We saw a variety of suitable description, We have solved various problems.

Here, too, these problems will continue.

As there is a difference alone.

Here the solution of problems of the I will give the basic steps, but

I'll leave you to solve problems.

Your solution for the given answers will have the opportunity to check out.

The first is with a paraboloid e kesiÅtiriy plane, and this plane,

between la paraboloid We expect to find the volume.

This usually we paraboloid center on the z-axis were choosing.

Here, the central minus one and selected for a negative paraboloid

x and y plane as z Not axis and numbers,

for the number of non-compliance such a plane in a selected state.

As the geometry, so that the schematic See the divide have a paraboloid.

You're taking a nap with a plane.

Of course, this paraboloid with a plane Estimation of the interests of an ellipse, cross-sectional general.

This section of the x and y plane projection figures

was chosen such that it is going in a circle.

Now this, how do we find this section?

Parabola, the paraboloid z in We synchronize with z in the plane.

Each time the intersection of two functions Whether you get in the plane in space that

're doing so, the coordinates We equating any coordinate.

But here at our easy to synchronize where z are the equalizer.

Here, the first function of the paraboloid,

As you can see the square of x plus one year plus one is equal to the square of z in here,

z If we take the right two x plus two years plus three is going on.

As you can see here there when you open x squared, but there is a bi two x.

X, the right of the two left takes two x one another.

Here again, a square and y but there are also two years.

It takes two years for the right two.

See here an there was a plus.

If we take them to a Sagyn remains.

On the left the x squared plus y remains a square circle.

Here Buc, E, circle the cylinder on

coordinates to make the process more it would be easy to see here.

certain limits on z.

This parabola with the level of this plane surface up to the limits.

So, the answer given I'm waiting for you to do.

The second volume of bi t.

TR, we saw previously in a circle

In the center of the circle b receive, circle of radius a.

If you choose a less than b, Such a structure is going.

That is because it is further away from b does not cut the circle of the z-axis.

This circle of the z-axis Are you turning around.

This top view image on the right.

This is a radius, the circumferential both

is a view from above of the harness.

And wherein the three Ã§alÄ±Åcam in size to show.

TR steering wheel of a car or like a wheel,

we have to take to the streets As the shape of the wheel,

a car is a bicycle in such a way that the shape of the tire.

They are also used in machine elements.

E this, Or wrap a wire on such as electromagnetic fields can produce.

These are not so much like fantasy.

Earlier we saw the surface of tori

but the surface of the two-parameter used to define fi and tetayl to.

Circle E, grandson to arrive to the surface wherein any point on the center,

from the center of the circle While connecting distance ro,

There will be a surface is equal to the time we rode.

Therefore, there was no surface ro.

Because Ron was a constant.

Here are a de roi.

Now a curvilinear them coordinate team creates.

The first volume of work to be done with them, find infinitely small volume.

Here grandchildren equation given here, a point in.

Limited went from zero to one ro We cover all things time.

For two pins of the reset goes.

Between theta pi goes from zero to two.

Grandchildren in the following pages of this surface account wherein in step

step-grandson of the surface has been Calculating the volume we want here.

Because the volume of natural There are three variables.

Or by calculating a first step Jacobin If you say I'm more discerning geometry ro,

f and in the direction theta The trio will fetch vectors

product, taking into may be negative here, but the volume

For plus value should be We take the absolute value of it.

In this way,'s.

If you say that there is no wi'm on dealing with directly Jacobin writer,

so this is the first derivative of x by ro line, the second line of plug-derivative,

In the third line of the derivative with respect to theta wrote that the determinant is calculated.

Mixed terms as may seem in a way that's very nice sine-squared,

a theta-theta plus cosine squared is such as sine squared plus cosine squared for

Just as a very large studies for simplification As a result of such a graceful turns.

This integration is also not a very difficult task.

But it is important to be able to remove them.

The results presented here.

Wherein v, that these descendants axis moment of inertia around.

No axis moment of inertia I I do not care if you say you

an integral calculus.

Here is x squared plus y squared and now you can find here

integral can be calculated.

The second example is the granddaughter volume

z, and theta cylinder coordinates to account.

This surface area accounts We have to also use it.

Of course there had two coordinates.

That the cylinder

coordinates of course volumeters we will have three coordinates.

The software to its volume,

We give the details, but the two Let's see what options are available immediately.

Because we have previously seen, easy integration on theta.

Two pie from scratch, going from rest theta involved independent of volume elements.

But before we can get our integral on on or before r integral can get.

R on here before We show the way to integration.

See all the details given here.

There are Ron.

there r.

r ro connected but no longer ro give up because

Instead ro r, z, and theta will be.

When we do these terms integrals in this way is going.

Although it seems like a lot of confused easily terms of this integral coming down.

Second e, options cylinder on z coordinates ago

After making the integral over r, always already leaving on theta.

Here on the surface of the grandson of theta

When none of z'yl r the relationship of this kind is happening.

Indeed, this is the circle in the order of z When you look at center b, which is

hence r squared plus b minus z

square is equal to a square equation of a circle.

Here you can solve the z and Before this integration over z

whether by the integral over the r varabilme look forward to the same results.

But in these difficult Do not bother you a bit,

which is a good thing because the only course anyone with problems resolved

socialists have to submit yourself too You need to solve some problems agonizing.

This is a pleasant problem I think.

Certain answers.

Thus, our class is finished.

This department is finished.

This section is finished.

Hitherto various We calculate the integral.

On the surface and on the volume.

If the vector fields and integrals along lines

On both surfaces, and volumes on the need for integration.

Many of you is to duymuÅu.

Of Green's theorem, Stokes' theorem, Gause as theorems are theorems.

These are the basic equations of nature Removing the largest vehicles.

The last of these features in our department,

The fundamental nature of this theorem We'll see how the equations given.

The basis of the following equation I mean:Continuous

ie mechanics of solids, liquid bodies and

gas fields of the environment that the dynamics equations.

Here is the mass conservation, There momentum conservation.

Heat conduction equation.

This protection of the heat energy mainly using.

And electromagnetism the still these equations

Using theorem is achieved.

Of course all of you in all these areas You are not going to be experts, but

What you'll see that mathematics combines these things up.

An important physicist, Nobel Prize-winning In many places these words of physicist Wigner

has echoes of mathematics staggering physical origin so as success hakkaten

You can even look the same in different areas theorem using the same methods

From one side of electromagnetism equations can achieve a

machine elements for use in hand, used in gas dynamics, fluid

Or equations used in mechanics even used in heat transfer denkem

which is used in quantum mechanics equations can achieve very similar ways.

The fundamental nature of the issues Some even conservation

Even istatistikd total a chance happening.

There is also a conservation there.

Obtaining the equation we will see that we have seen all

a curve on one integrated alarm on a surface of an integral,

in volume integral, i.e. in one dimension,

in two dimensions and three dimensions of the integrally one obtained by using a

and an apex that is will be issues that we see.

Bye for now.

This new section to discuss.