Doğanın Temel Kısmi Türevli Denklemleri ve Sayısal Alanlarda Korunum Yasaları: Kütle, Isı ve Elektrik Yükü

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En provenance du cours de Koç University

Çok değişkenli Fonksiyon II: Uygulamalar / Multivariable Calculus II: Applications

9 notes

Koç University

Çok değişkenli Fonksiyon II: Uygulamalar / Multivariable Calculus II: Applications

9 notes

Ders çok değişkenli fonksiyonlardaki iki derslik dizinin ikincisidir. Birinci ders türev ve entegral kavramlarını geliştirmekte ve bu konulardaki problemleri temel çözme yöntemlerini sunmaktadır. Bu ders, birinci derste geliştirilen temeller üzerine daha ileri konuları işlemekte ve daha kapsamlı uygulamalar ve çözümlü örnekler sunmaktadır. Ders gerçek yaşamdan gelen uygulamaları da tanıtmaya önem veren “içerikli yaklaşımla” tasarlanmıştır. Bölümler Bölüm 1: Multivar 1'in Özeti, Dairesel Koordinatlarda Entegraller Bölüm 2: Türev Uygulamalarından Seçme Konular Bölüm 3: Çok Değişkenle Zincirleme Türev ve Jakobiyan Bölüm 4: Uzayda Yüzey ve Hacım Entegralleri Bölüm 5: Düzlemde Akı Entegralleri Bölüm 6: Düzlemde Green, Uzayda Stokes ve Green-Gauss Teoremleri Bölüm 7: Stokes ve Green-Gauss Teoremleri ve Doğanın Korunum Yasaları ----------- The course is the second of the two course sequence of calculus of multivariable functions. The first course develops the concepts of derivatives and integrals of functions of several variables, and the basic tools for doing the relevant calculations. This course builds on the foundations of the first course and introduces more advanced topics along with more advanced applications and solved problems. The course is designed with a “content-based” approach, i. e. by solving examples, as many as possible from real life situations. Bölümler Bölüm 1: Summary of Multivar I, Integral in Circular Coordinates Bölüm 2: Topics of Derivative Applications Bölüm 3: Chain Derivatives with Multi Variables and Jacobian Bölüm 4: Surface and Volume Integrals in Space Bölüm 5: Flux Integrals in the Plane Bölüm 6: Green in Plane, Stokes in Space and Green-Gauss Theorems Bölüm 7: Stokes and Green-Gauss Theorem and Nature Conservation Laws ----------- Kaynak: Attila Aşkar, “Çok değişkenli fonksiyonlarda türev ve entegral”. Bu kitap dört ciltlik dizinin ikinci cildidir. Dizinin diğer kitapları Cilt 1 “Tek değişkenli fonksiyonlarda türev ve entegral”, Cilt 3: “Doğrusal cebir” ve Cilt 4: “Diferansiyel denklemler” dir. Source: Attila Aşkar, Calculus of Multivariable Functions, Volume 2 of the set of Vol1: Calculus of Single Variable Functions, Volume 3: Linear Algebra and Volume 4: Differential Equations. All available online starting on January 6, 2014

À partir de la leçon

Stokes ve Green-Gauss teoremleri ve doğanın korunum yasaları

Düzlemdeki Green teoremlerinden uzayda Stokes ve Green – Gauss teoremlerine geçiş. Uzayda Green – Gauss ve Stokes teoremleriyle yüzey ve hacım entegralleri. Diverjans, rotasyonel ve Laplasyen’in anlamı. Uzayda Green – Gauss ve Stokes teoremleriyle doğadan temel korunum denklemlerinin elde edilmesi. Kütle, elektrik yükü ve ısı enerjisinin korunmasında uygulamalar.

Doğanın Temel Kısmi Türevli Denklemleri ve Sayısal Alanlarda Korunum Yasaları: Kütle, Isı ve Elektrik Yükü25:47

Rencontrer les enseignants

Attila Aşkar
Prof. Dr.
Matematik Bölümü (Department of Mathematics)

Hello.

We have come to the last week.

Stokes, Gauss and Green this week important theorems

We will see applications.

These conservation laws in nature and basic field equations.

What are these?

These continuum mechanics, ie rigid body mechanics,

body fluid mechanics, gas dynamics, heat conduction and electromagnetism.

They are the foundation of classical physics Gauss and Stokes fields Green

one of the most important applications of Theorem one in the derivation of this equation.

Some of you may say that I Or you'll be an electrical engineer

solid continuum with What do I have to object?

Or heat conduction What to do?

Those of you who say you can.

But you'll see, that's the power of mathematics here, from various sources

Although the physics of these issues are coming, a common math mathematics completely.

