Hi there.
We start a new chapter.
Volume integrals in space.
Section before proceeding begins bi
integral we have seen so far Let us consider the types of bi.
Let's review.
As well as in advance what you are after Let integration are going to do.
Integrals we have ever seen following:the Cartesian Plane
coordinates, the two-storey We saw the integral over x and y.
Circular coordinates in the plane again We saw a two-storey integral.
on r and theta.
In the surface in space situation was somewhat different.
Although two-storey If the integration of this
curvature surfaces in space , desirably surfaces.
Therefore, bi plane surfaces fit.
Therefore, these surfaces x, defined in the y and z'yl surfaces.
Of course, as such, we curvilinear coordinates,
for example cylindrical coordinates, for example, spherical coordinates or
more different coordinates We see the opportunity to work.
And you'll remember so many schemes, We have seen that the integration scheme.
So far only We saw a two-storey integral.
Two of them on the plane.
Bi grain surfaces in space, but bi is two-dimensional in space
b of the two-dimensional geometrical shapes because it is a two-storey integrals.
WILL already see later In the Cartesian coordinates of three
storey integral in the Cartesian plane a generalization of that of coordinates.
Circular cylindrical coordinates, triple integrals.
This is still in the plane circular coordinates in the two-coord,
z ilavers integrally with the two-story bi, where r and theta only had
You will recall cylinder There r and z coordinates theta.
integrally involved in our three-story addition.
Triple integrals global coordinates have been done.
Requirements following from this, is happening:we were interested in body
The geometry of a sphere, circle, or any other
contains curvilinear coordinates Cartesian coordinates are not suitable for them.
With Cartesian coordinates can be written as but the most practical methods of calculating these
and this is because it is in curvilinear coordinates curvilinear coordinates in the two most common
type cylinder coordinates and spherical coordinates.
BI also saw the rotary bodies.
We have seen in the surface of the rotary surfaces.
Wherein a shape of the same taking bi axis
surface of revolution by switching occurs, or We can consider it in the volume.
Before the surface two-ply We saw in the integral surfaces of revolution.
Here he filled the rotating surface.
Has a volume.
This account.
BI in general curvilinear We will see coordinates.
Already them to each other Almost looks like.
Of course, the general curvilinear contains all of the coordinates.
Bi in the previous section, We saw in the previous chapter two.
Had Jacobian.
Volume elements obtained in Jakobiyan'l could be infinitely small volumes.
They are all mutually Topics will be completed.
Now suddenly moving to three variables I want to make a reminder before.
With infinitely small area in the plane.
Because space in the past when thinking exactly the same.
The simplest coordinate system following in Cartesian coordinates
We're building:x're getting.
x of x plus a little enlarge We're taking the delta x.
It consists of two lines.
We take y.
We take y plus delta y.
Consists of two lines here.
The intersection of these four lines to us b gave the infinitely small space.
Circular coordinates We're doing the same process.
There is a circular coordinates.
r is a fixed ring.
r is slightly larger than the delta plus.
That is the correct x, where x is equal to a constant
plus delta x is a right occurs here as two circles.
Similarly, when we say theta constant Combining this right to the center occurs.
When we say theta plus delta theta occurs again, right this second.
And thereof, the same as in this case these four lines,
thought exactly the same only There are geometrical differences.
Four lines generated from we obtain an infinitesimal area
here, like the curved edges we obtain an infinitely small space.
This area in circular coordinates
To find the length of this publication We measure the change in r.
In Cartesian coordinates The length of the horizontal delta
X, and we stood there in the delta y delta areas, we find infinitely small space.
Also here symbolically, i.e. the differential d * d y d is happening.
So we are changing in deltas d'Progress.
Circular coordinates did you know that this curvilinear E,
the approximate area of ??the rectangle here is the publication of the delta theta
direction is multiplied by the length Delta is giving us this space.
That the gene differential When we enter the deltas
When you change to d r d r We were getting d theta.
And in this field, with an infinitesimal area We're going to the two-storey integral.
BI in one year only d.times.d written.
Function of r and theta that the function d
time is just a d d theta r d r d theta times not.
Where b is the minor point I would like to draw attention.
Area multiplied by the length of two means.
Although this is, d is the length of b.
d theta angle.
No size.
This is therefore,
we like very simple in terms of logic b shows that things need.
That was in the plane.
When we go to three dimensions There are fantastically friendly jobs.
However, before this simple I've done geometry.
BI vector it we can do with the product.
Because we know from vector analysis The cross product of two vectors, the
of vectors parallel gives the edge of the area.
Of course, something very simple.
Here also we do not need vector.
Both in Cartesian coordinates that the circular coordinates
We found the infinitesimal areas.
But whatever we do with vectors is as simple as making it hard for a job,
right ear with your left hand as you know show Even supposing that the basic principle is simple
we are forced to solve geometry issues will come in handy.
Here we define the vector bi the size of the delta x and x
because it is in line with the axis i have been multiplied by the unit vector.
Similarly, the delta has y'yl j.
Both of these vectors Multiplication of these two pins,
formed by the vector parallel where the rectangular edge occurs.
Karen seem like a delta x delta y are not necessarily equal.
EUR vector multiplication when we received it,
and the absolute value but also to take Delta because I got to be a number.
Here the vector Remove from the bi vector multiplication.
On the absolute value thereof, the length, There norm requirements.
Delta x and delta y numbers that To come out of this vector operation.
j to i'l product.
j of the multiplication of i'l, We know that the vector k of the product.
So, in the direction perpendicular to it.
We know that a length k.
As you can see from here the delta x, delta y involved.
Yet we know the size.
