Our previous session Cartesian coordinates integral
being able to account, triple integral calculus We were able to see the main principles.
Infinitely small element with the delta z Delta area is multiplied,
that a delta delta area x We know that the delta y.
In cylinder coordinates in fact we have seen, but to here,
I also want to make a reminder.
Cylinder coordinates thoroughly To adopt.
Why is it important?
See also because of the nature, technology offers
Always see the cylinder geometry commonly used things.
Vibrations of a cylinder rod, Or passing through a cylinder of gas, water,
major fluid stream or a coil Do Sarmis cylindrical,
Because they are so important here 're standing on in detail.
Most in curvilinear coordinates those used with cylinder coordinates
are the spherical coordinates.
In nature, this way there, There are in the art as such.
One way or another with the the way we have modeled.
For example, the shape of the world is exactly what sphere, but the world of b
In the sphere by modeling a variety of events approach can explain important event.
Now we have seen the eternal how to find the smallest element.
It's a bit more open well, let's understand the geometry.
A point in space, a point P x, y, z coordinates were later explained.
You're getting the x and y plane projection.
Center coordinates of this point We call on the distance r.
This straight line connecting the center point x theta angle with the axis of his call.
Bi this point in the vertical We call on the height z.
So it is with our theta We determine the position of the point.
How the hell are they?
x and y are in the plane of the x r We find the cosine of tetayl.
This because the projection on the x-axis When you get r times cosine theta.
y is at times sine of theta.
z in the z-side.
Here we do not have anything extra.
Now this infinitesimal element, How do we find the volume?
Please remember circle.
We r is fixed at the apartment.
r is constant gives a spring.
Gives a circle.
r r is r plus delta count of that circle b.
We have seen this in two sessions.
You have received theta.
You have received theta plus delta theta.
These two give accurate.
This plane is going in three dimensions.
We have also reviewed.
Now, of course, where the resulting prism
it is a rectangular prism but it in terms of infinitesimal three
When multiply edge We find its volume.
One of its sides bi B spring.
b broadcast radius r, Open delta delta t and t is time.
Another is the size of the change in direction.
When it comes to plus delta r r r from where r is the change in the delta.
Bi changes in the z-direction.
When you get a fixed z is equal to You can find a horizontal plane.
This is where the horizontal plane.
When you say z z plus the delta gene bi but slightly above the horizontal plane.
The difference delta z.
Here geometry thoroughly b To determine the name of the publications.
changes in the direction of a c r.
is a change in the z-direction.
These are the time we hit the delta multiplied by the delta delta theta r z,
Two sessions ago as a much more practical that we also give the size.
Now you can open it a bit more.
His four different methods for We will calculate said.
Of course all the same result but I will give a little bit of geometry
To understand what I'm doing.
Now, in the area where the base of the
these areas a, b??, e and c are the oluÅuyp.
b e c base height field We get hit by the volume.
EUR circular area at the base We know from the coordinates.
r r times the delta theta delta times.
It hit the delta z'yl bi you'll find the previous result is clear.
Again, this area also had a delta.
This cylinder, Imagine that kind of look pipes,
in this disc s say the delta area.
This area is also the delta h b b in a vertical line for spring
gene f d arc and consists of a d height.
This area A, B, F, d the depth of curved surfaces
As for a car in that radius
we stand for is the change in When you're getting the same thing again.
So this is the order in which the triple product We'll get the same thing if you hit multiply
but these two geometrically When you combine the two kind
in terms of meaning to see it come.
Third, you'll remember by vector multiplication.
Binary vector multiplication was given a field.
We have seen that from vector analysis.
We have seen from vector algebra.
Ternary mixtures, mixed in the product vector multiplication followed by a
with the cross inside the
These three vectors u v w vector We were finding that create volume.
Now we call Delta radii
formed on the r vector for.
You'll recall Cartesian coordinates the delta x
on butting of i'yl conceptually equivalent.
The edge of it here We'll obtained by multiplying.
we know that the vector in space of functions of one variable,
based on a single parameter vector of functions Get derivatives were finding bi tangent vector.
Means the partial derivative of other variables,
ie, theta, and z are held constant The only variable that bi here.
This gives the vector.
It hit the delta r'yl this 're getting on the edge of this vector.
Similar things based on theta and z 're getting with derivatives.
We take these three vectors.
See where x position vector by r
wherein R in the derivative of a will give the cosine theta
will be given here to the sine of theta but by r derivative of z is zero.
Because r, theta, and z from each other independent coordinates.
Similarly derivative by theta b aldıÄımz this is doing the hard tasks.
Cosine sweat, minus sine derivatives.
Costa, sine te, cosine derivative.
But according to the theta derivative of z is zero.
Because the arguments again.
Finally, the position vector, that we're here
vector z from derivative zero income from the first two components.
Because no where z.
r and theta is doing the hard tasks z because it is based on the arguments.
The first two components involved zero zero.
Thirdly, z z A derivative is giving.
That's when we hit them, the trio Of course the length of the vector multiplication
changes in order to r, theta change, as the change in z're getting.
We found this b page before e, xin by r,
x and x to z by the theta derivatives based on the first line,
The second line, third line when positioned're getting it.
Here the easiest third If you open the determinant by column
once in this kosinüsl r is multiplied by the cosine
cosine squared minus b minus the There is here plus sine squared theta.
Therefore r times cosine squared theta r is the sine squared theta plus
stay back.
This is the delta, delta z, We stand by our delta.
So it turns out is from the determinant and This is the delta, delta, delta z's.
We found with geometry We reach the same conclusion.
Now here to differential We turn to d for deltas.
This symbolic value as and you're getting.
The prior that we find same here and this
is the determinant of the Jacobian We need to just observe.
Already it at all to these accounts he came forth without the triple vector
is not Jacobian jakobiyan in the variable
by multiplying the volume of changes, endless We know that we will get a small volume.
Two episodes ago Jacobins We saw in the section.
Or even better, but forgot done; I gave it to consolidate detail.
Here we find the same thing jakobiyan'l.
The next call in this section b I want to continue giving.
'll Pass on spherical coordinates.
In this geometry similarly We understand a little useful.
Bi Or to completely jakobiyan'l triple vector obtained by multiplying
a size that we can.
Useful to understand the geometry.
But the difficulty in understanding the geometry If you are partial formula
HES derivatives take, from jakobiyan infinitesimal volume calculated.
Of course, in the apartment, e, need to think a little more subtle sphere
but you will see it easily something that can be achieved.
Bye for now.
Attila AşkarProf. Dr.
