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
Uzayda Vektörler, Üçlü Vektör Çarpımı ve Geometrideki Karşıtı
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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
Uzayda Vektörler, Doğrular ve Düzlemler; Vektör Fonksiyonları
Uzayda eğriler: tek bağımsız ve üç bağımlı değişkenle vektör fonksiyonları. Düzlemdeki temel eğrilerin hatırlatılması ve uzaydaki bazı önemli eğrilerin tanıtılması. Düzlemde yay uzunluğu, eğrilik ile teğet ve dik vektörlerin hatırlatılması. Uzayda yay uzunluğu, teğet, dik ve ikinci dik (binormal) vektörleriyle eğrilik ve burulmanın tanımlanması. Uzaydaki yörüngelerde hız ve ivme.
Meet the Instructors
Attila AşkarProf. Dr.
Matematik Bölümü (Department of Mathematics)
Hi there.
So far, two-dimensional vectors 've seen.
So they were vectors in the plane.
Only one of them in the transverse groove drawing
also be able to tell everything is fine was trippin.
Able to draw in three dimensions, but supremely difficult.
Its already started in two dimensions.
There we better understand the concepts of drawing I think.
In fact, when we consider the two vectors vectors in space
After all, this creates two bi plane.
Therefore everything again download plane but possible
too much so easy to draw in three dimensions it is not.
I already give the team shapes.
There you will see.
But one of the purposes bi three events as shown in the three dimension size
To generalize a bit at the same time
three dimensions to understand, to understand plane easier.
Although three in three dimensions just living size
so much so easy to revive not all the time.
But this is easier with a generalized algebraic can be obtained.
Not necessary to draw it up.
B instinctively understand things better.
But not essential.
Now the situation was as follows in the plane.
had i and j vectors.
These unit sizes, length of the vector.
Any two vector x a x We tanımlıyo with components.
I and j, by putting it in this way a vector multiplied by a number of
again the product of a vector and a number of We have obtained as the sum of edebiliyo.
But I also do without the use of i and j we prefer.
Because more economic, more economical type of writing.
Go to the three dimensions of e, i and j are still there, but
When we think of any vector bi There is also a third dimension.
For him the third dimension zero We're putting.
j j there again.
Zero and one, but in three dimensions, the third dimension zero.
K as similar to the plane thereof in the
component perpendicular to the plane of zero, but a component.
This time with two count any vector We can not write with a trilogy.
Gene and its equivalents vector summation
In this way the shock count We can show.
But generally speaking, i, j, k to avoid We're trying.
Unnecessary to carry a parasitic information the.
Important information for accounts that's busting but You do not need it.
In general, we think in space Let's point.
One-to-point.
x is a coordinate in the x direction.
x in the direction of two years.
x three in the third aspect.
Of course, the artist is doing here now bi We're trying to do.
Leonarda Da Vinci three-dimensional shapes getiriyo it was obvious two dimensions.
Now here we are trying to do it.
He had little difficulty in regard to the fact that x b coordinate.
y, x two other coordinate.
x coordinate of the third of three.
Here we hand them as bi appearance
When we look at them from a plane send, I am able to convey.
Looks like manner.
He does a cartographer.
A plane of a sphere in three dimensions map reflects spend on it.
From two dimensions to three dimensions, are trying to understand.
We also do our ideas, thoughts This level studies.
It's straight to the point, x A x two
When we get to the plane of projection of bi Whether base.
Here we find the points D and C now.
These are the lengths of x one here.
x two here.
The projection of this point on the x-three we received
an x three boyuş also here, size occurs.
We show with two base.
So this projection projection plane,
to the horizontal plane of projection with base We show.
Such a gene x, x two x three such an Is point.
We're taking the projection to the horizontal plane.
We take x three-axis projection.
These X a, x and x is two to three call lengths.
Pythagorean theorem, Or some Pythagoras prefer.
E, here x is the length of the base
by that one of two square of an x x plus the square of x is two.
So two of the three dimensions of the problem of two dimensional plane are downloading.
Therefore, the size of the base of one of x two squared plus x squared.
When we get here in the vertical plane bi base edge
right triangle, the other to the base of its edge.
So the length of the two prime squared.
Here bi rectangle, triangle occurs.
Current distance to the base of the two base
, OA here because it is equal to the Base
From the right triangle to the point
coordinate the team from the beginning of the this vector x
When we say e, x is a component,
x is two, the length of x three-point We find adding a third.