Similarly in the circular coordinates E r is the vector r in the neck.
r'yl gets hit in the delta vector e r here, we obtain the vector.
Gene is a unit vector along the arc e theta.
We multiply it by the length of this publication length of time to these publications,
We have seen that there are times delta theta.
But the radius r of the delta angle theta is r times the delta theta.
This is also largely the product of two vectors gives us the same result.
to the product of two theta E r'yl Because the vectors in the plane
k gives the height of a Because thereof.
As you can see here We reach the same conclusion.
This second method we have learned b.
Although this plane You do not need but that
the moral of the bi-dimensional space method 50 Although able to implement.
then 50-dimensional Do you work with vectors.
Of course, in that case these lines, would not be able to use geometry.
The best part geometry we can see that,
so that we can keep more helps us to understand.
But the more abstract methods
very complex geometry enables the decoding work.
Gene jakobiyan'l forever We saw a small area.
Here, too, were doing the following: U saber in space,
held constant when V u the line has to be unstable.
To secure the same as r.
Wherein u little When you change the second line b.
Similarly, any line and any surplus With Delta V line here
gene with the curved edge we obtain a rectangle.
Here is the delta vector.
Delta vector tangent to this line.
How the Cartesian coordinates lines
Siya very simple b tangent and though i was Ji'y.
wherein a tangent on the curve
vectors by variable By differentiating we get.
But here's the bi Delta so there bi exchange.
The difference of these two Delta located:This is a vector.
This numeric changes in parameters.
Vector nature of these x Receive derivative of u, gives.
In the delta there like we do.
These two vectors from the multiplication of these two units x
According to v and x are ua product is obtained.
This is before us the two parts If you remember jakobiyan gives.
For example, the rules of this circular Jacobian Let us apply and Cartesian coordinates.
A point in Cartesian coordinates, position is given by x and y coordinates.
The third coordinate is zero.
Because we are in a plane.
Here we take the vector x of Delta wherein X gene, a derivative with respect to x.
derivative of y with respect to x is zero.
Because x and eat each other independent variables.
As we can see here i vector occurred again, Cartesian unit vectors.
the x direction, zero in the other direction.
Similar thing derivative of x to y When we received the course
where x is the derivative with respect to y will be zero.
will be a derivative of y to y.
As you can see here j output vector.
I here say their vector product, j, k the writing of the first vector
write here, write here the second vector
where d x d of d * d v is the equivalent.
But u x, there was also the year that you get them for.
Here you can see the work A zero zero zero one
this vector to zero out the same take lines like writing.
This is the absolute determinant open When you find value only,
a direction to go.
This would be the value of a y'yl in the delta and delta x.
As you can see much more of our t we know of ancient delta x
delta y small rectangular area In this curvilinear coordinates a
but this corresponds to jakobiyan ediyo Jacobian Jacobian supremely simple.
On the diagonal of a The Jacobian of a bi.
Moving on to the circular coordinates circular coordinates of a point
position is the position vector cosine theta, r sine theta, third, again zero.
Yet they're taking the derivative with respect to r.
We're taking the derivative with respect to theta.
We are writing to take.
This time we take the absolute value of i See expansions by zero is seen.
Because if you hit reset sinus, This second element reset again.
ji'li, too.
r times when we opened K, .delta cosine squared theta plus this minus sign
One more plus point will be changing, r squared times the sine of theta is involved.
We calculate it When the Delta gene
geometry we found 're getting in magnitude.
As you can see there more than one method.
Of course in the plane jakobiyan We also do not need much.
Circular coordinates and While in Cartesian coordinates.
Vector multiplication hardly needs to because there's a very simple geometry.
Now take the same concepts in three dimensions will apply.
So that was the purpose of this reminder.
There is nothing new bi.
We have done in only two coordinates works in three coordinates will do.
Now the small area of the plane in space How to infinitesimal volume is geçill?
D had in Cartesian coordinates, d x d y.
So here's a very practical As I'm showing.
If there is next to it With the arrival of a de d z.
Circular coordinates There were r d r d theta.
When we passed into space There are obviously three coordinates.
We have also reviewed.
They're bringing the next bi de d z.
In spherical coordinates many it's not that simple.
It is not difficult but it is also to be seen.
How it is now I do not want to write.
In curvilinear coordinates the Work done two Jacobian
Variables with u and v in belirtiliyos in plane
spots corresponding to the Jacobian of x,
x of y u to v, V y of its derivatives,
so that x y in jakobiyan Type u to x, y to u,
to v of x, y derivatives of the V. The determinant of the account.
When we passed three dimensions this There are also three components of x b.
There was only the x and y plane.
In space x, y and z have.
But the parameters is increasing, there is going w.
Here is the account of it when jakobiyan
jakobiyan'l in the same context, see here
jakobiyan'l possible change of variables where u and v are the d u d v.
If you're here again, but this Jacobian Jacobian of b from the upper level,
obtained from a matrix triple which is the determinant.
And here the only two who can change Not one, bi is the third.
As you can see supremely simple can go with a metaphor.
Now this, I leave here
I want to remember a little, so that they can revise.
E, then, again, these With it, the same thing
This time a bit more geometry We will show the account opening.
Additionally rotating at Bi volumes we'll see.
BI will see the global coordinates.
In fact, we have seen it.
We're in the bi z Cartesian added.
We're adding cylinder bi z.
Global hardly something so simple bi in addition, but not so difficult.
Gene bend, We have seen in the general curvilinear coordinates.
Jacobians are account.
This time the trio Jacobian are account.
For now we see here in this in the space of the fields in the plane of the pilgrimage,
transition to volume comparisons Even though we have seen it through.
Results also easily found.
Our next session We will fill in for them.
Goodbye.
