This qualitative in nature You have to say, we have defined

surfaces of surfaces some basic Let us examine the surface.

Of course the simplest surface x, y, z was confirmed linear

with only the first force that is the surfaces.

We have seen this bi plane.

But here, derivative, integral hardly we do not need.

Because of this, the vector with algebraic I have studied.

This bi-linear remain our top structure

x and y are quadratic functions is.

Yet beyond this bi, this time at z is a quadratic function.

We take this as implicit functions.

Please zoom ALSA because he can.

But as the square root function will appear We prefer to handle it in this building.

Now how the surfaces of these surfaces Let's try to understand that.

The simplest of these surfaces at a y one is taken.

As you can see x squared plus y equals z it most easily in the frame y

we take zero, y, z is equal to zero when you get x square.

Such a vertical parabola and the vertical section bi parabola facing up.

In contrast to zero we take x z, y plane a parabola facing up again.

Horizontal section at this time that we take to z sabitlesek

z you can see when we fix the hoops as occurs.

Where A and B are different from each other if the numbers would be ellipses.

Thus giving the surface of name If you want to name it the best

Stressing that consists of parabolas E elliptical, emphasizing Paraboloid,

we say paraboloid, the parabola generalization wherein the horizontal sections

circle but an ellipse would be more generally would be for the elliptic paraboloid.

See also this second type of plus minus increasing Let's take minus.

The third type is also a plus b minus

As we can get b minus b plus can get.

If we move again, we can get zero y.

y is the number of zeros in front of You could buy.

y could not have thought.

As you can see just where z of x

b means that function of its surface will be.

Similarly, we suppose here do not y where A and B coefficients were so.

We get to where we're a B minus.

If you're here on a B minus the

If we've achieved this zero function is going on.

Now let's examine them one.

Second state

z is equal to

In the minus x squared minus y squared, see wherein

In x squared minus z If we give constant values

As you can see again, ellipses, dai, Hoops will occur.

However, at the other, as there is a difference.

The minus sign C to be Hoops for negative values will be.

Real solutions for positive values 'll never find.

This is also evident from the figure.

Here when we get C minus We can cut the surface.

But if we take positive values C, the surface not cut.

Here this is equal to a value line Hoops is happening.

In the following sections we take it to the vertical with theca, income counterparts.

ALSA y is zero, you get zero mean y y is equal to zero the

you can see that the vertical plane x-z plane Nothing like a parabola will be.

This downward facing parabola.

Able to understand that in this way surface We're going.

Here you also understand the perspective drawing We're going.

Now a bit more complicated situation when we came

If we take z equals x squared minus y squared See

here we give a zero to y

When z is equal to x squared ie upwards, a parabola.

Here Siya these types of b.

Here, too, of course, more of this type built yet We did not, but

See also here that we get zero if x z

i.e. downwardly facing square minus y parabola.

This y, z plane downwardly facing a parabola.

In the first two in the other you always Öbürkü

Lookin down was all you all up did you look up.

As you can see here is different.

That means that if we combine them x upwardly facing direction

downward parabola y direction The parabolas.

This is earlier than the

Qualitative attributes for we say if the surface becomes.

Now how are we to get it?

It's easy.

See if z is constant and b.

Please assume a've given.

If x squared plus y square of it We know that hyperbole.

Because it goes to infinity x and y are the x and y

In addition to these small z goes to infinity which will remain for a finite number.

Y wherein Y is equal to plus or minus the asentot will come.

reset when there is a part of x is equal to z is absent.

An x y z is zero because the solution does not mean you do not cut the y-axis.

But whereas if y is equal to zero, ie,

cutoff two along the x axis We'll find.

z equals plus or minus one.

So here ter, who accepted this asentot and one and minus

z is the z-axis intersecting an an Remove the worth of such a hyperbola.

Maybe we should give it a negative inverse of z would be.

Current varieties to hyperbole, branches were formed.

That's it, it equals four, and if z for example, still

y is the same including asentot zero

on the X axis plus or minus two values were cut off.

There are two values plus or minus two value segment.

But it is valuable for the y-axis z plus There is no cut.

Say here that such a second 're getting hyperbolic.

But, conversely, if we say that z is equal to minus four There is no time to cut the x-axis.

These curves are obtained with the same asentot.

Hyperbolic curves is obtained.

I got to know very well these three species.

You just have a twin.

So in this sense.

Karey plus x squared minus y in the previous did.

Here squared minus x squared plus y.

Again with the same thoughts this hyperbola Remove surface.

But as you can see here that y is zero i.e. along the x axis

z, x plane, this time facing down would be a parabola.

Conversely, x is equal to zero, the upward-facing bi would be a parabola.

So it's surface, if you take the inverse of b as if you would turn.

Is equal to the equivalent value curves, the same geometric equivalence curves

structure including only negative here values on the x axis cuts.

z-axis, y-axis on the positive z values.

Thus, in this way, bi previous ninety degree would be frozen.

If you bought it in the space opposite the as if you had turned occurs.

Gene E, e, we want to give is here As you can see the parabola.

Bi at the same cross sections in the horizontal plane sections of the hyperbolas.

This is he, it is called a paraboloid.

But it's called a hyperbolic paraboloid.

See BI recalled in the previous horizontal plane

yl sections generally ellipses was trippin.

For this reason, these elliptic paraboloid we were saying.

Here, the hyperbolic paraboloid diol, we say.

Z is finally we take x y.

We do not see here x squared y squared.

But that has x and y to the second degree x and y in terms of function.

