 Hi, I'm Zor, welcome to Unizor Education. This lecture is about cylindrical coordinates in the three-dimensional space. It's part of the advanced mathematics course presented on Unizor.com. That's where I actually suggest you to watch this lecture from because the lecture contains notes and the whole course contains lots of problems and exams for registered students, etc. All right, so previous lecture was dedicated to Cartesian coordinates in the space, in the three-dimensional space. Now today I will just talk about another system of coordinates and it's called cylindrical. Now why do they actually invent different coordinate systems? Well it's because certain properties of certain geometrical objects are easier to express in different coordinate systems. That's it and I will just make a couple of examples at the end of this lecture. So what is cylindrical system of coordinates? We can approach it as follows. First of all we have to have a reference plane in the space where we will establish polar coordinates. Now on the plane polar coordinates are defined by the origin which is called the pole. And the polar axis fixed direction on the plane and then any point is characterized by two things. Number one the distance from the origin which is raw and an angle from polar axis to a ray which connects center the pole all with our point. This is angle phi. So these two characteristics, an angle which is called azimuth, a polar angle and the distance from the origin. So that's how we can define any position on the plane. Now how can we extend it towards three dimensional space? Well basically the same way as Cartesian coordinates on the plane were expanded into Cartesian coordinates in a space. We just add one more axis which is a perpendicular line which goes through the pole all. It has a positive direction and we will also call it z exactly the same way as in the Cartesian coordinates. And all it adds up to this picture is that I can actually take any point and then projection to the reference plane can be defined by these two coordinates, raw and phi, the distance and the angle. And then I will add the height of this projection. It's positive if it goes towards positive direction and negative if it goes to a negative direction. So three coordinates, raw, phi and z are defining a position in the space. It's called ap which is projection. So again, what's required? A reference plane on which we establish the polar axis and perpendicular to it, it's called Longitio-Dahl axis. Usually it's pictured as a vertical axis and the reference plane is usually horizontal so if I will use z axis or vertical axis it's still the same thing as Longitio-Dahl, yeah I think that's Longitio-Dahl, whatever, axis. And in the reference plane we will have the polar angle sometimes called azimuth and the distance from the pole A of the projection of our point onto the reference plane. So from every point we can go into its projection and then establishing raw and phi, we can obtain coordinates and from any coordinates we can actually construct whatever the point corresponds to these because we can always using raw and phi construct the point ap and then perpendicular to it and we will use z to establish the position of A. So that's basically all about cylindrical coordinates. This again one-to-one correspondence between all the points in the three-dimensional space and certain ranges of these parameters. Now what are the ranges? Well, raw it's a distance that's always positive or equal to zero if we are talking about the origin itself. And by the way for every point on this z-axis on the vertical axis raw would be equal to zero and everything outside obviously will be positive raw. Now the angle now obviously points on this line the polar axis and the plane which goes through this z and polar axis for these phi is equal to zero, for all other phi would be positive counterclockwise and it's less than 2 pi or 360 degree. Now I put less there greater or equal here but strictly less here because I don't want to include the 2 pi since 2 pi would be exactly the same as zero so it would not be a one-to-one correspondence. So and z has no restrictions from minus infinity to plus infinity. So these are the restrictions on the coordinates. Now as I was saying before what's interesting about other systems of coordinates not Cartesian is that certain properties of certain geometrical figures are probably a little bit easier to express in these coordinates. And let me just explain this using some kind of an example. My first example is a side surface of a cylinder, so this is my reference plane. Now this is my O. Now it would be like this. I'm talking about the right circular cylinder. This base sits on the reference plane and its height equals to H and the radius is R. So this is R, this is H. Now I'm talking about side surface only. How can I describe using the cylindrical coordinates all the points which are lying on this side surface of a cylinder? If you forget about the bases, if you just talk about cylindrical surface. Now out of these parameters which parameters can be anything? Well if I'm talking about the surface of a cylindrical surface it means it's going up and down infinitely so z has no restrictions. Now the rho it's a distance from here and it's always R. So rho is equal to R. And phi also has no restrictions because it can be from 0 to 360 degrees to everywhere we will find the point on the cylindrical surface. So basically this alone defines cylindrical surface. If I would like also to have it restricted by two bases I will have to add additional. Z should be from 0 to H. Because the