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Hi there.

Our previous session of this of functions of several variables

location, general mathematics education We had a brief look inside the place.

In this course, we're doing two parts We remember what we did in two parts.

The reasons for this, the first of my people we give the basic concepts,

I mostly second person applications that do

newer methods We remember what we see.

The main reason for this, if you derivatives at the end starting with

If you go up to derivatives of a section, then come to the integral,

If you take away integrally has two drawbacks.

Once differentiation and integration complementary concepts.

Derivatives explains the differences.

The difference tells the story of change.

This exchange calculations two main landing and take action ratio.

Already four treatment Both are doing this job.

In a collision integral and We know that collection.

That is in fact what we make not different from the four operations.

Limit of one course a very important concept have the concept of seeing these concepts

then until the end of e, derived gone to integrals relationship is forgotten.

When it comes to the integral of the derivative relationship is forgotten.

However, seeing these two successive complement each other, how the concept of

, and even better seen I think you can understand.

Therefore, these concepts We see together.

Then take derivatives such to go to the end of the subject,

Instead of going in one place There's a pause as benefits.

Do you understand the basic concepts.

Without understanding the basic concepts without thoroughly internalize the very

so when you go to wide application in my experience to date,

for others, even myself as a student disappearing people in a place.

For this reason, we distinguish between such.

I showed you earlier in this list.

The topics in this second part and section titles.

I told them in brief in the previous session.

Now we are entering our subject here.

The tenth episode.

You'll recall in the first part do, There were nine chapters in the first section.

Circular coordinates two-storey integrals.

Already circular coordinates consists of two variables.

Related integrals clear that the two-storey.

The two-storey integral Cartesian coordinates, albeit circular

coordinates, albeit more general curvilinear coordinates, albeit a basic approach.

In the first step would be locking a variable.

Other modifying the test trades are variable.

The subsequent step is already in the problem is reduced to a single variable.

Univariate as we know it coming to integration of functions.

So what we're doing at each step Univariate make transactions.

In partial derivatives to such two-storey, in an integral way.

50-storey integral, albeit

49of them before you make temporary Are you keeping a constant

So all of them are changing In step one variable are trying.

In partial derivatives such 50 even if you have a function variable

here, of course, partial derivatives 50units will be partial derivatives.

49can we fix temporarily, Are you getting the derivative with respect to a variable.

Then you can leave it to someone else variables can make.

It is estimated that as many This account does not have a great difficulty

Once you understand the concept, Once you understand this method.

Why circular coordinates important that we learn to account.

Because the circle of nature and technology spontaneous way.

I told you last logged in though a bit exaggerated

civilization with the discovery of the wheel There are allegations began.

Here in the art such as

In the ideal nature of a circle emerges as an illustration.

He also important, in some cases, Even if the circumferentially-

circle appeared likeness cykan in ways we do it.

A model that corresponds to the says there is a way.

For a two-storey integral forever have small area requirements.

In Cartesian coordinates, what are we doing?

We choose an x value.

Here are passing vertical line of x.

x plus delta x we choose.

That this is a sabitleyin Coming opposed to vertical lines.

Similar thing y and y plus In the delta structure this year

from the intersection of the four lines here an infinitesimal area occurs.

Delta x of x because its base We're going to x plus delta x.

Height due to the delta y this area is going delta x delta y.

To this differential ie, to the infinitesimal

When we take as symbolic delta delta y from x to y d * d have varÄ±yo.

Circular coordinates the the same approach as we did.

r is a fixed ring.

the circle of radius r.

he radius r r plus delta a circle so hard.

How these two lines gives as two lines here.

Theta equals a constant DC.

Starting from the center and which slope towards theta.

If theta plus delta theta horizontal who coordinates this with the right angle.

Thus two lines There are two lines here, too.

The intersection of these four lines to us again describes the infinitesimal rectangular element.

When this area of the account, This is roughly a trapezoid.

R times the base delta theta.

Current base.

