Hi there.

Today we begin a new section.

I mean parts very close to each other The infrastructure consists of the departments concerned.

Here are three sections.

These vector fields on the plane, general vector fields,

Stokes and Gauss also Green with theorems in nature conservation

the laws of nature and the obtaining equations.

Now you've seen it before vector fields.

But we do introduce the definition of a

whether he is a memoir of I want to address again.

Vector Fields said.

Now that we have ever seen What is the relationship issues you might say.

The answer, in which a multivariate functions.

However, a species that some features.

Other vectors FUNCTIONS very The relationship between variable functions

What we thought was very variable functions,

as well as the independent variable x that vector, i.e. multiple in numbers

y are defined by the dependent where the vector of variables.

Of course this is the most general form would not be right to start.

For this reason, we have received the most simple two-state.

This is a not a vector of x If you mean a vector component

so that a number of the job to kolaylat To facilitate representation

From this x single junction t call the number formed.

y instead of x we say, but see this bold because it's a vector.

Why does this have?

Because he usually shows the time this

kind of functions in the implementation of x indicates a location.

When these two get together these vector functions or

vector functions and space-qualified in They show a geometric curve.

When we get a vector x of the second end y is a number, but we're going.

This is the exact opposite of this situation.

Here was a vector x, y, sorry x It was a count, has a vector y is.

Here is a vector x, y number.

Again for ease of writing No. small vectors of two components

and only because it is said to them x and y instead of y in the z component comprising said.

These x y z in the space of three-dimensional space,

two variables digital functions are showing.

This anti-geometric surface in space.

Now we are going to do in functions Y is a vector, a vector x.

As there is a difference alone.

y usually physical a qualified function.

E.g. a force, such as an electric area, for example, a heat flux vector.

And usually it again at this x within the physical geometrical characteristics

Thank x y z position consists of coordinates.

t the time.

That is a kind of expansion of these structures, so that instead of x

where t the only place you're getting but also can t x y z.

This is physically qualified but The small number of components,

having three or four components,

Although the strength of such a force three Is the physical size of components.

Although an electric field There are still three components.

As a guest at the three locations at all times once our work is not necessary.

We will see them, but suddenly where the need is greater.

If one is already in space orbit, the opposite comes to a curve.

A more general nature, than one of x that many components of a vector.

the plurality of y component of the vector.

But here is another thing We need to simplify.

Because where x is made ??easy.

Here y made ??easy.

Where both x and y in the ease of many does not have too much difficulty, but no.

50For example, one component does not exist.

Not that the 80-component.

Physical requirements sufficient to There are number of components.

Where multiple Each of the numbers is ouÅuy

but most of the vector f We take a simple function.

The most basic functions that is a linear function

the only independent variable first force visible.

Then y is a y components, y two, y n,

If you say y me, it's that x, x a, consists of two components, such as x.

Then can be written as follows.

forces the first coming years, x forces also comes first.

Joining them call the numbers a i j.

Because both carry the information here x in need, as well as the information in y.

As you can see, these two-dimensional a vector in a sense,

Not that the vector simulation as saying.

They are creating a matrix.

This is not our concern, not the subject of this course.

But again, this multivariate most of the functions

simple function but has multiple components.

If you pay attention here In each particular

Thank you x in the first two y, we have the simplest.

Here we take the simplest function.

The vector fields somewhere in between.

Yes consists of multiple components of x and but does not occur from too many years.

For example, the matrix of a thousand thousand dimensional matrix can be.

It does not force us.

In this derivative transactions do not have an integral, with algebraic operations can be made.

This linear algebra creates issues.

This is not our topic.

Now if we go a little at first Do you remember our approach.

We work in three-dimensional space before I would like to understand what the issues on the plane.

Once you understand the issues on the plane usually easy to get in three dimensions.

At least conceptually simple,

calculates the course more different things will occur.

As a vector field Let's try to define.

It's such a precise definition, but also in which physically dependent variable

location and time independent which is also a vector variable,

but physically the vector a vector We call the vector field function.

Or two or three in most of the physical applications

as four independent and dependent is a vector of variables.

We'll show u'yl areas instead of years.

What could be things like u?

Speed. But the velocity of a particle is a vector of three

is a vector of size.

Rate multiplied by the mass of momentum.

Force still requires three dimensions.

Three dimensions that moment.

From these mechanical issues.

The stream of gas may flow, a fluid may flow,

There may be a heat flow.

Electrical current may be.

Issues such as the electric field requires gene vectors,

gene vector magnetic field requires gene vector.

As you can see classic In physics will limit,

Finally, because this physics lesson, but a To feel the relationship is worth.

T. The often physically t represents time issues.

x y z shows the location and the internationally recognized

To us he has seen other We do not use a display.

We have said that the x, y,

t z and t, but that there may be There are significant problems.

So when independent events We call these static fields.

Electrostatic, magnetostatic, in strength, mechanical, static,

heat conduction time as independent static events.

t is found in their we call dynamic range.

Their opponents know.

Hydrostatic time here would have been would hydrodynamics are independent

Or would electrostatic would electrodynamics.

When this dynamic show.

Let's see some examples.

One of our daily life often Do you often see on television,

Turkey's road map here.

They show the wind currents.

Here at each position in space a vector defined.

All of these vectors a vector field is generating.

Shows the currents in the seas have here.

Of course, the wind will be onshore wind though.

This is a droplet of fluid at the time of falling down

The current generated in shows or here are a constant wall,

where a fixed wall, a forward plates are placed in a liquid.

When you take this in the middle of the plate,

formed therein shows the vector field.

This vector field at each point comprising speeds of lipid particles

or their speeds at which shows The issues encountered in technology.

A fixed shaft such as a cylinder and rotating it around the track.

These arrows within it such as a fatty shows the movement of a liquid may be.

Here it is defined at every point in space vectors forms a vector space.

The magnetic field of the natural magnetic field of eye We can not see, but to be able to visualize

In all these experiments for high school We did put a magnet

If you put around the iron particles, this way we see that the lines.

These are magnetic field visualization, each point a

If you thought it was a tiny magnet that the magnetic field inside it

If you put them all vectors consists of the vector defined in point.

Or consisting of a GÃ¶kden such that a lightning

a visually We realize the event.

Here's the underlying electromagnetic visualization of the area.

A magnetic field at any point This work consists of an electric field

going so well visualized.

Here a vector of each point by putting You can define the vector field.

Heat flux; Of course you still can not see the heat.

Substances may be carried by a heat,

we can see the movement of the substrate but the heat can not see.

But at the same temperature nok,

if we combine points with a line wherein t is zero, t is one, two t,

t the same temperature points in the three so-called obtained by joining lines,

perpendicular to their orbits indicates the heat flux.

You can put a vector to each point.

Here it is a flow of heat we obtain the vector field.

This is a mathematical thing.

There is a physics course, but under this In mathematical notation.

Now I want to stop here.

I need a break.

These general terms of culture events A visualization, understanding,

why they are doing them a We had to arouse interest.

Now then mathematically them What problems occur as

their interests and how these problems we will try to solve vision.

Bye for now.