base, the lower base restricts z from below so it's not less than 0 and the upper base restricts from above it's not greater than H. So if you add this you will get exactly the surface of this particular cylinder with this particular height sitting on a reference plane. But if you're talking about cylindrical surface nothing can be easier than that. So that's why actually the whole thing the whole system of coordinates is called cylindrical because cylinders are very easy formulated using equations with these cylindrical coordinates. Now if you will add to this let's say Cartesian system and if you will put Cartesian system something like this. So this would be y and this would be x axis. Then all these points on the cylindrical surface, let's forget about bases for a while, it's those points which are equidistant from this 0. And if you have these two coordinates then this distance is supposed to be always r. But if the coordinates of this x and 0 on this Cartesian plane it means that these are obviously right triangles this one and this one so x square plus y square is equal to r square and z can be anything. So basically this is the same as this. And this seems to be much easier, much simpler. So that's why cylindrical surface is better to represent. Guess what? Cylinders. Alright, another problem. Another geometrical representation and it's cylindrical equivalent. Okay I'm talking about the following. So this is my reference plane, this is my vertical z axis, now this is my polar axis. So now we are defined completely right? This is z, this is o. Now what I'm talking about is I will take a plane which goes through this axis. So it's basically perpendicular to the reference obviously. So it, let's say something like this. And obviously it intersects my reference plane at some line, right? So I know only one thing that this particular angle within the reference plane is equal to capital phi. Now the plane is actually infinite, I mean it goes all the way. So my question is how can I express using the equation with these coordinates all the points on this plane? Well, let's just think about it. The only thing which is actually restricted is an angle. Everything else that can go up and down on this plane or further or closer to the point of origin. So there is no restriction on r or on rho or z. So basically this is very simple expression of all the points lying on this particular plane. And again, whenever the angles are involved, let's say you have a motor which is rotating, the rotor of the motor actually is rotating. And you would like to express rotation, you usually express it in angles, let's say, certain angle per second. Because if you're talking about number of rotations per second, it's actually number of 360 degrees per second. So that's your speed in degrees or regions over there. Okay, and the third example where I will relatively simple represent my geometrical object is a cone. So again we have some reference plane. We have vertical longitudinal axis. We have polar axis. And we have a cone which is standing on the reference plane. And I'm talking only about conical surface right now. So how can I represent the conical surface? Well, points on the conical surface, on the side surface I mean, are not restricted by angle because whenever the angle is, we can always find the point there, right? However, the rho and z are definitely related. Because as I'm moving upwards on the surface, my rho is decreasing, right? And my z is increasing. So what's their relationship? So this is z and this is rho. Now if my cone is given as radius of the base and the height, what can I say? Well, obviously these two triangles, the big one and the small one, are similar. So rho relates to the r, this is r and this is rho, as these characters to these characters. This character is h minus z and the big one is h. So that's basically an equation which completely describes my conical surface here. I mean, I can express it slightly differently. For instance, I can say that z is equal to h minus h rho divided by r, right? From here, h rho divided by r, z minus, right, that's what it is. Or we can put h out of the rho divided by r. Now indeed, if rho is equal to capital R, which is here, then z is equal to 0, because we are here on the base. And as z is increasing, rho is decreasing, rho is decreasing. So we are subtracting less and less. And at the very top, rho is equal to 0, so the z is equal to h. So that's the correct formula. So this is the formula which describes my conical surface. Now if you would like to restrict the side surface only to this particular cone, you obviously have to put a couple of more restrictions like this. z cannot be below 0 or above h. So again, this is a description of a conical surface. It doesn't look very complicated. But if you would like to make this into, let's say, Cartesian equation, that would be much more complicated. I mean, if you wish, you can always try to do it yourself. What will be the correspondence between x, y, and z on the conical surface? Much more complicated than this. All right, so I was just trying to make a few examples where cylindrical coordinates are used. And no, they're not used as much as Cartesian coordinates for many different reasons. But they are used in certain cases. Well, that's basically it. I mean, what I suggest you to do is to do some exercise, for instance, in case of this or in case of a previous problem with a cylinder, try to express the same relationship between Cartesian coordinates as you have just expressed the cylindrical coordinates equation. Well, that's it for today. Thank you very much and good luck.