Ders çok değişkenli fonksiyonlardaki ikili dizinin birincisidir. Burada çok değişkenli fonksiyonlardaki temel türev ve entegral kavramlarını geliştirmek ve bu konulardaki problemleri çözmekteki temel yöntemleri sunmaktadır. Ders gerçek yaşamdan gelen uygulamaları da tanıtmaya önem veren “içerikli yaklaşımla” tasarlanmıştır. Bölümler Bölüm 1: Genel Konular ve Düzlemdeki Vektörler Bölüm 2: Uzayda Vektörler, Doğrular ve Düzlemler; Vektör Fonksiyonları Bölüm 3: Düzlem Eğrilerinden Hatırlatmalar ve Uzay Eğrileri, İki Değişkenli ve İkinci Derece Fonksiyonlar ve Karşıt Gelen Yüzeyler Bölüm 4: Özel Yapıdaki İki Değişkenli Olarak Karmaşık Fonksiyonlar, İki Değişkenli Fonksiyonlarda Kısmi Türev ve İki Katlı Entegralin Temel Tanımları; Limit Kavramının Gerekliliği ve Anlatımı Bölüm 5: Türev Hesaplama Yöntemleri Bölüm 6: Türev Uygulamaları Bölüm 7: İki Katlı Entegraller ve Uygulamaları ----------- The course is the first of the sequence of calculus of multivariable functions. It develops the fundamental concepts of derivatives and integrals of functions of several variables, and the basic tools for doing the relevant calculations. The course is designed with a “content-based” approach, i. e. by solving examples, as many as possible from real life situations. Chapters Chapters 1: General Topics and Vectors in the Plane Chapters 2: Vectors in Space, Lines and Planes; Vector Functions Chapters 3: Reminders of Plane Curves and Space Curves, Quadratic Functions and Variables, Surfaces Chapters 4: Special Two Variables Complex Functions, the Basic Definition of Partial Derivatives and Two Storey Integrals in Two Unknown Functions ; Necessity and Details of Limits Chapters 5: Methods of Derivative Calculations Chapters 6: Application of Derivatives Chapters 7: Two Storey Integrals and Applications ----------- 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
Teğet Düzlemle Yaklaşık Hesaplamalar, Sonsuz Küçük (Diferansiyel) Kavramı ve Zincirleme Türev
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Çok değişkenli Fonksiyon I: Kavramlar / Multivariable Calculus I: Concepts
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Koç University
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Türev Hesaplama Yöntemleri
İki değişkenli sayısal açık fonksiyonlarla tanımlanan yüzeyde teğet düzlem ve diferansiyel. Zincirleme türev yöntemi ve tam türev. Yöne göre türev. Gradyan. Koordinat dönüşümü ve Jakobiyan. Taylor serileri. Kritik noktalar, en büyük ve en küçük değerler. Türev hesaplamalarının üç ve “n” değişkenli fonksiyonlara genellenmesi.
Conheça os instrutores
Attila AşkarProf. Dr.
Matematik Bölümü (Department of Mathematics)
Hello.
Tangent plane to the topic of our lesson last two We care.
In the first part of the definition of tangent plane We gave.
A tangent to tangent plane truth generalization that
In this way a single variable functions We know, as seen
in which y is equal to a function such as f x
at any point in the curve A We can draw a tangent.
There are important applications of this tangent.
Here, more generally, the two variable in function
We deal with a subject line tangent, but it a surface plane is not happening.
Imagine a sphere.
Sphere which is tangent to it at some point plane.
We have seen how to remove this equation.
We saw a variety of applications in concrete on examples.
This is followed in this section and
sections of the tangent plane We will see applications.
Of the tangent plane itself but also important tangent
be removed from the plane, tangent plane There based applications.
One of them with a plane tangent approximate calculation.
This single-valued univariate function against the following.
We çizsek a tangent at the point x is zero
and x is a value of the function we'd like to find.
Sometimes these functions to calculate the value of It is difficult.
It is very complicated.
Instead, at this point, you receive tangent too far from the point when
If you calculate the value on the tangent good an approximation can be found.
From this point, two as in the x If you grow away from the mistakes we make.
