Hi there.

Open or closed curves in the plane

n t and u on the u We saw integration.

In particular, closed-loop, In practice very often.

Already in the complex-valued function on this closed loop,

On closed curves, cycles We have important information on.

As now, a little semi-speculative I want to start with something.

We have two such integral.

u t u n d s and d p.

If these integrals values ??independent from scratch,

This orbit is independent of integrals will be zero.

It stands for u t, To calculate D x, v d y.

u n d y, and d stands for n minus x

because it has components, In this way they are going to zero.

I also know:When these be independent of the orbit of this integral?

If u y v x is equal to the first type, If so there is x minus y equals zero.

In the second type, see u v's thereof, minus its derivative with respect to x.

This is the minus minus minus one there,

plus it also to y derivative is equal to zero.

We have seen them.

Cauchy-Riemann them anyway conditions as

complex valued function emerged.

We can think of this speculative as:I wonder if we take them

While we integrate over an area, Do something useful out?

V x minus Y obviously to be zero, which translate this c

If we say that the area of, so this will be zero by definition.

D will receive a zero on the integral.

Similarly, wherein the zero We'd take over an area integrated.

Now, what we have found?

That the first integral so that on cycle

The integration of this cycle turn on the field,

wherein using the components is an integral zero.

Similarly here.

Now that's one I so we s two is equal to that of a base I

so this is what we say n'l from Integral is equal to the integral of this.

Now this is a coincidence?

Or under it more Is there something deeper?

Do you have a more general knowledge?

Them or are they just Is it because resets?

This zero, zero more of here Do not say anything that comes to mind questions.

Of course, in science before already After beginning with the appropriate estimation

by proving progress has many time.

My yalaÅ this inductive, We say that the inductive approach.

Describing the deductive is told through but,

to see this sort of thing important from an educational point.

Now we have two examples Let's try it on.

The characteristic of this example the following:This is the first example

with x and y components minus We will see in the example the cycle,

t projecting cycle different from zero.

I wonder if that same value for the right side of Is it going to be different from zero?

We want to do this.

A second test we want to do more.

It is here that an x components with x and y in the plane vector fields.

They are coming out of here, for example as a cycle tornado

Or a wind cycle Solar storms, such as the

Gorse them online How we express?

An angular rotation speed If the direction of the k omega

in the direction perpendicular to the plane, Combining the starting point

Get the vector by vector multiplication See As we come to a conclusion:

Means multiplication of vectors i j k We are writing the first line.

In the second row vector, k vector subject to x and y components are zero, z directions

and x is a vector component in the plane, Because the x and y components.

The third component is zero.

I see it when we open component i of the row

We take temporary.

Columns are taking.

Zero.

If the product on the second diagonal y.

But we put it by changing the sign, minus y.

The second component, component j,

and the column j We take the line temporarily.

We start with a minus sign.

See the product is reset to zero before.

Zero.

Minus the product of the merger of x.

x. But there is a minus because of this jar.

Because the plus signs minus plus he is going.

Plus of minus two as we find it.

This means that the vector field indicates a rotation types.

If we draw here that at each point The angular velocity vectors from speed

As you can see judging as vectors In all of these have an angular velocity omega.

Apartment for her in this sector area There are parts of equal area.

But is increasing with distance The heights of these vectors.

In a second area, let us take the x and y, When we stopped at any point in this

extending away from a center we see that the vectors.

If that physical sample and, an electrical charge to such

here about putting the electric field Or water goes in this direction

supply and water if I put your fÄ±ÅkÄ±rtsa there is spread in the same way in every direction.

In an idealized situation, of course.

Now two of them,

we can even say that the opposite of each other Two field vector field.

Now let's get them.

We're taking this first vector field.

Here u t will take time.

I also had this one.

On this account, will this cycle.

What is on this cycle?

're Getting an apartment.

On the center of this circle, coordinates of the center

at the start of the radius a circle with a're getting.

This physical connection hani If you want to make this angle,

omega-times proportionally over time.

Under such a bi There is also a realistic model.

Wherein the vector field We have seen that be the case.

If we write it in circular coordinates, y is obviously

Once sinus t, r times the cosine of x t.

r taken out of this circle,

other than the circumferential direction tangent vector This is not something negative sine and cosine t.

So with such a return, with cycles are concerned.

This account is easy to do.

Here is a cosine of x t, y t sinus, d * certain, specific dye,

d x, take them to the point If there is repositioning or u d x plus y,

As you can see here first with the first,

with the first, ie with negative y

of this second sign when it is substituted, and