Hi there.

Coordinates in the previous session We examine the transformation.

This is partly under coordinate transformations We have seen how changes in derivatives

and coordinates the exchange of the team We examine two issues that we looked at.

Both asked the following question:this that's exactly what you'll need for the conversion?

This is important for the following:Mathematics I was a little I say jokingly do not like the uncertainty.

Of course, at the bottom of this topic in mathematics When you provide birebirlig

'd expect from mathematics to expect, ancy can not answer.

Because if one thing on the same Or if going to place a

You have found one ten different results If you come from where an additional

size and often it brings unacceptable in many conversions.

Making them as Jacobian We are faced with an interesting concept.

Both in the transformation of the coordinate system in the transformation of the coordinate transformation that

we have under the account of partial derivatives

a transformation which is a cut- appeared in size.

These two variables now We koordinatlard.

A binary matrix into two I have found the determinant.

Now initially the same jakobiyan appear to be much to do olmuyo

b in the environment as a sweet surprise We will see how to get against us again.

Now we have this from curvilinear coordinates

we mean a XI connected to v, also connected to v y u.

For example, R v If the theta circular

y coordinates of x r cosine theta r knew that the sine of theta.

So this is a function of u and v vector x.

E we e, u fixed prisoners that x vector will depend on a single variable.

TA at the beginning of the third part b

A portion of this course We have seen that curve.

So we keep u constant When x is drawing a curve.

Let's say you have this curve.

This one. V. In contrast, if we keep constant the

expedition will consist of curves has changed.

we are in Cartesian coordinates We were doing the following:x had.

We artÄ±rÄ±yo a delta x as x.

There was also a delta y y y so we artÄ±rÄ±yo.

Also saw or circular coordinates.

We artÄ±rÄ±yo as a delta r r, artÄ±rÄ±yo up t have a delta theta.

Now generally schematically When we think of b and v

a coordinate space and at the point kesiÅiyo

belirliyos this point of how the x and y u and v determine this point.

Let's take a u'y increment.

Let's say Delta.

This obviously will be a second curve.

How is it equal to r plus delta y r When the 're getting a second circle.

Here, too, a little at a different location We will obtain a second curve.

Similarly, let V bi increments.

plus there is the delta there.

Again we get the line b.

See these four line crosses bi 're getting curved rectangle.

Now these rectangles We'd get around the edge vectors,

Be units, For this length will not show,

here right now, our mind is coming:We have two vectors.

This is the vector product of two vectors gave us an area of ??absolute value.

That will be the Delta area.

Now it will conduct with the accounts.

Go with Delta that the vector

in the number of times it We show the change.

In its definition of the vector x According to the derivative.

As you can see, because when you have a fixed a vector function of one variable.

E vector functions the curve was gÃ¶steriyo.

Derivative of this curve to the variable According gives a tangent.

This delta to u'yl when it hits the edge of this curve

to the other, and a length that is tangential

a size close to it we find to have as similar.

We immediately for these infinitesimal After you find the e symbolic,

Or differentials forever were finding small icons.

Here instead of deltas d's putting're getting it.

Now we proceed immediately.

Here again, the case was written.

A, vectors comprising these infinitely This small area is the product of two vectors.

Is a multiplication of vectors.

Of course, a vector will appear here its absolute value, size,

length norm, each separately can be used, we need to calculate.

Ta in the first portion We know from the vector.

Vector product of two vectors created by giving them space.

Now of course there are other numbers that is to vector multiplication we get out of it.

So we get to this size.

But if we keep u in d u u separately, d * d u.

I.e., x of the x vector, the components x u of the vector x and y based on the derivative.

We yazÄ±yodu i j k for vector multiplication.

First vektÃ¶rol, We are writing to bring to the second line.

Similarly, the vector v d * d v.

We are writing her second.

Of course, we are in two dimensions in the plane The third component to zero.

He writes zeros are also these three let's build a determinant of threes.

We need to open it.

As you can see i and j components are zero.

If k'yÄ±nc d.times.d the following two components

duality becomes the determinant of a matrix.

This bi k'yl but the absolute value is multiplied, we account for the length of

this vector, k is a size for being also do not need to write it here.

