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
Vektörlerin Geometride ve Cebirdeki Tanımları
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From the course by Koç University
Çok değişkenli Fonksiyon I: Kavramlar / Multivariable Calculus I: Concepts
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From the lesson
Genel Konular ve Düzlemdeki Vektörler
Fonksiyon kavramı: girdi – çıktı, bir değerin diğerine gönderimi, çizit, ve dönüşüm gösterimleri. Çok değişkenli fonksiyonların sınıflandırılması: uzayda eğriler, yüzeyler ve vektör alanları. Düzlemde karteziyen ve dairesel koordinatların, uzayda karteziyen, silindir ve küresel koordinatların tanıtılması. Fonksiyonların açık, kapalı ve parametrelerle gösterilmesi. Vektörler: düzlemde geometriden cebire. Düzlemde toplama, bir sayıyla çarpma, iç çarpım ve vektör çarpımı. Bu işlemlerin üç boyuta genellenmesi ve üçlü vektör çarpımları. Bu kavramların geometrideki anlamları ve uygulamaları. Uzayda doğrular ve düzlemler.
Meet the Instructors
Attila AşkarProf. Dr.
Matematik Bölümü (Department of Mathematics)
Hello again.
This is the third of our coming together.
In the previous section that the general mathematics structure
issues which will be examined in the place of We tried to locate.
In addition, in order to be successful in this course Univariate
of the basic functions you need to know information we have remembered.
Now we start our main subject.
Vectors themselves a significant thread.
But one of functions of several variables greater number of variables that are defined by
The easiest way to identify the with vectors.
This serves two purposes therefore a section.
Do a lot of information about vector has.
Thus at the beginning of a little fast through the bi but it will be a reminder of some of the finer points
You may have forgotten perhaps learn I would like to remind forces.
In this section, we will see vectors and The simplest vectors
these are functions of the space to the right The simple vector
functions through the application of the simplest two variable
application functions to the plane We will see the application.
Plane and also verifies that next
we will see topics useful for will be.
Because again, your own experience
If you remember the univariate in function
more complicated than what you know functions
You will see before entering the simplest.
Vectors to identify and account There are three ways.
First geometry methods.
This supremely useful because the issue where was coming, why did some
and internalize what we're doing important to be able to feel for the event.
But more than that to go forward at not possible.
Because just because we can touch, we can see that we can draw
E, we stay at levels greater progress we can not do.
With numbers here, though, the number of first then with a pair of plane
and in three dimensions, with triple the number of n'l wherein numbers
will not need much in the size vectors will be gone.
And everything we do here in geometry We will see that we can do with numbers.
And it's much easier to transactions we make will determine one more time.
There is a third way.
With this premise, the axioms in Turkish The term is also used for or postulates.
Recommend make a start with
to discuss issues with the public, but this not our concern.
This four course I'm talking about a single variable function in the differential
account of this game, you have already seen supposed to be the first course.
The second issue multivariable our topic differential calculus of functions.
The third issue of linear algebra.
The third lesson.
We'll see.
This is another subject matter and then differential equations.
This almost everyone, quantitative sciences dealing
everyone needs to know the smallest team.
Course team.
We propose that part about We will enter the course.
Now he is starting vector.
We observe a number of points.
We monitor on a point.
In order to determine the location thereof an x We choose an x two-coordinate team.
This is a point ba, better place To determine the coordinates
e per team, with a line from right
When we get a vector merge We are.
Because the start point to the directed
accurately describes a vector.
Each vector has a length of.
Is the direction of a well.
Wherein the length of the vector, such as We show.
To distinguish it from the absolute value pairs We show between the lines.
If a single line, the absolute number value.
Here i, two pairs of lines
mean length of the vector length gösterince is coming.
Like any vector, indicating the direction of the unit length vector is useful all the time.
If you say this, the length of each vector
Divide obtain a unit vector in the same direction We are.
As shown here, the definition of As per unit length
and x is the vector of their longitudinal
If we multiply equation splits and, of course, impaired.
But where x is given by the length of vector
If you place the unit vector of each vector
say that the size and direction of the unit, We can define the direction vector.
Now from the point, then, of course bi We can not do much with the point.
Suppose we went to two points.
A and B points.
