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
Üç Eğrisel Koordinatla Sonsuz Küçük Hacım ve Jakobiyan
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Çok değişkenli Fonksiyon II: Uygulamalar / Multivariable Calculus II: Applications
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Çok Değişkenle Zincirleme Türev ve Jakobiyan
En büyük ve en küçük değerler: yerel, mutlak ve kısıtlama altında. Kısıtlama altında en iyiyi arama (optimizasyon) ve Lagrange çarpanı yöntemi. Kısmi türevlerin uygulaması ile değişimler hesabına giriş.
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Attila AşkarProf. Dr.
Matematik Bölümü (Department of Mathematics)
Hi there.
About two preceding session
valued functions We examine the coordinate transformation.
The main approach to this lesson As we always dreamed.
Two variables from a single variable transition requires some important new concepts.
Therefore we are always our first introduced We introduce the concepts of two variables.
But three of the two variables even much more variable
transition to the account number of variables, requires extra calculations,
but conceptually new You do not need too much.
Now here's the general We will see a situation,
The general approach of this general attitude We will see an application.
We have previously with X and Y We have seen conversions between.
Here, the x of y the next one is coming z,
v is an aspect of the coming w hence the three variables,
between three variables We're talking about transformation.
If we give an example of position in space A point defined by X Y Z
with the definition of spherical coordinates, Distance from start
Run geography major from the North Pole If we use the term meridian,
pH on the longitude angle and on the latitude angle
including the theta x and y z are connected in this way.
We've seen this before, but now followed in practice this
coordinates a bit more We will open as a reminder.
You know the two coordinates x and There was partial derivatives to y,
of the components of the position vector x y z
x by y, u and v derivatives I had naturally here in the first row
z as function of x and y task is coming, according to Gene derivative is taken,
z in the second line comes again x in the same structure and y'yl
added a third line going Of course w x y z as u have this time to
and its partial derivative by if x y z according to the
If we know the partial derivatives a function u and w
According to the partial derivatives of v and w We are obtained by multiplying this matrix.
Gene recalled in matrix multiplication As is done in practice.
This horizontal vertical take to bring the vector
where the first row opposite multiplied by the number.
The first number is the second number as the first The third issue with the second with the third.
We repeat the same process on the second line In the third line we repeat the same process,
and each time that at the end a we add up the numbers to a single number
and down, wherein a number, wherein a number here we find a number.
According to this x y z partial derivatives u by w allows us to have transformation.
Detail for two binary As we have seen for three trio
we can do the same thoughts Another thing to repeat
therefore not result than We do not mind going through.
Determined by a gene x y z point near
du dv infinite infinitesimal dw'n
dx dy small areas of Dz This gives the conversion,
it doubled again in binary We have seen in detail.
When we ask this question to those of us
that the partial derivatives du dv by DW
the x and y and partly z'l derivatives only sturdy?
Similarly, wherein the endless Is that a one to one small area
where x is undergoing transformation in terms of?
Otherwise important to one relation in mathematics If this is coming from five different locations
We will not know which came from these five What if you were going to go to different places
We will not know he respects This is an important one.
and to algebraic equations but here we see something derivatives
At the end of a number of them, a number of functions, even after all.
They are known unknowns them.
This is the only equation solution to give the
determinant of the coefficient matrix must be nonzero.
Wherein the same thing, that is, more concretely
as a criterion to be connected by a
this matrix is connected to an odd number determinant, the determinant of this matrix.
When we look a bit matrix See a different structure in year two
There as here x y z Following Getting derivative,
There are still X Y Z denominator in the second line, Getting the derivative with respect to v in the denominator.
Yet here x y z denominator, Getting the derivative with respect to w in the denominator.
There are cases where the opposite denominator only x u v w this time derivative is taken.
Here are the y, u have to w, There u v w z derivative is taken.
Matrix but different matrices not much different to each other
also known as the Turkish transpose Transpose A term is actually not bad.
The first of this second matrix changing the column lines
or following the axis of the matrix take is achieved by turning on.
The transpose of a matrix that, transpose is equal to the determinant of itself,
it comes from there because of a matrix According to a row determinant ha
According to a column've opened've opened ha Or any row or column
See here if you have opened by one two three one two three six different
can be calculated independently of the way Although the data is the same value as a result.
Here the matrix transpose determinant by column
We have opened the line becomes open by counter means, is therefore equal.
That the size Because we call Jacobian
German even called Jacobi 'Jacob' j y for what they say
that a scientist observe and understand the importance.
As symbolically shows d's not something where the partial derivatives
form a representation only.
Here you x y z u x y z respectively
to v w x y z in the first or x as ua
write to v by w taking the first derivative of the line,
u v w y by the same way and likewise the second line derivatives
including both the same number of shows that this determinant.