So here, as a general mechanic When you do all the time x
where a x b x add three to two.
Two had an x where x.
Just bring b x three.
One square of its length x two of a x There were square.
To this we add the square of x three.
In this way, we provide generalizations.
Collecting in a similar manner.
Want it prompts with vectors vektörlersiz do it.
Vektörlersiz we do, prefer to We are.
The X a, x two, with the three components x a vector y, y two, three components Y
when adding the vector opposing We take the sum of the components.
BI with the number still in shock
with this number of the vector components We bumped.
Only one in two dimensions x and x twos that is when the x three
we do all the operations x and y are three three By manipulating, adding're doing.
Here again the two dimensions in the plane x a and
x two x three, while adding only here 're writing.
Gene i, j, more economical k'sız software unless necessary for him, some
In cases have i, j, k for We prefer not to use.
When we came to the inner product inner product of two junction, while x is a
A year, two years, two of x, while that opposing
take the product of the component gathering again, this time from the opposite
where y is three for x trilogy hit them We are adding to it.
It's a simple generalization.
Vector multiplication have a bit more finesse.
You will recall in two dimensions x a and x two U, Y is a
took two years to write, and these determinants We have to k'yl çarpıyo.
Now here's the first line i, j, k are writing.
In the second line of the first vector,
In the third line of the second vector 're writing.
And these determinants are opening.
According to the first row of the determinant We are opening.
So i times where i line and throwing column
two years back that x three x two terms of three years remains.
Coming in second bi minus sign.
Three, three three-pointers in determinate You will recall
plus or minus signs, plus he has it going.
Now it's minus j times, where j row and the column j are taking.
X minus one year to three years back an x three remain.
We are also writing here.
There are of course Bi k.
k with a plus sign next.
where k is the line of milk and threw back
x A x the remaining two, one x two coming years.
In two dimensions, we know this already.
That is two-dimensional if the vectors x three
and y see how the three were not consistent would be.
x three zero two zero years, sorry x two zero but three years is zero, x triple zero.
Thus, these terms are zero.
i'l terms.
As for the term of three years j'l zero.
x triple zero.
Gene is zero.
But when we came here three years, and x three Since there remains the term
and totally two-dimensional vectors by multiplying
We're going to have to give a consistent product.
Now here from the inner product of vectors a We came to the number.
Vector multiplication of vectors by multiplying We're going to a vector.
Bi bi well as CE, multiplied there.
You will not see here, but as information I always me, as a student, I was wondering.
Here are the bi structure.
We're going to the number of vectors.
We're going to vectors of vectors.
I wonder if we can go into the matrix of the vector said.
There are three here, because the number.
There are three numbers here.
Therefore, if you multiply them one at a time We find the number nine.
This doha, indeed, such a
When the process is much more advanced in the I discovered.
Indeed it occasionally things that may be encountered b.
It's called the dyadic product.
Wherein a product of the two vectors A matrix can achieve.
Here you can not put a sign together.
Ie linear in this matrix multiplication algebra
In this course, you'll see nothing but already
As a student, I've always wondered, maybe
the wonder of you who would be I say.
Moreover, this beautiful structure in mathematics show.
There is always such a scheme.
This is in order from the first two vectors bi
are you going to the number of the two vectors of the B are you going to vectors.
So a lower bi identity at a time turns.
In the latter protects the identity.
More than a third in the upper identity going.
Vector comes from the matrix.
I'm also introduced to this opportunity.
Now two new operations in the plane than non- is there.
Because E, two, bi of the two vectors
After receiving the vector product of the vector b involved.
This third bi vector-vector We can multiply.
We call this triple mixed product.
You also can multiply a vector.
This is the triple vector product.
We will see him.
This is what happens when we do the multiplication.
X, we obtain a product of the new components were as follows previous bi E, section.
E, we have found that opening determinants.
Z z also take two
z the third vector of three components We will hit.
E now know what is the inner product.
The first component with the first, second component
with the second, the third with the third component We will hit.
We get something like this.
We also see that these three by three bi nothing more than the determinant.
In this way, when we wrote the third line Turn based.
According to the third line you get hungry, you when z
where once one of our column, row, you will take.
The rest of this binary matrix determinant.
That's the point.
Plus minus signs because you will die minus plus he is going.
Z minus two times you say.
Two of our threw the column and row x an y three, an x three future years.
This one.
Similarly the term z triple.
B very interesting thing.