Already Stokes and Gauss Green With theorem we

epeyl have made numerical applications, but most of these theorems

important applications of this digital account Beyond these equations is derived.

Already Stokes and Gauss theorems that Green

To generate equations As a result of deal has emerged.

In the pages of time before this theorem, long and decisive

These are non-derivation fine to hesitate in a clean way

and supremely easy place without shows the acquisition.

Of course, these can apply There are some physical concepts, but

Everyone knows them already.

iii strengthened and communication with them be established in another course.

Because of their mathematical but hardly an objective per

will provide an environment for many areas.

Now located before a Let's start with remembered.

iii frequently encountered and many different areas covering

We encounter.

Of the most important applications of partial derivatives One of these equations produce work,

is the solution of this equation.

These differential equations like this Our issues are beyond this but it

how it came equations showing where they come from and

It's just because of partial expression Not a fancy to

practical service requirements to respond to

You will see that emerge and will be fantastically easy.

Now we have them as a reminder I say we saw the wave equation,

In its simplest form the function of x and t The wave equation in one dimension.

c was the speed of the wave, also saw it.

This is the second derivative with respect to x, the second derivative with respect to t.

A second equation that also süz-

mA or leakage or heat equation is called.

See, while the second derivative with respect to x to t The first derivatives that are based on time.

The third type is completely position coordinates x and y.

t so there is no time bağmsız a physical situation.

Wherein the gene with respect to x According to the second variable

there is a second derivative, but the sign plus, See cons here.

These equations of events in nature They show many different types.

A sprawling nature of the events in information is dispersed locations.

For example, I'm talking to you my voice reach you.

Or is able to see me, you carries electromagnetic waves.

Wave equation.

This type is called hyperbolic,

As an analogy, because if you do x squared it to the second term and t

of course he can not resemble a square is equal to not necessarily literal metaphors.

See here for a hyperbola There are equations that for him,

Inspired by this, is called hyperbolic type.

Four of these two variables If we generalize variable space that

not only the coordinates of x If we take x y z, then these climates

according to the space coordinates iii becomes the second derivative.

How this equation is removed,

very easily We will see that removed now.

The second type into your tea in the morning

If you just put your tea spoon of the temperature After a while you will feel.

Yet there is a communication, but not like a wave communication.

You have gone too far When is this going to deflates.

Damping term for it is here.

There whereas the first order derivatives iii had the second-order derivative.

Yet the only space here coordinates of the three coordinate

If you remove the second order derivative of iii'l are changing.

Work in the case of elliptic functions.

See why they call it the elliptic where x squared plus y squared equals a

if you make an analogy like this one The equation of the ellipse is a circle, but the circle

this elliptical because it is a special case type is called, this type of equation.

Here you see here is x squared it with If we compare t with the same thoughts here

equals x squared with a simulation as t This is also the equation of a parabola.

For this reason, it inspired here, that has nothing to do with this parabola, the

therefore, to classify this simulation , to be used as a nice selection.

And we will take care of us here The classical world of quantum physics

other than the atomic molecular physics iii equation becomes defined equation.

See how it looks like x divided by There are second order derivatives with respect to x,

There are first-order derivatives with respect to t but here i have a good virtual,

has an additional term here.

But it looks a bit like structure with much of this equation

because of different complex iii valued function.

Its more that we We'll take care of the three species.

Now we have the conservation laws,

Some things in nature conservation law for example, the mass is protected is protected.

In classical physics, the mass would not will not be lost in the not created.

Iii Kaniuka This is going to very old, We see that the change in mass.

One of Einstein's mass iii other than that they work.

Electric charge, if an electric charge No, this is not all there, all the time.

Heat energy, if an otherwise interactions or thermal energy is preserved.

Other kinds of energy can be conserved.

With these numerical functions The quantities shown.

For example, in probability theory a again a numerical probability distribution

function it is protected.

Because the total probability.

But in addition to these numerical size vector magnitude, there are qualified.

Initiation thereof

If we consider the vectors Licá speed, but are rapidly converging momentum.

Momentum is protected, angular momentum is protected,

electric and magnetic fields are protected, electric current is protected.

Heat and mass flows have protected them the size of the work that these Stokes

Green Gauss theorem and how will be implemented and hopefully we'll see

what we have learned all of this a It may sound like the crowning

results is very important because these two is obtained using theorems.

Cause of the emergence of these two theorems already To obtain these laws

force to force people they find theorems.

Already Gauss in physics I know Gauss' law.

Green Gauss theorem we start from here.

If you remember a vector work with the unit vector perpendicular to the hit as a

If you get off on an area integral on the volume of this area close

comes equivalent to a three-storey integrals but here comes the divergence of u.

Similarly, a translation for t multiplied by u.

See previously wherein u as we discussed a

We're looking to accumulate over the cycle.