Height is the delta.

This time in our having hit two r delta theta, delta, we find r.

When we brought this to the infinitely small We find r d r d theta.

Thus an area of a curve

the region bounded by the curve d x d y on the field

r d r d theta integral or by integrals over occurs.

Now they Some applications without

circular curves Getting to know the coordinates.

If r is a constant R equals a circle like.

y is equal to x is an alpha

tangent to a right angle alpha does.

This is equal to theta b.

Because of this, this all on the right

connects to the hub the right angle is always the same.

Tete.

So theta equals b gives the correct one.

As a curve is often encounters.

Radius of the center as a shifted on the axis,

An example of a radius taken here.

In a radius of using this center 're getting circle drawing this circle.

This shifted the center of a circle.

After that we will see a bit of the equation but they are remembered.

r is equal to two times the cosine of theta is happening.

Had a radius rather than a two times the cosine of theta would be.

Current varieties growing e, drop coils Or are spirals equals

i.e. R is A theta or theta tetayl r linearly connected to each other.

This shows such a helix growing increasingly radius.

Theta is initially being zero is zero.

Where pi divided into two theta When r is equal to pi divided by two.

When theta equals R equals pi pi.

As you can see theta also increases with increasing r.

Here such a helical or 're getting pop-helix.

Such a.

Curves are often encountered.

For example, a stone from a point, a water into a

If you throw a stone these types of waves generated Go to functions, or

From this point, the electromagnetic Please Ã¼retse area as an antenna,

With this kind of functions comprising electromagnetic fields can be expressed.

This kind of laughed at functions We call these types of curves are

as a cosine of theta is equal to the If n equals one already

I would get it but it is equal to r curves for the two of these varieties involved

How do we find it:Theta Zero r

here is a cosine Tetala As may also be sine Tetala

Does sine cosine of theta here Tetala Do you understand that you may want.

Tet r is equal to zero At this point of the theta

When you see grown as if you make here forty-five degrees

This will be the cosine of pi divided by four

Because the cosine of pi divided by two cosine of pi divided by two is composed of two zero.

So the forty-five degree becomes zero when you arrive.

That's a zero in the way of such as a leaf occurs.

We also understand that this curve.

Similarly, curves called heart curve can be any of the following structures:

or a minus cosine theta plus cosine theta or a plus

As to minus sine theta can be expressed in four types.

If we take the first theta The cosine of zero zero one

because it is a minus zero.

So we start from this point where pi divided into two theta

In that time theta pi divided by two is means we are on the y axis

cosine of pi divided by two, so ninety cosine of zero degrees,

mean cosine of theta is zero, r is going to say that.

Tete pie when it comes, that such wandering on the x-axis

When we arrived it cosine connecting this point to the center

towards the cosine of pi it's a negative cosine

hence one minus minus one or two going on, means that the two are going to say here

in this way the less the scale We come to this point is a convenient two.

That means that we start from scratch We went to a two-au found.

If we continue this, we see cosine functions like a symmetrical

The same is the minus Tetala will be're getting such a way.

Another family was a family of conic sections.

We know that even middle school since high school, a variety of cone

If you cut through the plane you Remove the limit of a closed curve in the

Remove the parabolic curve of a single open branched or cone bottom open curve

If you continue in a double-stranded open Remove the hyperbole that this parabolic curve.

All these functions circular coordinates shown in the following structure.

opposite to be the one that comes parabola,

E is greater than one opposite hyperbola income,

E is smaller than the opposed to the elliptical income

wherein a and b of a If we say that half the axial length,

If c is the coordinates of the focus and If we choose this way to coordinate team

i.e. at the beginning of the coordinate sets You can select a focal point.

This focal point O and B, namely

This works out to an ellipse equation where d is what we give.

ee coefficient merkezkaÃ§lÄ±k It's called eccentricity coefficient

and is expressed in this way is called.

This example of the world the sun movement around this

is an elliptical physical You've seen your course.