Here, too, the EU vector connecting A to B, in contrast
B. A vector obtained by connecting BA Thank you.
Now many vectors such as like them to be random in space
many accounts here and there to find plane does not correspond to.
These vectors parallel to them lengths
from the center so that the change start
per team point coordinates, beginning of the next
In this way the slide a vector 're getting.
For example, the EU vector in this direction, the direction of Because change.
Be it in the opposite direction vector.
A complete contrast to the direction is changed is coming.
The first thing you have learned here immediately We're going EU
vectors equal in length to each other, and Ba but the direction is reversed.
Now, of course, still a little bi colon Şiyar can do.
Three, five, ten, up to the point where you want Let the triple point b to go before.
A, B and C, you get the point.
Here, too, the A to B, returning the EU vector, B. C. combining
Gene in the vector bi BC
AC vector obtained by combining C We can.
As you can see here you AB to point C B after
With vector or directly from BC to C. gradually arrive.
Now these vectors parallel to each other again See let's slide.
AC vector parallel slide a current vector 're getting.
They have been drawn correctly.
BC vector, see BC vector x two side
but one is going in the opposite direction reversed x going.
BC get it slide a parallel vector We are.
Slide a vector parallel to the EU itself See also the EU in this direction again,
x and x is an increase both in the direction of 're getting vectors.
Here we see something very interesting.
The following peaks here shown
If we unite as an equilateral parallel e, e,
hand, obtained an equilateral parallelogram We are.
This parallel-sided as the EU bi
edge, the other edge BC and B, AB, BC, AC ie the
Directly combining of C point it becomes diagonal.
This is what we call here parallelogram rule rule out causes.
As I said at the beginning just how is
Why not let us see that it is need.
And this collection is going to rule.
See here because I go from A to D. When you go from B to C
Collect these two vectors would have.
However, this also directly from A to C. theca to go the opposite is coming.
Here, too, these two vectors in the EU and the BC When we gather
diagonal vector equal to the same AC that is coming.
Here the parallel edge aggregation rules we find.
This parallel ka, we found that simple observing geometry we found
Find more generalize the results Let's say that two vectors x and eat.
Where x and y are vectors of vector sheep that their sum
As parallelogram diagonal is defined.
If we take a vector addition of the C-fold for example, twice the
If we take two and a half times the five-fold is defined as follows.
This vector X.
To generalize it, get rid of B, Let's just say the vector x.
When we get hit with the C vector X vector we have the same
but the vector length in the direction of C-fold is defined as.
Here you have defined two basic operations We're going.
Soon we will again recognize two basic operations but
a bit of our instincts, our intuition to generalize
Let's move a little more.
Now A and B points in the plane When we give
combines them into vectors to baseline XA and XB he said.
I wonder if these vectors AB vector denominated find
Could it just with the aggregation rules proportional get.
Refer here to the EU, the EU XAa
If we add that equals xB from A to B. could go
directly, rather than arrive as broken roads When we got to the right we have the total diol.
So that while in the possession of X plus the EU, XB.
XA xB right side glance from the EU We see that the removal of XA.
Similarly, when we look at the beginning again
How will we get to the point here?
Before we go to B with the XB.
Donr A than B, we come to the ba'yl.
So in this way is the XB XA plus B equals.
XB XB minus happening on this side glance.
Thus, here again, the vector of the BA
points A and B, depending on how the vector see that.
Here you need to pay attention to right now.
I need to pay attention to the order.
Our intuition or instinct, maybe we immediately EU thinking we are saying
Let's take the second as the first can, whereas the opposite ranks.
From the second, the second point of the vector We're taking the first point of the vector.
Here also consistent rules.
The second point of the first vector We're taking the point vector.
Therefore, this supremely important.
When given points thereof us shifted to the center
in this process to find the vector I need to do.
This is compatible with the parallelogram rule.
Why is this so and why they were now he is useful both from everyday life
Let us emphasize simple examples.
As though we were a force and this force Assaka we had from point B to the ropes.
This force is distributed in this way we've seen.
It happens in nature.
Many have seen in your physics lesson anyway.
According to these aspects, these components components of the vector
is obtained from the projection and As you can see here, the x and y
vector sum of the parallelogram with the rules is achieved.
Similarly, in a creek a looking at boats
force in the direction x, the force in the direction y Suppose checks.