Generally be easier because x Do you write vector based on the
take derivatives components in the first row You put the second line, put them on,
You put on the third line.
This process is a bit more more difficult or
I need to find something inverse transformation You do not have that much to them.
Now here are a few applications will do.
First of all this so sweet surprises I think initially no
Perhaps we can not think of a making feature of the Jacobian.
Now in curvilinear coordinates as We think of space x is zero y
z is zero point zero is defined by the curve.
This curve has a u v and w fixed curve
in this transformation, keeping only
remains the only variable parameter is staying.
Derivatives based on these variables we get When we find the tangent here.
As volume in Cartesian coordinates We were finding a delta x as x
We're changing y up a delta y As far as we're replacing a delta z.
We're doing the same thing here u We're changing the line a little
We call plus the Delta, there is a bit of line We say there are increasing there, plus the delta,
w are also slightly increases As we're finding plus.
So here's the same Cartesian here As in the delta coordinate edge
delta delta x y z as the volume element Delta Delta Delta have here comes w.
We have only one difference here curvilinear edges of a hexahedron
There are a prism Cartesian case it was steep and plane faces.
Now that the derivatives We find when we get the tangent.
Tangent here gets hit with change Instead of the same curved edges closer
We find this vector length There are three of them.
Now let's remember almost everything in vector When the three vectors thereof
ternary mixture of these vectors of the multiplication We know that the volume form.
Let's apply this to our present observation that
it is always the numeric symbolic of the values
So we're going to value in terms but generally more easy going.
That instead of deltas d's The vector of the direction of putting the vector,
du'yl hit the edge of it in this size symbolically shows.
It is counted them deltalı d du dv dw is a symbol.
Since the product of these three vectors We find the volume of prisms.
This is the number of dv dw, not vector,
We take it outside and inside the rest of this triple mixed product.
This trio of mixed product determinants that
We saw te first part in the second part.
Then u x vector derivative of the
The components of the vector x x y z According to the derivatives thereof.
Similarly, based on V x y z because there are derivatives of vector v
where the derivative with respect to v of the vector x u v w kept constant condition.
W x y z in a manner analogous to genes derivatives based vector that is going on in the war.
When we calculate the determinant
We see again that the Jacobian and that I'm this sweet surprise,
initially unrelated a Jacobian from the origin,
In the first coordinate transformation had a volume of here
account of're doing but Jakobia we see that the subject be so much
because it is not a coincidence that we received every shows a vector of partial derivatives of a
shows the tangent vector for him involved here, and so this volume,
with infinitely small volume Jakobia we see can be calculated.
Now let's make a few applications.
Jacobin in spherical coordinates and let the infinitesimal volume calculation.
Spherical coordinates were as previously
I have seen a more detailed look very important because a coordinate set.
A space point, the point in space p are x and y and z coordinates,
Also the starting point of the When we combine the resulting
length of the first spherical coordinates We show great R'yl to call.
The projection of this point We take on the x-y plane,
As with the annular coordinates Combining these smaller centers have little r
wherein r is projected on the X axis with an angle theta is taking place.
One of them is an ideal cosine theta the projection of y on the sinus
theta will bring, but if this is See this region where the pH angle
Inspired by the term angle measured from north,
polar angle opposed thereto Because it was a sinus pH is coming.
R is large in sinus phi'yl r gives the product a little.
This is no longer here to circular coordinates tetayl seem to have come to the cosine
Multiplied sinus x we find y we find in tetayl gets hit.
Likewise, this large R projected onto the z-axis
See the side next to open pH is the cosine is coming.
Here x is the vector of the components of x
The pH y R z, We see the transformation of theta.
A three-variable three-dimensional vector as a function of this nature.
We in our previous general discussion u v w respectively, were also
The pH of R V, W. de theta If we choose these vectors, respectively,
According to R phi theta taking derivatives We will write the first, second and third row.
See, we take the derivative with respect to R. R is linear only first
Coming force, because it is a derivative of the phi and theta, wherein the hard task
his only sine cosine phi Tetala and these terms remain.
The second line of the theta constant R'yl We're getting hold derivative of phi,
therefore R cosine theta constant does not change temporarily
sine cosine phi phi derivative wherein There still a sine cosine pH pH
where pH is the cosine pH minus sine here.
In the last line of the theta We'll take derivatives.
R & R sinus sinus phi phi is here R & R sinus sinus pH, such as pH remains.
Minus sine cosine theta derivatives theta minus sine theta come here.
Derivative of sine cosine theta theta, he's coming here.
The last term, of course, also has a feature
here does not seem theta R and taken to pH constant
Due to the definition of partial derivatives derivative with respect to theta zero turns.
Now such a three by three determinants We found a place to zero if needed
According to that row or column to open the convenient, If you can do no less accountable.
I was here in the third column made by the initiative.
Once the two lines for R If we take them out of the determinant
value does not change, there is a R R, R square is coming here.