Here a surface is a steep one face to the other side of the surface

one face of said other limit measure the size of your face.

Stokes' law we know work here.

Gauss's why I'm writing before the first Gauss's theorem is taken care of so many things with.

By Stokes' theorem of electromagnetism some laws are taken care of

We will see them.

first, women

Let's start with conservation because it is the most concrete ...

Look at the size of our work When we see a mass.

There is a rigid body mass, a liquid There is a mass of gas, it is protected.

As typical approach in space, random We choose a volume, call it V,

Let's say that this limit Let's say the S off the surface.

This closed volume and surface random why choose a particular case

not always be valid here to the basic quantities,

bulk density that in this region there is a mass

density ro, an the outside an agent can come here.

First specific example of a gas stream You have selected a region, the region's

because there are varying in mass density may increase the density may be reduced.

A well outside this region There are the masses, have current.

This is necessarily a physical not necessarily in,

wherein a gas stream in general, A liquid stream Top

media is easily visualized You have a separate area for the telling.

Here a mathematical randomly select an area that we're

will write the mass conservation in the region.

If the nearby stream,

I'm sorry if you are determined with velocity v how much mass flow

If the density at the speed of its transportation is defined, we show that in the J,.

Yet the electric current j

We will also, because there is also a moving electrical charges.

Now inside the border Let the mass moved accounts.

See the border into j j multiplied with n, which is what

take the integral over all boundaries You need to take in order to find the sum.

Here we have once rode the vector perpendicular to the current hit

We're taking the integral over the surface, because we put a minus sign

towards the outside of the selected area, whereas this

he said ingoing mass a minus sign to come here.

Mass in the volume is changing,

mass, volume, wherein the concentration of material ro

t the time of this ronin According to the partial derivative is changing.

Here too, there is a time unit time because it is driven on the road, both units

coming into the region's mass equal will be the change in mass.

Conservation of mass is reduced it in the end.

So we take these two We equalizer from the outside unit

It is also the unit of mass equal When changing the mass.

See here for a pleasantness There are also wrong.

Pleasantness başlayaym ago from wrong, someone One integral two-storey three-storey

an integral, bringing them together not able to collect that state.

But pleasantness in the council, where ro v, n is multiplied with s,

completely Gauss, Gauss Or Green structures in the divergence theorem.

Therefore, the Gauss theorem it immediately applicable here, I wrote again.

Gauss's theorem to the first term of this If you apply Gauss's theorem what I was saying,

S, where n is the term hit You will receive divergence.

Here's where you're writing.

Now this is both a pleasant structure three on the left side because it came

story was an integral on the right side of a hence we had triple integrals

these same three-storey integral We can combine under.

See dt d ro's here, where there is divergence ro

and to have both of them combined dv'yl Get integrally involved zero.

See how easy it had found, Gauss's theorem if it is not

three floors of the two-storey integral We could not turn combine them,

I hope that this awareness of the importance of Do you realize, you can catch.

We randomly select a region were beginning V, so that's a random area

of the integral thereof to be zero, integral must be zero at every point.

Here this means that the total mass of the conservation in the region from a random

We're at the point to conservation,

that a partial differential As you can see it happening equation.

Now here's the heat equation I'll do follow the same path,

See also the physics of it is quite simple.

Heat a mean temperature is defined.

Heat energy physics, this bit but also a difficult task.

If you have a very dense mass and how where the specific heat C V shown in this career

specific heat, high specific heat something carried by the high energy,

the higher the higher the density, for example, a low air ro

C V that carries a low energy less than for the same volume of water.

This coming from a physics thing, If you want to deal with mathematics

If you want them a Think of it as completely coefficients.

Now we know that the heat going from hot to cold,

means that the derivative of the temperature,

gives the derivative of the heat flow.

It still comes with a minus sign because it means that you're going from hot to cold

we went from hot to cold a positive,

plus valuable to obtain a flux We're putting a minus sign.

There are also a factor, these coefficients according to the nature of the substance,

easy to heat the metals having but a small number of walls,

If y'all do Or plastic concrete wall A material temperature at this done;

then having a large coefficient will not be a small factor.

Yet here is a random 're getting volume, this random

random close volume has a surface.

Gene into that surface in the same manner This time the flow of heat flux,

we made a little earlier in the problem It was mass flow j, say

outward unit vector and in which Let's say small to heat energy.

See d with very simple things again We were looking to mass change on the inside,

wherein the energy includes We're looking to change.

V constant value ro, and C in this example, only the temperature change.

These include the total mass of the unit

If we look at the change in mass of this volume integral on the need to take.

Again, we encountered a three-storey integrals.

Now again inside the border We're looking to move heat energy.