Different aspects of the forces themselves, different.
According to the boat again this parallelogram rule The total is going towards.
So these events in nature In order to calculate,
In order to define, for observing vectors will be a useful tool.
Already it became important to.
After starting the much more abstract here
As I said at the beginning of the parties before Our observations of life
with observations, but afterwards we construct currently
but we see that we construct are helped to understand the events.
Let's do an experiment like this again.
Can we put a weight on a pedestrian.
To order an X, X is extended to we see.
Assam to twice the weight twice We see that space.
Here we also correspond to a vector
operation of multiplication by a number brings.
Again, we can do the following experiment, suppose a I took a horse cart
When the two forces, of course, completely same network, power
but may not be able to find strength at experience these thoughts
If you think a reservation here two jump income doubled force.
We saw that a vector of the previous hitting a C'yl
leads to the isolation of the process.
Now here so all geometry We have talked about.
Geometry is useful to pass algebra.
There currently benefit from care.
Because there is always the possibility that CEZ do not have.
However, with the numbers always a simple
simple calculations with a pencil on paper we can do.
To do that, but also with geometry We need to be consistent.
Earlier, we found consistent with the rules we need to be.
Equivalent, co-structure theory need to develop.
I need to be useful for Our development.
We first that x A, x i and in both directions Let's define the unit vector j.
Component thereof in the first direction, length a, a length in the other direction is zero.
Similarly, we define a ji.
Whether this is the second coordinate direction.
This, of course, in the first direction
component is zero, the second track in the direction A component.
Now we have learned the following process.
We have learned to multiply a vector by a number.
I multiply x merger.
X in the same direction but of a size that is large enough we find a vector.
Similarly jiu x multiply by two.
Two in the same direction and length x is large enough to we find vectors.
And when we collect them parallelogram According to the rule, we get a vector.
These numbers, x and x is one to two the number of these We call the vector components.
This x plus x two times once i ji
As we wrote like this geometry
than to show the consistency soyutlaşarak
I will come and they also constantly jie important, but they
Not a great feature, and it always hard.
Hence i and jidan, they are known more efficient than ever to write,
more economical, affordable way to write we obtain.
Thus, a vector with two numbers We have the chance to show.
Similarly, we collect two vectors time
them and gather them from the geometry of this 'll do the parallelogram rule.
This number is similar here again and gather them with triangles
equivalence follows from the edge of the parallelogram
that the projection of the edge, since it is his wife here
For a year, the difference is now all thereof
and where i can find the equivalent
ji'siz very economical way to write are exploring.
A number, a vector of two, a number We show with a pair.
Yet a number of gene pairs with the latter Let's show.
Jima their sum and to the i but without the need of their
that give the same results, giving them The results are consistent
to give a first number of collection among its members
We collect the second number to which We collect a combination of components call.
I would get the same results.
Now I did not particularly stringent, but to You can see for yourself that
We draw spouse, similar triangles is the result.
Similarly, multiplying by a vector C say, and here again I ji'lersiz
We write C with individual components corresponds to multiplication.
The length of a vector from the Pythagorean theorem We know.
See here if we go back the length of the vector
Pythagorean theorem, x is an edge from the other x means that the two sides
one of the x squared plus x two of the diagonal emerges as a frame.
We can do that too.
These individual longitudinal dividing the vector By dividing these numbers are achieved.
Thus we have obtained by drawing all
the opportunity of obtaining the number of operations with we find.
This is extremely important because progress to draw both
is not easy for anyone and multi-tools I need edevat.
Not so easy to draw properly.
Instead, all of these things with numbers we can do.
We can do with it to number three We can generalize in size,
The dimension in the fifteen-dimensional space We can generalize.
Fifteen-dimensional space is also possible to draw it is not.
Therefore, do not go into numbers, algebraic
method of adopting the rule here is coming.
E alone geometric, approach a There he rule
In order to internalize the subject because better able to draw something
important to be able to understand the issues more In order to better understand important.
Now these two operations today
After introducing bi ie, two vectors sum
and one of the vectors E, multiplied by the number of first
parallelogram rule, the second
vector extending in the same direction or shortening a little more than
If the number of hits by changing the length of we can find.
Now we will see two more with the transaction.
Would suffice to give the title here.
Our next encounter these issues We will examine.
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