Wherein the first term We're taking the cosine of phi
where the cosine of phi rows and columns of the dislay
these two backward binary matrix and its determinants remain.
See here gets hit by a sine cosine phi phi comes every
pH The pH of the sine cosine on both sides.
The second diagonal changing the sign so you're getting both
sign plus is happening.
So before sine cosine phi If we take phi see here
There was also a cosine cos phi phi phi square sine now it is coming,
In addition to these common multiplier cosine squared plus sine squared theta theta is coming.
As you can see here a sweet simplification consists sine squared plus cosine squared
one is for you.
The second term will remember the signs
goes as plus or minus doing this minus into a plus.
There sinus pH in this line second line and when we exclude the third column
See here had sinus sinus phi phi There thus comes sine-squared pH.
A sinus pH here I had sinus pH becomes cubes.
Gene likewise cosines multiplied by the cosine square
gives the product of the sine-sine square, it is still more simple in a nice way.
They became a cosine square Tetala Tetala sinus and square.
Refer back to the common term sinus pH If we remove a welcome back again
simplification comes, cosine square plus sine-squared phi phi is happening
sinus got one of the cube We also receive the sine of phi.
Because they give you a cosine square pH plus sine squared pH supremely elegant results,
R squared is simply the sine pH.
This Jacobians, Jacobians If this volume, infinitesimal volume
d u d v d of the Jacobian of w'yl d r d f d theta that is multiplied by multiplying
If you look at how much We can calculate it easily.
And it's infinitely small volume three
We will use the nearby storey integral.
I want to give an assignment.
The volume of an ellipsoid ellipsoid francalı somewhat similar to our work.
In one aspect, ellipses, such as spheres, but each elliptical in cross-section in a direction that the coefficient
c being a öbürkü b öbürkü We find such a way as to.
This way the coordinate transformation as showing promise.
This is a b c for half an ellipsoid axis gives the length.
And they are not equal When will the ellipsoid.
Even if they are equal especially if one takes this
is already turning into a circular coordinates.
The presence of this Jacobian Following the presence of the infinitesimal volume
extraordinarily similar, such as global global example we did a little while ago.
This is to calculate the partial derivatives I want to find Jacobin.
This is to you to check your accounts I'm giving here is guidance.
These provide a Cartesian coordinate.
As the volume of Cartesian coordinates We know that d * d y d z.
Cartesian coordinates fantastically simple
z is the coordinate system x u v w y so this is as much an individual
from simple coordinate transformation We're talking about transformation.
Now we are going to do this Taking the derivative of x by u.
Take the derivative of u Remove the first number is one.
The second third of zero is zero.
So on the first line There is a zero-zero.
Because u v w from each other independent coordinates.
In the second row the same x that x We will take the derivative with respect to v.
U v w is again independent When we think of u according to v
derivative is zero, a derivative of V.
According to v w derivative zero again.
So on the second line zero becomes zero.
The same logic as in the third row thought to be zero zero one.
So the unit matrix occurs.
This is also a determinant.
Because it is a determinant We convert this very simple to behold
but in the event of a reduced conversion know and understand the terms, already
providing a view of what we know I think that is useful.
D u d v d w it is, but u
v w x y z because we know d x d y z we provide.
Last in a paper cylinder coordinate a coordinate transformation.
Is nervous, cylinder to use, As coordinates.
P is a point in space again.
This point p in the x-y plane We're taking the projection.
Start to run the distance.
The angle of this line with the x-axis theta.
And at the height z.
The first part of r and theta links As in completely circular coordinates.
And it's happening z added.
So x y z z transformation from the r-theta.
Gene have the general structure r and z instead of w theta
Once you needed derivatives take the Jacobian is calculated.
Jacobian of the supremely well again will find that simple.
Where z r and theta from This first is independent
we take the derivative of the line by r zero time derivative of z by r.
Again when we take the derivative with respect to theta z derivative of theta zero again.
The second line means last third element zero.
In the first row.
The third line of this We will take the derivative with respect to z.
Because it is independent of z e r theta The last line will be zero zero one.
Teta'n turns out very simple.
You and your calculation To see your reply
I give here the accuracy of the results.
Now move on to a new topic.
Before going it alone appropriate to take a break.
And we have seen examples of this
two coordinates of the three coordinates Please remember to switch a.
The complex functions is actually a bit like two
because they are functions of x and y variables.
x and y are functions but x and y is interconnected
x plus two for good this year variables can not move as well.
y if x how to act, the y by watching her behavior.
Is there a relationship.
Here are taking advantage of this feature but that the two variables
In focusing on the complex functions We will have a specific outcome.
I The square root of minus one because that is the unit imaginary number.
we often complex will be valuable.
Here are all of a We will see in the next session.
Thus, related to derivatives sections will be completed.
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