As you can see here again Jacobian occurred spontaneously.

x and y components of the position vector We're taking the derivative with respect to u.

Yet the same position vector We're taking the derivative with respect to v.

That was Jakob.

As you can see the infinitely small space Di Di is equal to V Jacobian times.

You have to start planning.

But it arose spontaneously.

So it is nothing but a miracle appeared as a pleasant surprise.

Now we come to it there are three dimensions We will also see an important application.

This will occur in the same way again.

My two examples of a kick start.

We know the simplest coordinates Cartesian coordinate system.

u to say x, y have to say,

in the first row of x and u We will take the derivative with respect to v.

a derivative with respect to x u.

derivative with respect to v is zero.

Because u and v independently.

Similarly, the second We will write to the line y.

We will take the derivative of y.

because he is the derivative of y va zero.

y is a derivative of V.

As you can see Cartesian one involved in the Jacobian coordinates.

Go to the Cartesian Cartesian already have should not be in the field for a change.

So ja, d u d x, d v d y we know

areas, such as in Cartesian coordinates, infinitesimal

field value as a simple geometry In our results we obtain.

Yet to provide bi geometry we obtain circular

coordinates in the infinite Let's for a small area.

Gene circular coordinates r cosine theta, we know that r sine theta.

When you write them as vectors the first component, the second component,

As you can see, this vector depends on two variables.

Jakobiyan dependent on r and theta means to calculate this

of the x component of the vector r

According derivatives.

by r derivative of y components.

That is here to go cosine theta sine theta is coming.

This time, the same vector theta we take the derivative with respect

When subject is not affected cosine of theta minus sine theta,

sine cosine theta theta comes from we obtain the determinant.

Now if we open it, see cosine squared is going on,

is a sine-squared is going There are negative but the second element

second multiplication minus the second term changing sign're getting

r squared plus cosine is therefore sine-squared is going on, sine cosine square

for a given sum of squares is the only We know this because the area remains and

If we say that J du dv J, is found here,

did u r v theta, if it is happening is that theta

geometry obtained by simple thoughts The results we are providing here.

The main advantage of this method is governed such a simple geometric operations

We can not, for example in cases elliptical coordinates as a

This coordinate system is roughly circular coordinates of genellenmiÅ.

Coordinates like a genus consists of an ellipse perpendicular to them

comprising the hyperbolas There is also a coordinate team.

This is fine in some cases something that works.

On this subject you tried, but it also can remove

Our goal is not he already do not have time We give here as guidance.

x u and v as shown Y is denominated again and v

If little attention in terms of interesting to see,

hyperbolic functions are also There are trigonometric functions,

There kosinÃ¼slÃ¼ together with sine There are quite a nice thing together.

If we Panini account that the Jacobian

've found this value to remove Waiting for homework.

The case wherein

made on finite values??, but When we take these curves ellipses

u come in for constant values

v is constant in the two curves in

As you can see it happening hyperbolas such an intersection is revealed.

Yet an interesting coordinate team are the parabolic coordinates.

They are facing up downward facing parabola with family

family consists of intersecting parabolas, How the x and y

If another cut coordinates we're here takes a certain value,

we're slightly increase the value of a van we've received, we're a little boost V.,

As you can see four of the curve again Create a curvilinear rectangle involved

From simple geometry we think them again You also can easily find.

Using them

When Jakobia new account it turns out quite a nice structure

I have squared plus the square, so Request this item also belongs to the field

plus there is the square of the sum of squares du becomes who is possessed of dv'yl.

Gene coordinate team There it hyperbolic.

We passed a little quickly, but elliptical, parabolic, hyperbolic coordinates teams.

These are given in this way again.

These are the simple geometry you can not not work.

They are a parabola, a hyperbola family

In this way the bisector Those who accept asymptotes.

Another second family of hyperbolas x and

considered asymptotes y axes and is thus a network is knit.

See if you imagine it not much difference of x and y.

They also have a rectangular Building a network in which

creates a rectangular network They are subject to the curved edges.

Here again, the same idea in If we were doing a u-value u

increased a little, a van value taken have increased slightly

obtained from such a rectangle here It is a rectangular curved edges.

Gene're already edges so