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Hello.

In the previous session of two variables We have examined the function.

We saw most basic of these structures.

The most basic structures, an upturned cup or as a

such as the topography of the lake, down

and these two types of species facing, from parabolas occurs.

Therefore, they are called to the paraboloid.

And in a valley between two hills Think.

Combining the two hills of the valley here a parabola facing up

Although there are hills forming the main lines of the downward facing parabola.

Therefore, facing up to parabollerl The downward from the merger of parabolas

involved and the type of surfaces thereof in a We have seen that there are hyperbolic paraboloid.

Then again, with second-degree xi,

y's the surface and intermediate compound in the second degree

the spheres and cones only

The hyperboloid-piece or two-piece 've seen.

Now I see them significant, a We will work with the team and that the surface

the surface of what it is, what do we work at

There are at least he recognizes the benefits I think.

Here it is no longer numerically to concentrate on the representation of functions

how their differentiation and integration in We'll see how the calculation.

Once you understand the functions of two variables After it

generalized to three and n variables is quite easy.

Therefore, the first basic step univariate function of differentiation and integration in

function of two variables, and the derivatives ensure the transition to integration.

We have said before, a derivative

differences criterion, an integral accumulation of the criterion.

So show the difference in the derivative There subtraction and division,

In the collision integral and showing accumulation There are collection.

So the fact that this process of four occurs.

Among them, of course, the most important concept, the concept of addition,

you learned in middle school added four processing The concept of the limits of the concept.

Now let's see them.

I remember before the formation of a surface.

In a univariate function y = f (x).

Now whether the dependent variable z x, y functions can get.

Again three variables three independent

variable, three free variable functions could imagine.

x, y, z are independent variables.

These data function in w.

And more generally, a y, a digital value

x1, x2, xn from variables such as We can.

How this operation is once again Let us remember.

If we consider the situation in a univariate We have one argument,

We choose it over the x-axis, for example, let's choose a x1.

Y1 value that a function f with We are sending.

This is on the y axis at the value of y1 's.

Descartes made extremely simple but

This ingenious approach works in our grasp supremely

groundbreaking approach to these two axes perpendicular to each other to get together.

In this case the x-axis on the x1, y1 y

when you get a point on the axis in the plane 're getting.

We also point to the coordinates to them.

When we change x, of course minus You may also go to the function value

appropriate, according to him again years will vary.

Here are the spots to go to different places for

for example, we obtain a curve of this kind in the plane.

In the same idea, this time with two variables Let us apply to functions.

Our independent variables x and y from the occurs.

I.e. z is in function f (x, y) To create a function that again

point in the plane of the point f We are sending our point with.

Here it is defined in the function as such.

However, by applying the ideas of Descartes independent

variables with the dependent variable 're getting.

We take arguments in the horizontal.

We're taking the dependent variable in the vertical.

Of course, here comes the problem right now, we have three dimensional

treatment, x, y and z of a point comprising We show the plane.

We have previously said that a painter, of a camera

as does a map maker

a point in three dimensions in the plane We show.

Here is a perspective representation of it 're getting.

It out of the plane where the x-axis are interpreted as if carried.

Functions of one variable is the same in here as the independent variables

When a zone change in the z will also undergo a change.

That's when we draw them a surface 're getting.

The same concept of the three variables, the variable

but of course at that time so we can broaden boot does not have a chance to stay.

Even in this function, because at least the x, y, our argument and

which the dependent variable that we defined

function even in the four-dimensional space needed.

However, we are living in three-dimensional space.

Therefore it is not possible to boot.

We've seen this before, several sections by trying to understand.

Now, our purpose here these surfaces After reminding the existence of a single

variable functions and what we do I remembered

How we will expand on two variables find out.

Here, too, there is a very basic approach, two multivariate approaches in the following functions:

One of the parameters temporarily We will freeze, so the function

Univariate coming down a structure, namely y y0

In this year when we no longer freeze variable

does not seem a relation between z and x occurs.