If you will be close to small errors.
Why is this important?
Because of the mixed function instead of calculating values
equation of a line from a function value will account.
This supremely easy.
Of functions of one variable that already We also see.
Now in its function of two variables We know that obtaining the tangent plane.
Using this approximate values calculation.
Meanwhile, the fact that I wrote in big letters say approximate
I do not know, sometimes I need to pass an event five digits after comma
not much of a value in knowing about as its
can be much more useful to know the value, can be much more meaningful.
In this regard, the importance or just a not to find value.
Now here's an example to start immediately I want.
Sample this:given a function like this:z is equal to the square root of
As you can see, this function x squared, y but we also have a square root squared.
It forces a split second as We can write.
x and y is equal to a comma eşittit three three In eight commas.
Of course, this small calculator in your hands You can also account with.
You can find the exact value, but it about
find the value would be much more useful.
We're doing it this way.
Tangent plane, we know how to equation is written.
x is zero at the point y is zero and z is zero partial
We find derivatives according to x and y and x y and
because it is the first in terms of our forces this equation, we find that a plane.
We saw this as a clear function We did the calculation.
A function can also be turned off, it was very could also be a function of the complex.
Here is a simple to follow good We start with the function.
The first work to be done to find the tangent plane.
Its zero for x, y, z zero paused We find zero.
That's why we choose them?
Close to values close to three and four because you can see the bottom of the square root
precisely because an avalanche as it we find.
Here the function of the x and y
According calculate partial derivatives, it calculation
This is the easiest way to a karekökl work forces split second.
Provides a derivative minus two divided by two times x
but that would be for one-half dilemma is gone.
One of it, minus one-half
ie, the square root in the denominator of the first force that force.
When the account value in three and four as we find it closed.
Similarly, according to the partial derivative of y and also As you can see in the tangent plane of x,
the first force with y and z We find the equation.
Where x and y is very easy just putting account
because x is the value of z can do three a comma.
A comma is a comma minus three three zero giving.
Three decimal zero eight minus four minus two giving.
And a half when we collect them we find.
See a number of very pristine.
You can find the exact value.
It is a small computer in your hand but a little
the square root of it, you will get a square, You will receive this frame.
One of three comma three comma eight from 26 'll remove.
You will also find the square root again.
Then such an irrational The number will appear.
One point three, nine, six or something this thing was going.
One and a half is not, of course, the exact value
but as you can see in the near extraordinary value.
Seven had made an error in 100.
We know this because in this error expect.
This is because it is easy to draw in one dimension I'm showing it.
This tangent to the curve on the real value on
value here as the difference between the tangent plane to that of
slightly different values on the actual surface
In many problems but the important thing will be There is a value close to it.
As noted here and in capital letters
I gave it does not end just business application.
We're doing the calculations with computer and in bilgisaya
The method of calculating the basic strength This is a method of tangent plane.
For him it is much more applications It is a method.
Now we move on to a second application tangent plane.
Tangent plane we wrote:z it function x
The value of y is zero is zero, it is zero at z We can say.
Here is the first variable x minus x Reset partial derivatives by.
According to the second partial derivatives of variables.
As you can see x, y, z in terms of linear the equation of the tangent plane.
Let's write it slightly differently.
Delta earlier in mathematics we encountered a token.
Shows the difference because the differential In the West say the word comes from.
Bud, it is the Greek letter delta.
Now the negative y where y is zero.
The difference between the negative y y y is zero.
This can be shown by the delta y.
x minus x to zero delta X, We can show.
Or for this z z is zero minus minus f
With our zero-delta or delta with f We can show.
This is therefore the equation of the tangent plane so much
able to write in a more compact way I might.
We saw earlier this differential the concept of functions of one variable.
This is the delta symbol we put in place, it We call infinitesimal.
Turkish is called a differential in Turkish also be used.
In this way it is equivalent to f or d z, There is a difference between minor.
The value of this function, in that z value difference.
As you can see, this partial derivative with respect to x d * is going to hit.
According to the partial derivative of y to d.
As the immediate attention to this difference shoot.
Here delta f x the number of them.
The value of x between x and delta When the number.
However, where the number of d's, but a icon.
