curve will be.

But in space, when we consider all theta are

To say that these curves are the same.

Because independent of theta.

This is equivalent to come.

Take the z-axis such that the curve

rotate around here is you said as we have seen.

These surfaces get this right take the z-axis

by switching it around a rotational surface creates.

For example, if you receive a circle where it

circle sphere surface obtained by switching b you will.

So a spherical surface of revolution.

Here, if you receive a straight line, vertical bi straight line, it

Even this is rotatable about a z-axis circular cylinder is happening.

Therefore, as you can see circular Also in a cylindrical surface of revolution.

Would be useful to know them because in the end from such surfaces

to understand the nature of combining work I will try.

We want to technological devices, tools will try to do.

Any function of x, y plane and this time we take

This kind of revolution if we returned on y We get the surface.

I.e. mA axis rotational axis of rotation y one.

Here would be the third axis z from z independent.

Similarly y is again f, x This time function x

While we obtained on the same function When you look a little different this time

A surface of the rotation axis x, but this time is obtained.

These special surfaces of the second kind.

İki Değişkenli Sayısal Fonksiyonların Görsel Olarak Anlaşılması, Fonksiyonların Açık, Kapalı ve Parametrik Gösterimleri, Silindir Yüzeyleri ve Dönel Yüzeyler

多变量微积分 I：概念

Meet the Instructors