Similarly x temporarily When we freeze or at the point x0

When we get z, y and z between function relationship occurs.

a vertical plane, the XZ plane, YZ plane, that in a vertical plane.

Now a single variable functions Let us remember what we did.

To get a first derivative at the point x0 We stand, we choose a point close to x0.

These two points p0 and p0 with a true we obtain a beam combining.

Distances, between these two points x parallel to the axis

delta x, delta y along the y axis as have defined.

We are following the process, p0 to p1 We're closer.

Although found in every point see here this time

P0, P1, combine with a right The slope of the beam will change.

Getirdiğimz this, we bring, we bring

When the direction of the tangent at the end of we find.

In cases where the tangent is shown.

So the delta x, 0 in the limit tangent We arrive.

One of the important things we a

We choose the delta x, delta y immediately connected to it turns out to be.

Therefore, delta y, delta independent of x not at all.

To make this work, two approaches is there.

We are standing at a point x0 is an ideal and

When we calculate the slope number we find.

The second point that accept a variable x0 We are.

Near this point, the slope of the variable We found each point for

When we found the slope to where a We're going to have to give the derivative function.

In the first, a number of late derivative function,

for we do with the point of a variable We find functions.

Let's see it more closely.

As we find the definition of the slope; y

The change in direction of the x-directional exchange ratio.

When we stopped at the point x0 function value

f (x0), in a nearby point f (x) 'skin We find the difference.

In this, we divide the difference in x and that in the limit

In x0 we have shown the way to the x df dx a

point that the derivative at the point x0 We find derivatives.

This is a number because a number of point x0.

As the need to limit easy We can see here see

above we put x0 instead of x f (x0) - f (x0) = 0.

We put x0 instead of x is 0 at the bottom again.

So 0/0 uncertainty involved.

This is almost always in all derivatives the conditions encountered.

Here is one of this uncertainty limits

specificity of the process for obtaining himself.

Derivative at a point where we're talking about but we of x0

variable x at the point where we do that x

as a point near the point We can define.

Then of x in a slightly different place point again

To find the change in the function y x

+ Delta x is the value of x is the value of We're taking.

This, we divide the change in x.

This is already the delta x as the basic variable took.

Base de fr it to the limit as x We show.

Where x is a variable residue, so we obtain a function.

Again, the importance of clear limits.

Directly to the delta x 0 and Herzegovina, the denominator will be 0, we will find a 0 in the denominator.

Therefore 0/0 will be uncertainty.

I need it for work already limits.

Specific values of these uncertainties, certain To remove functions.

However, functions of one variable, multi- variable

of the functions that you have received prior of course you know

but we assume it to be very accurate in everyone

a time to remind you know varsaymayıp These are issues we have seen the benefits.

See wherein each spot

defined point x, i.e., the function available

that at every point we can define x

f prime x function

When you change to a function that xi x0'l this at point

we find the value.

Near its more independently

We calculate concepts but independent, directly

we calculate at the point x0, it is a derivative in point

We say, show that the number equal to possible.

This is fine base under mild conditions for x is continuous

ie, the derivative df dx at the point x0 number

When this function is calculated from the number equals.

Briefly the main lines of this derivative be reminded.

Integration with a similar concept when it comes to is achieved.

How were the two points there delta x

indicating the distance between where the There delta x's.

Here we call the delta because each interval xk equal

also not necessarily equal, if taken occasionally friendliness can be provided.

But in general need to take equal is absent.

Within this range of xk'yl between xk + 1 range

We choose a point xk, the function We find value here.

This f (Xk).

Our goal is, the curve y = f (x)

of the curve from x = a to x = b, which is

range of the area under the curve finding.

We do not know how to find these areas.

But here's the big differential calculus success here in the first step

a small range about the curve We find the area under.

This area of the base XK delta height we found when we hit the field.

When we collect them in this curve

The area under approximately we find.

Already from where this zigzag, broken of lines

we can not remain under the full curve in the we see.

But here again the differential calculus, infinite The success of small accounts with

When we take the limit to zero delta x This total gives us a certain value.

