Hi there.

About the preceding session We prove two theorems.

One of them is quite We have seen in detail.

This projection of the t integral along a closed curve.

You can see the expansion of this integral D x plus y as any other.

The two-storey integral to the have proven to be equal.

Similarly, on the projection of the n gene expression also appears here

and the gene thereof in a two storey We have seen that the integral is equal to.

Mathematically, a work cycle integral, one cycle integral,

that it can now.

Less if you say u to v, If u and v in the first turn to.

Similarly here Remove the identical two-storey

integral, but in the back of their physical meaning is different.

This is one of the on the tangent vector

i.e. projection on walls this integral accumulation.

By this a second wall One measure of a side passage.

Because it is the vector perpendicular at each point of the surface

Upon projection of the unit vector 're getting and we collect them.

This is the u n d s point gives the integral.

Open their software.

Now we want to see a few examples.

So we also we do on account of the cycle,

we do on the field at the same We need to find a result.

Our first example of this theorem is a useful, show n.

Such a linear integral given account.

As you can see quite mixed terms here.

Coordinates of the center of this team in that in the center and edges

the circumference of a square of two says on the accounts.

This is not something that can not be done.

But on four separate correctly 'll do the math.

There are quite complicated terms.

Now will you do it with respect to x, it, is not so easy a job.

x cube times sine squared x partial You'll make them with integral stuff.

This is quite a complicated job.

But we are now using this theorem, We prove theorems of Green,

instead of calculating it with the same result We can return to an area integral

and perhaps this area of integration We would be easy to find.

Now first, following by Green's theorem.

Integral given to us like that.

Green's theorem there is x plus y in D there.

So the first term where u x cube term.

The second term this sinus The term also has hiperbolikl.

Theorem tells us to make this account Is there xi instead of ALSA, that u

y If you leave the field integral on the even if you can find the same result.

See, there comes a time that we get x.

when u get y a minus involved.

But in theory a minus sign because it is a minus minus one.

Both times the integral over the area, that area twice.

The two sides of We know the area of ??the square.

Four square.

There are also a couple of factors here.

So eight square just as involved.

Of course, this sample a little of this highlight the usefulness of Theorem

designed for one example.

But in many cases this can be difficult.

So the integral cycle can be difficult.

Using Theorem space We can do the integral over.

Or area of integration It can be difficult to calculate.

Area integral was asked to do.

Integral to this cycle turning to easily account

brings in the possibility.

But this is not always going to be a very useful I'm guessing that there is something.

You can see.

That the direct integration A very long to make,

a lot of mistakes is an example to be made.

A second Green theorem we can do well with.

Because the second Green's theorem us says the following:Negative v times d *

plus u d y times, he says.

Means that the term is now y'l as we will describe.

y'l term is now here:So x, Our hiperbolikl term sinus is happening.

As there is in the negative x We will determine the multiplier.

Therefore it it has the minus sign will be.

As you can see here has changed In case you have entered the sign of a difference.

But the result will be the same, because here will take different derivatives.

These two variations of each other take the results will be the same.

We take the derivative of u with respect to x.

A derivative of u with respect to x.

Plus, you're getting v is the derivative with respect to y.

Refer to the outside where there is a minus, minus the inside.

This is the first term contains only x Get them to the falls derivatives.

Similarly, the terms herein y'l When we take the derivative with respect to x had fallen.

Here comes a a.

Again taking twice the area eight frames is achieved.

There is also an application like this, a generic application.

We uu minus one-half of x, y and and we take it first vector comprising Green

we apply Theorem, see D x plus v d y, v x minus y will be equal to u.

So once d x plus x minus y

d will be equal to y times two-storey integral to the right.

v where x.

there is a future of x.

Less than u y u got us here.

from Y minus future.

Here is a minus will be given for two.

So a divided area twice both times.

A means of finding a ground Find the general formula would have.

Let's make an app like this:Corner a, team that is at the center of coordinates.

A corner x axis on a wide.

One corner of the y axis on the b-high

triangle area Green Let's do a checksum with the theorem.

Because such a triangle We know what that area.

One-half times the length of these edges,

Because the height b, b is one-half times a is open.

But we of Green's theorem As an application,

Our hands are used to doing it.

Now we've found the following formula:Area A minus two times x plus x divided by y d d y.

If you get on the wall We find its area.

For example, if you receive map of Turkey,

if you make such a point of corners with may be subject to thousands of points.

If you want to calculate this area, This formula gives it to you.

Hes wall, When this integral on the account,

And when you take little pieces You can zoom in with them correctly.

This quantitative analysis is entering quantitative analysis, but also

their origins in basic information.

This too is a formula that can be used.

Now let's do it.

We're going from A to B.

Its width, base a.

Then we go from b to c.

E the right of the equation can be written easily.

where y equals we've already given.

y is equal to b divides a.

Because of the slope, b is the height of a base, according to

b divided missing, but because descending anything.

For him, have a negative x.

when x is equal to y is equal to zero.

Right.

when x is equal to y is equal to zero.

We are at b.