Why are we doing this?
Because it's pretty algebraic operations with symbols
as it makes splitting, collection actions such as we can do.
Now it follows immediately after diagnosis We are going to recognize.
We're going to a full definition of derivatives.
See rule chaining derivatives.
Now that we have done.
This infinitesimal expression in the function infinite
small change, in terms of partial derivatives representation.
DX take this great divide d x, ie d x to straight.
We divide d x.
d f d x was divided.
Where d * d * cancel each other was divided.
Here, too, is divided by y d x e, y base this We know.
See here for two types of X, d f d divides encounter.
Bi one flat d, b is in one of is curved.
This flat is fully derivatives d'lier call.
Because of this partial derivative with respect to x plus y
partial derivative of y with respect to x according to Multiplication of change is happening.
We call this rule chaining derivation rules.
That X and Y because the function f but y x is also
When provided as a function that There are a chain of care process.
With respect to x has BI.
have to y.
y is the derivative with respect to x.
From this point of chaining rules are called derivatives.
d f d X, divide separating these partial derivatives for
d f d x of the divide is called the total derivative.
Because d f y when x is divided by a constant We were holding in the definition.
However, with respect to x changes here and there y are based on the exchange.
In this function the sum of the two
all of x changes according to full substitution giving.
This is an important concept and its We will see applications.
Also very common rule chained derivatives derivative is encountered.
Also in this single-valued function There are anti.
I will remind them that you When its turn.
Now let's talk a little bit more clearly.
We f x y z equals the two have variable functions
y is equal to g x but also function have been given.
It also means the geometric görecez.
We can do the following:As long as y g * known
the most natural thing that comes to mind instead of y g x is put.
This time we put these two
valued functions of one variable function goes down.
I showed it to X, great fun.
Thus, derivatives of z with respect to x is no longer
the only variable is the only full- derivative.
This large derivative of f with respect to x.
I mean, but I can not do it all the time.
Here the maintenance more often f x can be a complicated thing.
Also derivatives of the term in this chain and
separate individual meanings of the term I might.
When you do it as d f d *
When you put x to y in terms of x You're losing this information.
However, this may carry important information.
We assume that we have now.
as a function of x given y.
But in many cases the implicit function If there is a relationship between X, y
or x-parametric, parametric years If you can not complete them as a given.
Then preferably chain derivative
or in cases where the only solution is has.
You'll also see examples.
What is the meaning of this?
Now we know.
f x y z equals the mean surface.
This x y z coordinate teams of these surfaces Let's show.
Whether such a surface.
We draw a simple on the surface easy to follow Whether he.
this time we give g * y is equal to x
y is the equation of a straight line in the plane y is equal to g x.
But we think in three dimensions at any time At one point z does not appear.
when z is independent thereof We have seen that the surface of a cylinder.
That is any point wherein the on the truth
at all points on the night, this provides equation.
Because z does not appear here.
In another sense, the review is:such bi
Let vertical line and the curve on the We can run it.
A roll surface where the surface thereof draw happens.
Now we have both the surface and the cylinder
When we say the surface has both not have to say.
The existence of both of these two means is the line of intersection of the surface.
Cross-section is.
So these two functions of the data in We take a cross section of it.
Now we have this year is equal to E g x f x y function when placed in
The function of x means that only comes out z'yl a function of x occurs.
Of functions of one variable means that We reduced.
x z plane.
This intersection curve we draw such a section, we obtain such a curve.
This time we take the derivative with respect to x
we know that the derivatives on a curve it is the slope.
So we rule this chain with derivatives we account
The surface of derivatives given a full year
is equal to g x function defined by on the intersection of cylinders,
on intersection curve, the slope of the we're found.
Please assume that in this geometry, but the this is a
Although the function of the work performed, similar thermodynamic
between the temperature and pressure of the work done E, if g is
it so much, much larger than geometry There applications.
A second way would be:Gene us our given surface.
But this one instead of the function y equals x in y line was describing.
x is equal to x t, y t y
parametric representation, and we see Let's first bi account.
We df infinitely small partial derivatives with respect to x
is multiplied by x, y is multiplied by the partial derivatives was y.