In some cases it may be forever, but something get them to understand that it

we have reached a certain fixed value Let conditions.

The field is going.

This icon who made it for the first time Leibniz, the German science

people, summa in German (zoom in) or p with the letter written

which means that the first letter in the word total s pulling

gained by extending an icon in it We call the integral symbol.

Again we see the importance of limits.

Is 0 if I put delta x instead of directly

Under this collection will be 0, 0 for the will collect.

But, the number n, the number n on the range to B

When divided by the distance to n when n When you divide part of the

This time the number of the delta x goes to 0 time will be gone forever.

Then you are faced with a situation like this, 0 as the sum of the infinite.

This is an uncertain thing.

Here again the limits of this uncertainty remove certain

concept that allows us to find a value.

How is it fixed in derivatives

value by calculating the derivative of a derivative in point a number

were finding where a and b of the integral given as

found between the values of the value of a areas are account.

This is a number that certain integral we say.

The same variables near a point derivative As we calculate,

we obtain a function, such as the upper boundaries, for example, variable

we do, then, depending on the upper limit We'll find a value.

When we change the upper limit of this value

then obtain a function to be changed We are.

We call this integral uncertain.

This area under the curve remains is

where x is found, but there area we find.

If we find this little area here.

When we came here from a curve

So far in the field of we find.

This function sets.

As you can see in a number of derivatives We can find a spot in the differentiation.

Integration can also find a number two fixed

value area between this specific integral call.

Again, the variables in the derivative at a point We find derivatives.

Where the derivative function call

If you want the integral function We could say.

This means that integration more acceptable uncertain have seen

A term is a function of this.

Now let us remember them.

These two variables now We will generalize.

An important integral of the concepts one at a, b, we select the range.

In the meantime, the function must be defined.

Allow the discontinuity of the function We can give

We could not let discontinuity in the derivative.

You may also remove eternal values.

But this is the limit of infinite values life is not always that we need to do so

not as easy or pretty, but not necessarily it will be a value, not

but again, if we look to the generosity of life our

We tried almost all of the functions the problem of discontinuity

does not create, even if the piece indefinitely continuous testing

place, but the type of jump discontinuities may be.

The function of this single jumps everywhere will be defined exception.

Although all of the infinite value collection may finite values.

If the infinite value out of them a so we can keep it under control.

On the basis of differential calculus

the steps outlined here, we derived a putting side by side integrally,

both small increments We're trying.

In derivatives which point there is in the vicinity delta x for

're getting near that point, a delta x one can get to the left side to the right side.

In the integral from A to B until We divide the range of many of the delta x.

But the basic idea with the small delta x

strive both common concept.

In both cases, the limit still more common concepts

Before the beam inclination of about

In the slope on the curve are determined.

Again, the curve of this product under

delta x in the range of small rectangular area as we see.

Here delta x in the range of the beam I could see the slope.

This, in common in both.

In the second step about shared values We find values.

Taking the absolute limit in both 're getting value.

Precisely at the point where the slope of x we find.

Here, too, between A and B definitively We find space.

Therefore, as you can see their derivatives and integration

complementary and fully co- working with ideas.

Of course there are differences.

Derivatives measure differences.

We measure the difference with two handle.

The distance between the two values and proportional

As the differences in proportion finding.

Wherein means for measuring differences subtraction and division operations are used.

Measuring the accumulation of the integral multiplication, how many collision

I've saved the fold would mean a the collection, it also measures the accumulation.

As you can see the two work two four operations not find the derivative and integral.

But we can not do something is not the limit, 0/0

uncertain as the sum of infinitely many 0 there were cases.

That saved the limits of uncertainty.

Now you've done so far a reminder, We make new progress.

I hope many of you on this subject remember, do not remember

also an opportunity to remember here, or no Even learning done;

information allows us to progress our will give.

Now we pause here, because the only Valuable

derivatives of functions of one variable and We remind the integral.

Thereafter two of the same concept

How to multivariate functions We will see that implemented.

These concepts with a very simple idea We will expand to two variables.