No one ever repeated index.
Here you have one.
There are two or three on the inside.
No term cooperation.
Here there are two.
Contains no binary terms.
Terms and have a triple.
There are as similar to the triple indicator.
The term does not include triple display b.
Only one and two.
Refer to have such fine structures.
These structures already do this regularly
At the end of his account for simplifying the show.
This triple product has an interesting symmetry.
I told you in the preparation section.
When you change a determinant lines changes sign.
If you look at where e,
z to get it when it hits the x y'yl We were not.
y'yl hit z to x to obtain
Refer to say it to the third row means.
Here again, the same result if expansion Do you find it.
Thus, according to this symmetry, the same triple ediyos get your product.
I say this because here plus x, y, z Do you write like a triangle and trigonometric
In return, plus, if you return to X, y by multiplying multiplied by z
y'yl z to x by multiplying the product of the vector z'yl xin
by multiplying the vector y and K, the internal product gives the same value.
Similarly, when going in the opposite direction We have seen that E,
z'yl mark by changing the order of x changes.
See the first vector of x and z product.
Wherein the vector product A x opposite z'yl is marked.
When it hit the opposite sign y'yl income.
D plus, we said it.
Remove these cons.
Similarly, each y times thereof z tert
z in the negative z times y minus sign gives.
Therefore, they also correspond to the x, y, z ' hit but minus
Returning trigonometric direction X, ie z
z'yl inner product vector multiplication multiply y y vector
All our product inner product of these minus sign would be those here.
Become equal among themselves.
However, the previous one becomes minus sign.
This symmetry is also interesting to b symmetry and in the accounts,
frequently encountered in practice and work the utility.
Ternary composition triple
multiplying the resulting vector in the x hit y We puffin.
Because the vector product.
This b and the third vector inner product we can get
vector-vector multiplication as the third bi materialize.
This is a procedure less frequently.
Here I give the formula and you bi homework the promise.
It provides two-component AmAlArdA sister,
more simplified, to behold the third components are zero
I would recommend it to provide homework in vector basis.
I would recommend that you do this.
From here alone following these geometries are extracted.
y is multiplied with X perpendicular to this plane.
z'yl sense to be perpendicular to this plane impinged steep.
Therefore, back to the plane of the rest income.
If y'all understand it a little bit now go home Think for yourself.
Accounts already show it.
This is a jargon used less.
But to complete, in order to Tamla here I'm offering you.
Now we come volume account.
In triple product, an area in binary multiplication you know that we have achieved.
I.e. the product of x and y with X vectors y given.
One of these multiplications, A, D, B of the plane gives the area.
It is perpendicular to the field vector data.
Let's do it now.
Z'yl take it to multiply.
z'yl to the inner product.
See what's happening?
She OC n times the product of the base area.
a projection of the inner product of n times OCR.
See OCR n times the length of the first one.
Second OC times the cosine fi.
Cosine of this angle fi.
So the angle between the OC.
From the plane C in the withdrawn the same angle with the vertical.
Because they are parallel to the edges of steep When you take it to n
This means that in parallel with CH of two parallel vector.
E, multiplied by the OC kosinüsl, cosine multiplied by f gives the other.
So this gives height.
CH gives height.
Now let's return.
This floor area gets hit, this height Remove the volume thereof.
So in this trio of mixed product has an important meaning.
This gives a volume.
As this volume of applications, we can do we can find.
x, y, z vectors has been given to you How do we find the volume generated by them?
E is the determinant of the triple product 've seen.
So the volume of these determinants we can find.
Another embodiment is this.
Three vectors in the same plane or not?
If the three vectors are not in the same plane As you can see here a volume that will occur.
But if the three vectors in the same plane, then its volume is zero.
Thus formed by three vectors v If zero is understood to be in the same plane.
On the contrary, these three vectors in the same plane
This volume is not answering the question: is calculated.
Çıkıyos volume to zero in one plane.
Unless you quit is not in the same plane to zero.
After that we will do examples.
Because again, we have seen a significant process.
The first of these operations from two dimensions to three dimension generalization.
BI gene in three dimensions and vector inner product multiplication
Although bi similarly continued third more output process.
This time, the mixed triple product.
He has been giving bi volumes.
Now we have to this, a, let's cut it here.
Because I have quite a few details.
I'm sure you thoroughly them one more time yourself
Read the video passing through better 'll understand.
The next session together UmUzdAysA examples will do.
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