So let us now talk about different coordinate systems, and first of all, I want to introduce to you
the Cartesian coordinate system, which is the standard coordinate system that all of you know.
So in Cartesian coordinates, you have a position vector that is characterized by three components,
X, Y and Z.
So if you introduce these units vectors here that point along these directions and have the length of
one, then you can write your position vector as X times X plus Y times E Y plus three times easy.
And this gives you the typical vector notation X, Y, Z for the three components.
Now, here you can see that everything is really uniform.
You have these lines, these grid lines, and the grid looks the same and along every direction at every
point in space.
So it's really uniform.
And this leads to very simple line elements for integration.
Very simple surface elements for integration and very simple volume elements for integration.
So you can really separate your line and three dimensional space and equidistant elements and you can
separate, for example, the volume in cubes that have the same size.
And they are characterized by the length deep divide and easy.
So of course, for many problems, it's very, very useful to consider Cartesian coordinates.
So if you want to do some integration in three dimensional space, you do not have to think about what
the volume element looks like.
You just write it down like this and you're good to go.
However, there are, of course, other problems where it is not useful and a very famous one is when
you try to calculate the volume of a sphere.
So in this case, if you consider such a cubic volume elements where you introduce many, many cubes
that are very small in size and you try to fill up this cube, there's the sphere.
I'm sorry, then it's very difficult to do so because what you would have to do is you would have to
parametrized a sphere in Cartesian coordinates.
What this means is you must have an equation for the surface of the sphere.
You have to know how to express this in terms of the X, Y and Z coordinate.
So you get some square roots and everything is quite difficult and you have to be very, very careful
about what you are doing and you have to think very carefully.
So in this case, it's probably not that useful to consider such cubic volume elements, and it's much
better to use volume elements that already are in such a circular shape and already account for the
sphere in terms of the geometry.
So for such problems, we can use alternative coordinate systems.
And next, I want to start with the with the polar coordinates, which is one of the most used coordinate
systems, but it's only in two dimensional space.
And later on, I will generalize this in three dimensional space.
So in polar coordinates, we consider such a, um, circular coordinate system.
So instead of considering these coordinates, X and Y, we consider the length of our vector R and a
polar angle five.
You can see this is the polar angle.
Fine, this is the vector.
And the length of the vector is, of course, are when you do not write it as a vector is just a scalar,
it's not a written bolt.
And there's of course a transformation.
You can calculate the length of the vector R by calculating the square root of the X component squared
plus the Y component square.
So this is just because you have a rectangular triangle and you can calculate the and the polar angle
fine by using the Yarkas tangency function.
So typically what you find is you calculate the arcus tongue and function of the Y component divided
by the X component.
But that's not uniquely defined because it's very easy to think about it when your point is.
For example, somewhere here you have a positive X and Y.
So this ratio here is positive and when your point is here, you have negative X and Y, and so this
ratio is also positive.
It would give you the same angle by calculating this.
So you really have to account also for the sign of X and Y.
And in most programming languages you have for this, um, a predefined function that is the second
across its function.
It's most of the time it's written eight, 10, two, and all it does is it takes the result from the
Archos Tongans and for some points it adds another PI to account for this sign.
OK, so we can write down in this coordinate system our position vector in terms of unit vectors as
well.
However, here it's a bit different.
You do not add up three different unit vectors.
So here it's really just a single one.
This is because if you look at these unit vectors, you can see you have the E.R. unit vector pointing
along the R direction.
So if you want to express this vector R, then you do not need this vector here.
You only need this one.
You have to take this vector which has the length one and then you want to.
By the length of the vector art itself, and that's the whole trick, you really have already accounted
for the geometry.
So, for example, you have a function that only depends on the radius.
So then it's much better to use this coordinate system because you only need to use one vector to express
the position vector.
That's already a great advantage.
Now, of course, you also, for some problems, need the five vector.
For example, if you want to calculate the change of our velocity, then it can of course happen that
it has some transverse component where you need to see five vector.
So this is also very much needed.
And like in Cartesian coordinates, if you calculate the scale of product of these two vectors, it
gives zero, which means they are perpendicular.
This is because of this minus sign here.
So this means we can just take this vector, multiply the length, and this allows us to express the
position are in terms of this two component vector.
