Hello again, this

in the previous session on a surface We calculate the slope.

According to the x direction or y direction thereof Depends on what went

According to the partial derivative with respect to x and y as has partial derivatives.

Our approach is based on two variables function

only one of the variables, holding variable

such a reducing function of y y = 0

When we select a curve in the x-z plane 're getting.

The slope of this curve partial derivatives.

Similarly, the vertical y-z

We are in the plane of the slope that accounts According to the partial derivative of y.

We use them only when videos derivative of functions of one variable.

Now the only two variable function variable in function

Using the basic concepts of integration We will proceed.

Let's remember again maybe You will get bored by them by

but how they are so basic that it If you saw a little bit more so will strengthen.

As in the univariate function were doing; From A to B as x

this range in the direction of y = f (x) curve We begin by calculating the area under.

Our approach to this range equal and must be

x1 x2 x delta delta delta delta x non- We divide intervals.

xk xk + xk +1 this delta between the range let x.

We were doing it in a single variable.

And within this range any

selecting a point height calculation We are.

For those that do; We know

at a time of the area under the curve We can not calculate.

For her work with the infinitesimal, the basic approach of differential calculus

this, that a small range of approximately We are above the range area calculation.

This account the area of a rectangle We know it is coming to.

When we base the calculation of this area, The difference between x,

Select a point on the way here We find the height of the function.

This is the product of two infinitesimal us that gives the area.

They are also about the time we collect

as the area under the curve we find.

In the past it to the limit time, the delta x's

terms, the exact value of the field We find that the transactions.

So the range of small range we divide a range of

finding function of height from the following We find the area of the rectangle.

Collect them at the bottom of the curve We're going to the field.

Two variables we also fully the same.

But the scope of the two variables.

Ax in the X and Y from A

b, where y is the go to d from c Let's define the region.

This is going on a rectangular region.

In this rectangular region x, x1, x2, xk

the discrete values

and similarly we choose y axis Let's pick on discrete values.

When we combine them, see One area on x

y with the boundaries of the range on limits

As this is a small way, from the intersection rectangular occurs.

Let's say you to this point xj and yk.

In three dimensions, considering what is happening perspectives

representation variables x and y and z in

When we receive the variable in three dimensions such we obtain a surface.

Here again, as in the x direction and y we predict the direction of the small range

here is a little to the left in the picture with grain , shown here as the area is made.

Here we choose the surface from the point of going on height

We find an infinitesimal volume and wherein 're getting.

This in more detail in the image on the right shown.

between xj and xj + 1 XJ of the PricesLa point we choose.

However, between yk and yk + 1

with QoL in the region in which we point we choose.

When we combine them, with xj yk'y this leftmost

shown in gray of the region, wherein horizontal perspective

rectangular region in the plane formed 're getting on a volume.

This is easy to find the volume thereof.

Height function f from the xj and yk It is at the point of account.

With the delta area of the base XJ tbsp.

As you can see on the x and y different indicator because you need to use them

At the end you will change from each other will change independently.

To find the total area of these two kinds we can do.

First, let's find the approximate value.

This small area before the k free holding, i.e.

yk'y free on holding x Suppose we proceed.

This is happening right term.

We are here to collect on j.

Then on the collection of k 're doing.

If you notice, j, k of the border limit does not have to be the same.

Left to collect them in reverse order 're doing.

Before we do collect on k,

After collecting on the j 're doing.

This comes opposed to this; x of the first path we're constants,

year on year on year equals from k1

m up, then come xi We're changing again

K from 1 m up to collect on 're doing.

So this rectangular region vertical strips If we're scanning.

On the right are doing the exact opposite,

We choose a fitting on the collection of yk j 're doing.

I.e. the first strip on the horizontal strips collection on the second strip ...

Of course all these rectangular here infinitesimal

These two past fields of collection gives the same result.

However, in the case of single variable functions As we go to the limit

so the delta x and delta y time limits

When we received this approximate volumes total volume

the actual (absolute) gives volume and this one by integrals we find.

Wherein two of these in the collection The integration of different leads.

Wherein x is a constant in the first integral

Retrieving like, in a variant of the same as such.

Function of two variables temporarily We narrows to a single variable functions.

of functions of one variable y on We are integrally account.

On the value out of here, but x is a remains variable here,

We take this time integral over x.

Or, conversely, by fixing the x on y We take the integral.

Wherein Y will remain a function of y will remain.

Then y on it after finishing We take the integral.

That is a multivariate approach function in the first

Starting from two variables, we see a variable

We choose a fitting temporarily, on the other derivatives

or the integral operation, which If you want, you're making it.

Then again sequentially as shown here We are an integral single storey account.

The transaction is a more do it.

xj held constant before xj'y on j = 1, j = 2

including, i.e. on the vertical strips We take the integral.

After holding y constant horizontal On strips

is integral to each other, and we both equal.

In this picture on the right as the geometrical We see that.

've constants x, y variables, means y

We find a strip of the same as shown here.

Or y is fixed, this time x We find in the direction of a strip.

We find in this volume.

These strips, that small strips

the sum of the volumes of the total volume we find.

In what we do and how that Clero

theorem based on the order in which the derivative If we take appropriate conditions, let

if provided, it is a simple continuity It was necessary, where these integral

If these two integrals are equal is coming.

We call it a poor outcome.

This is not a result of the weakness of the term that Accordingly, rectangular

we received a special zone for the weak theorem is called.

It also has a strong theorem.

They want to pass.

This is just the analytically as follows:

statement, please look at them to understand Try.

There's nothing you can not understand.

K on the first two steps on the j whether the collection of

By combining shapes just saw

of analytically expressed by the formulas.

Now the next part of this weak

as a result we become stronger result will bring.

This means that only the rectangular region not on any region

on the results, we will.