Wherein z is equal to the constant values See the time we get

If you see a constant y z equals this constant divided by x.

If z, and c are positive Or z values, plus

hyperbola y is equal to the following types of values consists of a pair, an example if you receive,

y is equal to one divided by x, the following parabolic branches is double.

Less if you receive the following parabola this time, branches would have achieved.

As you can see here on the asymptote hyperbolas.

Between that of a previous hiperbollerl The difference of 45

When rotated a previous're getting.

Because in a previous form asymptotes 45 degrees.

This axis c, i, where the minus sign

by plus and minus values of c role, still here

z z plus the ones in the works, this time minus values is.

They are not difficult to understand.

90 degrees so that a prior be rotated.

Now we come to the cylinder surface Let one of them.

For example, say that y than y is not found will be independent.

So all the sections in the xy plane parabola will be.

Here he took a piece of paper a parabola but along the x-axis, the

When you edit, if it was negative again x parallel to the axis

A family of parabolas, which in this because we say parabolic cylinder

because the function of the cylinder in the third

No variable or a second independent variable.

Also, because the parabolic sections

parabolas that, in analogy to the y there is not time x

Although z equals y squared, then the y-axis which would be parabolas.

So the major surface of such surfaces is going on.

I'm just saying these respects the most basic: Linear function immediately

A second-order functions beyond Because in terms of variables.

The above results we found To summarize, we have x squared plus y squared.

These Artillerie or sour.

There are x squared minus y squared.

They saw Artillerie or minus.

There is currently no y's z is equal to x squared.

Means that there will be no roll.

This Artillerie and sour.

These surfaces, all of them will be paraboloid.

More precisely, the first of the second kind.

But first a circular elliptical Or the paraboloid,

rotation paraboloid can say that, because x squared will be r squared plus y squared.

The hyperbolic paraboloid them.

Here the parabolic cylinder.

Open in all essential functions, z first degree, second degree of x and y.

This can be considered a generalization.

Here is the numbers 1 to 1 divided by the square b Although there would be elliptical square sections.

Hence the overall shape, elliptical

paraboloid, the overall shape, hyperbolic paraboloid.

This paraboloid cylinder.

Of second-degree power function As there is a difference.

This time not only the x and y second degree z in the second degree.

Most science area of the sphere equation, of course.

In the sphere of the equation x squared plus y squared r Thank surface of revolution obtained by writing the square.

Z because normally, the function of theta could be.

In the absence of theta for all theta Show 're in constant,

that a certain fixed angle with the axis of x and y, ie

When you think of it in terms of three dimensions the vertical surfaces

sections because they are always the same exit surface of revolution happen.

Here follows a systematic approach in is there.

This is x squared plus y plus z squared plus the square.

Then we are changing signs, x squared plus y squared minus z squared.

There are three possibilities here.

x squared plus y squared minus z squared, x squared z squared plus y squared minus the same thing again.

Zero, one or different types to be minus one brings to the surface.

See goes like this; rotating all of them surfaces because the x squared plus y square

If the r squared, r squared equals z squared.

Thus z is equal to r.

But it also means that you can plus or minus 45

degrees angle z is the equation of a line in the plane.

This is a vertical plane.

Theta independent.

This means that the 45 degree line You can take the z-axis turning around.

This way you know what it was, a the truth

If you rotate around an axis of a cone get surface.

So it's a cone.

I earned it, plus when we do Valuable see this side of the same, this is

plus x squared plus y to understand it Let's say the square is square.

See hyperbolic equations involved.

Here will be a hyperbolic surface.

But when there is a hyperbolic axes one

does not interrupt the other cuts, which here cuts

Let's see, if we r z is equal to zero We find a part of plus or minus points.

r minus z squared is equal to zero, we do a no way.

It did not provide a z.

That means a cut z-axis paraboloid.

So it is a branch of a branch above which follows.

Here is the negative of this function minus throwing left

transferring the negative r squared plus z squared, If r is zero as you can see our

would be plus or minus a cut, so and it will be cut here

On r R x y plane of the will not be a part of.

If we look at them, that x squared plus y squared plus z squared the ball.

Order here if you had a zero point would be

If he had taken such a negative real surface I would not.

For this reason, here are a single species.

Here may be the surface of a cone.

When we receive a zero on the right side.

On the right side you can see when we get one There is no such a cut with z.

There are only a segment x with y, and that hyperbola

z 'be such that the cutting of branches and And because it is a surface of revolution

When it is rotated, the resulting in daily life this way by car

or something when some power plants

on the side, next to the factory cooling You can see the towers.

They removed in this way.

Nini flip side if you look at a variety of modern You can see shapes in architecture.

You can see the machine elements.

They are not really that works well shapes.

Similarly

if you receive a negative right-hand side, then As you can see in our sector there.

In our hyperbole when cutting branches in this way will be.

Two branches.

That we rotate around the z axis will be again in a hyperboloid surface.

They will always say hyperboloid surface because it consists of hyperbolas.

But it's a one-piece hyperboloid surface we say.

Because it is a single piece.

However, the couple split the hyperboloid Known as the surface.

This is used.

He'll be here a kind of magnet, a magnet

is, becomes an arc, the center comprising You can use temperature, such as.

These things always used.

Whether a shape is always idealized sphere to examine

because it allows almost any science used in the field.

The numbers in

1 minus 1 will take place on a b c

If we had not received sections hoops

then they would be horizontal ellipses

sections of the rotating surface, apartment Hoops will be

Although it would be ellipses but elipsl hoop attributes

a similar way off in terms of shapes.