If you throw a rocket a critical

quickly, when you focus is on orbit

you can go away from similar as hyperbolic as you can go again.

Here they frequently encountered applications and functions.

We want to start a simple operation, r is equal to a circle

limiting the areas where two storey integrals we want to account.

Of course this result we We know radius r

which is bounded hoops circle's area to the square of the pin.

Now let's calculate it.

d rd d theta is We have seen the area means

r dr d theta two-storey with the integral over this space.

Now write these limits is important.

We will see this in writing soon.

Moments to account m * in which we say we'd like

from the first moment of the x-axis with distance from the

but this distance is defined as Or first hunch years

where x is called instinct seems wrong to me but we think we are using y, then x

as the distance from the axis defined this this are obvious.

That in Cartesian coordinates that we are writing.

We came to the circular coordinates When we know d rd d theta.

If y is such a point

and the determined point tetayl x and y coordinates of these

When we get to this point projection wherein when the horizontal angle theta

is the length of the vertical cosine theta length r sine theta.

With this Cartesian circular coordinate transformations between.

Instead we say that y is also a sine of theta

There is also d from thus becomes r squared.

This kind integral us x relative to the axis of the first torque data.

Similarly, according to the y-axis this time in the first moments

it happens x times d Cartesian If y'all saw the coordinates but forget

also an opportunity to remember the y axis If the distance is determined according to X,.

Still sounds like the opposite instinct that this is precisely the right

We find our logic, because this from the axis of the torque arm are looking for.

x is the cosine of theta is Put here again unless

r r squared one occurs r dr d theta one is from

dr d theta to see integration as a two-storey.

Here the x-axis center of gravity and on the y axis on the

The coordinates of the center of gravity in this way is given.

When we come to the second moment in the x-axis

The second moment by the square because y from the infinitely small element's here

The x-axis by the length field The second moment is the square in y years.

According to the y-axis, again, these infinitesimal area from x

this is complementary in the sense here comes the x squared.

They coordinate transformation we again in terms of x r y r sine cosine theta

As you can see when we put theta here is an r squared cube is made here

where d is coming in for a we obtain the formula for this kind of work.

Let's make an application to the apartment Let's find the area and moments.

We know that, but we know on a sample

again, we have developed tutorial apply theories.

R and theta is important here limits on a letter

In order to complete the circle r will go up to scratch,

Up to two pie from scratch theta will go so hard limited integrals.

Fixed limited integrals where R is d

hence theta integral comes these integrals can be separated.

Is the first integral is zero until a subsequent

the second integral zero Up to two pie on theta

separable because integrals the borders fixed and functions

r r function as a cross sets a time d theta

r is once you see it when you have made we obtain the pi squared.

d theta pi comes from the two.

r r squared divided by two from the integral because it is a square divided by two is coming.

Twos will cancel each other out.

pi squared involved.

When we look at the first moment of the first moments would be zero already

I can say, because the center of gravity zero point zero in a circle.

Indeed, previous We also do conversions

m x them as circular coordinates worth remembering.

Home to the x axis when we get recognition The first moment x

from the axis of the infinitely small The distance is determined by the area.

d y times.

When placed instead of y r sine theta sine of theta is an integral and

square of the integral occurs.

is the integral of the square zero but the sine of theta

integral from zero to two pie zero Do you think because the sine of theta functions.

Plus the area minus the area until Or has the formula

If you look at this integral will be minus cosine theta.

A cosine of theta two pita.

A zero.

Will cancel each other out as a minus.

Will be zero.

cosine to n in the same way the integral of theta

Up to two pie from scratch is zero.

Because under the cosine of theta plus much negative space in the area

each other, but it's still there and is taking

If you want to numerically the integral sine cosine theta theta.

Sine of theta zero in two pie.

Ie at zero zero zero minus zero occurs here.

Its going to zero.

Find second moment If we want both to this,

in mechanics, strength by movement of a body

Or about the way related to the change in size.