The heat is transported inside the jar once enerjs showed that this flow per unit time

The heat flow transported, it

vector multiply, but inwardly but who wish to account

The vector n is outward bring to a minus sign.

Entering from the border to collect all it is to find the integral.

Now, this heat energy conservation What does it mean, from the outside

changing temperature heat inside each other to compensate, but have observed the following Fourier,

As we've done just that j T minus K times that of the gradient.

I take it's place.

See Gene was just on the sample j n j

Putting in place the gradient term cons took another plus was that.

See, we meet again with the same structure, The DS is a vector multiplication.

Now again, just like a beautiful We have encountered the opposite thing, but almost Gaussian

Using the divergence theorem, the term would come and triple integrals.

Here comes the divergence of the gradient

But the divergence of the gradient of t gives Laplacian.

Where the coefficient k was attached to this article coefficient reflecting its properties

're getting hard, and he came out If you want you can put them inside.

Now here's the conservation of heat energy very simple, include changes in the external

is offset from the other kinds of energy There are already problems behind something,

Thank you from the outside is changing inside We are writing a balance that is coming.

On the inside from the outside to change After you apply Gauss's theorem

triple integrals, we again turn to because we can remove the integral

This applies to all V, so again, We're taking a partial differential equation.

See again the heat equation We found that pop here

At first he kind of parabolic I showed the heat equation.

Here's obviously a factor that reflects the business, but mathematical physics

When we look at it before we have seen from the parabolic type.

The heat conduction to the equation equation is called.

The material into the soil at spreading accordingly; a

if you throw it into the water took a paint paint, Or if you throw a salty substances,

Or, unfortunately, polluting if you throw substances, oil in the ground

If you go by filtration tapers, Examining this equation always all that.

Conservation of electric charge:This is a completely same, the same philosophy, the same account.

Gene of such an electric charge is distributed, in a place where a random volume

We choose:Included in Volume V Limit SA The density of electric charge Q.

The same material as in the distribution.

We, of course, that in physics They observed; Gauss going up.

Already the law, the law will be called the Gauss' law.

It's called Gauss' law, electrostatic law.

Gives an electric field inside.

This electric field that produces electric charge.

Gene N unit vector outward.

Now let's write them all.

Total electric charge in the volume, whereby the power density per unit volume

charge quantity Q, that the total volume of is integral on:triple integrals.

Moved inwards from the border As electric current,

electric field generated by current: Of course we do not know it,

Physicists observe it at the end of a long They saw that such a relationship.

Wherein the boundary portion of the electric field is the sum total electric current.

In still other cases, both of these to compensate each other like.

Nature rules are as follows:one says Epsilon has a coefficient of zero,

The total electric field but it moved inwards total

electric field inside says the electrical load is equal to.

Yet here was tasteless at first appear, one of the integral two-storey,

One of triple integrals We will ensure equality.

But that is sweet sides There divergence here.

Divergence of the law can be used N e is the inner product.

Right here in Green Gaussian We can apply the theorem.

N comes from the divergence point E E, This three-storey integration.

Two of them that equalizes When the divergence e

Q zeros equal to multiplying and epsilon.

In junior high school, but maybe not this You have seen since high school Gaussian

electrostatic is the law.

I want to give an intermediate knowledge.

About the last week, About the previous week, an electricity-

a potential of a vector field,

the vector potential of a We have seen that separation.

Here are divergence EUR.

So if we take the divergence EU,

See:Divergence in radians Laplacian will be highlighted.

The divergence of a curl is zero.

We know this from the vector features.

Divergence gradient properties.

R means drops.

So we see here that electricity load only the relevant parts of

electric field regarding the potential part.

And R is not here.

Because when you get divergence The term of each relevant R falls.

The divergence of the rotational zero.

And this is the electrostatic Fi potential call.

That means electrical charge Q in Gauss's law From a potential laplacian coming.

Refer to the longitudinal laplacian come again.

See what type of equation was that?

Electricity was equations.

See where D dx square frame F, dy square, dz are square.

T here no static.

Therefore, electrostatic call.

And an elliptical equation was kind.

If the right side of this equation is zero Lapras equation was going to say.

To the right side is not zero This becomes a Poisson equation.

Go early if you'll remember now.

I gave them to remind.

See, we kind of following equation We have:Laprasy are equal:Q.

as a function of XYZ.

This kind of elliptical work.

Before such equation, We found that the heat conduction equation.

Wave equation a little more complicated but that both çıky

See an example of how We also found easily.

Now, at last we were here: Gauss' law of electrostatic

is from a potential electrical We see that this accumulation of charge.

This important discovery course of civilization,

For the development of civilization One of the important steps.

For now I leave here.

Then we will do,

vector magnitudes issues related to conservation.

Goodbye until we meet again.

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