Now, this was just where the polar coordinates in two dimensions and the three dimensional generalization
of this coordinate system is called the cylindrical coordinates, because here what we do is we take
our polar coordinates from 2D, where we had only the X and Y axis, and we just add a uniform component
Z.
So this means we extrude this circle to a cylinder and transformation's for X and Y and the length in
the polar plane, which is this time called row and depolarizing of it's exactly the same as on the
previous slide.
So if you know about polar coordinates, you also know about cylindrical coordinates.
And you can see since we have this this uniformity along this direction, we do not have to transform
this lead component.
It will stay the same in Cartesian coordinates and in these cylindrical coordinates.
So these coordinates are, for example, useful if you have a wire going along the Z direction and then
you want to calculate, for example, the magnetic field that will look the same for every Z component,
for every cut through the cylinder.
And then you just have to look what it looks like in this X Y plane.
So therefore you can use cylindrical coordinates.
And one important comment is that here we introduce a length rule, which is not the length of the vector,
are it is the length of the vector if you take this vector and projected onto the X Y plane.
So this is why I did not call this are here because we are in three dimensions would be X squared plus
Y squared plus the square and then the square root that we do not account for the Z component here.
So the position vector in this case can be expressed like in polar coordinates.
We have the length in this polar plane times the unit vector in this polar plane.
But then we also need the Z component.
This is why we add here the second term.
And when you look now at the unit vectors, they look like this.
So you have a very real vector which would point along this right direction here.
So it would just extrude this line here, this dashed line.
So it points along the direction of the vector R if you take it and project it into the X Y plane and
then it has the same shape as previously.
So yeah, here's a small mistake here.
This must be zero and not e r then we have the EFI vector which is also the same like in polar coordinates.
You could put this vector also here and would point um, tangentially along this, this gray line here.
And then we have to Z unit vector, which is the same as in Cartesian coordinates.
It just points along in this direction.
All right.
So now you may be wondering, why did we not use here a different generalisation?
Why did we not?
Right.
R is equal to three dimensional R times e r if we want to do this, we have to consider spherical coordinates.
And this we can do if our problem is really sarika symmetric or rotational symmetric in three dimensional
space.
An example in this case would be if you have an electron or some other charge and you put it in the
center of your coordinate system, then the potential or the electric field, they would only depend
on the distance.
So it wouldn't matter at which angle you are looking, it would just matter for the distance.
So in this case, we would have some rotational symmetry and we should use spherical coordinates.
So here we really have the, um, component are, which in this case really is the length of the three
dimensional vector.
So it is X squared plus Y squared plus the square and the square root.
However, this time we do not have a uniform Z component you can see here, it really did.
It really looks different when we go to different spaces.
For example, here, it's very dense, but if we go here, it looks very much different.
And so this makes it a bit more difficult for the other two components.
OK, we can introduce the poll, our angle once again.
So this is the same as previously.
We take this vector here and calculate the polar angle, which is the angle to this X Z plane.
But then we add a second angle, which is this detail here, which is the angle of the vector with respect
to the Z axis.
And this can be calculated by the arcus cosine function.
So what this means is we now have a very, very simple position vector.
It's just the length of the vector times DENR vector, which in this case looks more difficult.
So it's always some advantage and disadvantage.
So here we have a very simple position vector, but the unit vector looks more difficult because we
have no two angles here.
However, in terms of geometry, it's really easy.
So these this vector here are it's really pointing parallel to this right back to our here.
And then we have the two vectors, Eevi and Edyta, and they point along the perpendicular directions.
So once again, if you calculate the scale of products of these vectors and these vectors and all of
ETR and Eevi, it will give zero.
So this means they are all perpendicular to each other.
So in this case, we can write the position vector in these three components, X, Y, Z, and these
are of course these transformations here.
All right, so now we have learned about four different coordinate systems, but most of the time is
at least for physical problems, we will use either Cartesian coordinates or spherical coordinates,
at least if we are in three dimensional space.
So since these spherical coordinates are very important, let us talk about an example and let us I
want to show you something where these coordinate systems are useful.
So I want to show you how to consider integration in spherical coordinates so we will calculate the
volume and the surface area of the sphere.