In electricity, this magnetic have to do with couples.

These x and y instead of the When we put values

because i x x y squared sine Does the integral of the square.

Here, here too there is an r squared.

is the integral of the cube is coming.

Here the integral sine-squared immediately In terms include double integral.

Here comes the bi p.

Here r divided by four from four comes in four divided by four.

Similarly, the i i y and x y.

where b is interesting because i x and y sine cosine bi coming.

Get it by the integral Or two sine cosine times

is twice the sine of theta pi Thinking

of the integral of this zero that emerges.

Or numerically If the account of sinus

If the cosine of theta d theta d be.

Divided by the square of the sine of theta than two above zero in the limit.

Down to zero in the limit.

Therefore zero.

In this way we come to conclusions.

Half of the same problem as you homework I want to find on the circle.

Upper limit only change.

Here you go up a pie from scratch

If the limit for All other accounts are the same.

This is a useful exercise.

Wherein both of the results obtained in some cases complete circles

in some cases twice again you will see that similar.

In particular, let us note that m *.

m x on the x axis resultant torque from a distance.

I.e., x-axis by the weight thereof where y is the average of that center.

It is clear that it is zero.

Will be here in one place.

This is roughly zero 45 times the radius turns.

This is an important integral integral.

In this first part we did.

I'm here again.

This probability calculations encountered in Is the integral of the Gaussian bell curve.

This curve something like that.

And this also heat distribution problems and

In many more problems It is an integral encountered.

The only integral to this closed account storey integral impossible.

Single integration of the easy to calculate the square.

And that circular coordinates can do.

A nice application circular coordinates.

to square minus one-half years we take the integral from minus infinity to plus

value thereof at this forever as the integral over x.

Because at the end of the x and y finally lost integration variables.

Therefore, as we both Putting Since we find i frames.

But when we combine these two integral minus one-half to two

x squared plus y squared infinite plane is transformed into an integral over.

Because x and y from minus infinity comes to plus infinity.

Now we circular coordinates If we use x squared plus y

r squared frame will be.

d * d y r d r d theta will be well.

This integral is easy to calculate.

Current maintenance is easy:minus one-half r squared minus d r d r u would have to say.

But there r d r exceptionally nicely.

Also integral to the variables separable types.

Once this function d

teta'yl r d e r squared minus r can be thought of as integral.

To cover the whole plane circle with infinite radius b.

So the bi previous apartment problems of the same boundaries b

difference goes to infinity the upper limit.

Gene theta zero to two pie.

upper radius is zero as well.

What is the greatest value of this radius here?

Endless.

Of course now the first term allows us to two pi.

The latter is very easy.

This variable transformation by di going to try later.

Here you will find a value immediately e to u because u e to the integral.

BI is coming here because of the minus sign.

BI also give zero when r u is zero.

But we give r forever When going minus infinity.

Less zero at infinity.

If the value of a.

Therefore cons of this

transformed to the second axis plus two're getting the value of pi.

I is the square entergal whereby two pie.

i is the square root of the integral of it.

Two is the square root of pi.

This is an extremely important integral.

Here is a generalized Gaussian from here function as we can.

Here we've worked divided by two.

However, where b sigma sigma were square, this square

The second moment is easily found.

This shows the standard deviation.

These little ear for now for the promise of fullness.

But you'll see them frequently.

Here an algebra is a cross- are faced with the function.

A higher algebra functions.

This error function is called.

So this integral from minus infinity not forever

from minus infinity to x if you receive this

integral to the interior of X, x ayrÄ±d Let's say for example that x to the base.

This would function b.

is a function of x.

This error function is called.

Because of this error probability function accounts of encounters.

Bi in this way in the heat equation we encounter a integrals.

Now I want to call b.

I have seen the general concepts circular coordinates.

Now a bit more complicated by the I can say a little more

issues to dominate We will see examples will require.

I now adjourn.

The next session goodbye until your opinion.