Divided by d t take them all.
but no f in x in t has t.
t exist in y.
Therefore d f d t d x d t and you can see d y d t arise.
Again, we find bi chain derivative rule.
So f t by the direct
accordingly, we do not know what happened, but it the derivative
we'd like to indirectly calculate the full derivative because it has bi single variable t.
Then as the chains of f with respect to x derivative derivative of x with respect to t.
This bi showed partial replacement.
But according to y for f y'yl de change
partial derivative partial derivative of y with respect to t is going on.
We are seeing that instead of this chain derivatives as a function of x t know.
as a function of y t know.
Take them again if we settle bi As in the previous case, the z
ie parameter t is time in terms of terms
obtained as functions of one variable We can.
It looks natural.
And its derivatives can get.
But these functions are usually open As with the function
as when y is equal to g x No generally
t be too complicated for a function and it
take the derivative quite get it without making mistakes becomes a difficult.
Fallible human beings.
Also again, as in the previous one parts, the
The addition of these pieces again a total derivative Recognize meaning.
Change the situation in this respect to x, that y Go, because the current situation shows.
As always slope geometry derivatives
or more generally a substitution measure.
This is the measure of change with change with respect to x According to y by the changes taking
For additional information that is available here The second way is a better way.
Is a more useful way.
Accounts also easier to make a way.
We'll fix it with examples.
Now let's look at a few special cases.
First, f x equals y f x y z rather than for one variable, such as x and y equals zero,
only free variable, but closed Suppose that a given function.
Now here it full derivative with respect to x If we take just the
will follow the formula wherein z equals F x As with y.
Its derivative with respect to x, y-derivative, the derivative of y with respect to x
but this is not our right to be zero for side is zero.
This is such a time right-hand side is zero the slope of the function when it is got a bumper
change of y with respect to x from found your say it may not be easy to find the slope.
Because here solve y in terms of x, but You can calculate d y d x.
However, without solving this here many
In case you can not solve this chain We're derived account.
Here we look at the d y d x f x Put this way.
Less d f d x.
D f d y divided again obtain this formula you will.
If we look at this software to x less than fractional
A portion of the derivative of the partial derivative according to y becomes less marked.
This is an important application.
I hope you are feeling.
This can be a very complex function.
Here y may also be able to solve.
We will see examples of it again.
Z is equal to a second path may Fi
x and y are also a function of I might.
We called this compound functions.
In a univariate function it 've seen.
d f u d y d to f
multiplied by the derivative but here this with di time depends on x and y.
In a univariate function of only x would be connected.
d f d * know when you make one variable in the function
zi, chain derivatives univariate compounds in the function
chain derivatives of f and u according to derivatives with respect to x is the derivative of u would have.
Have the same thing here.
When we divide d f, where d x d divided by D x remains.
As there is a difference alone:both u and x y is a function of f
that X and y are indirectly
Because function wherein partial We're putting derivatives.
We're putting the partial derivatives of u.
Because of both the x and y depends.
Similarly, in the same condition d f d y occurs.
However, the compound functions of one variable Had
Without y d f d x d f d u here again would be like here.
Here it is but plain is divided by d *
would not it would be shaped like derivatives would be flat because of one variable.
Here, the partial derivatives.
Now given x if y g
See it on this one variable, though 're doing.
d f d u d u d *, but multivariable
function, the function of two variables d D x is a full derivative.
d x of the previous saw di where f u put in the function instead.
After the end of the derivative with respect to x to y
According derivative derivative of y with respect to x at happens.
As you can see the difference from one variable is there.
In one variable Y is independent from this term would not.
d d x is divided by the univariate would be flat to d.
So this formula would be.
That two, three, more independent
The variable functions If you generalize
With this same method that accounts You can.
In general, though, such an implicit function it has two free variables.
There are two independent variables.
Here x and y f are large
With this in mind based on the same partial derivatives is obtained.
Now that a little time to digest I need to spend.
I give him a little break.
After that both of these problems resolved We will reinforce concepts as well as account
our ability to make our cars tendancy will increase.
Bye for now.
Please review these issues together.
Because once you resolved the problems and them more
You'll watch comfortable both immediately I'm hoping to be strengthened.
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