 Hello everyone. Myself Rohini Mergu talking about cylindrical coordinate system. Learning outcomes. At the end of this video, you will be able to define the volume element in the cylindrical coordinate system. You will be able to write the equations for differential length, surface area and volume in the cylindrical coordinate system. You will be able to figure out the minimum and maximum ranges for different coordinate axis. Before I move further, can we just revise and can we just list the types of coordinate systems? So, types of coordinate systems are Cartesian or rectangular coordinate system having coordinates x, y, z, cylindrical coordinate system rho, phi, z, spherical coordinate system are theta, phi. In this video, I will be talking about cylindrical coordinate system which has coordinates rho, phi, z. So, cylindrical coordinate system consists of two planes, which planes phi constant and z constant planes. So, one plane is phi constant which is shown in yellow color and z constant plane which is shown in violet color. Then it also consists of one cylinder which is rho constant cylinder which is shown in pink color. The cylinder is shown in pink color. So, these three together forms the cylindrical coordinate system. One cylinder and two planes, the cylinder is rho constant cylinder, planes are phi constant and z constant planes. The animated cylindrical coordinate system is shown. Here you can see that this green one which is changing is nothing but rho. This is rho, green one cylinder is rho whereas this red one which is moving around is a plane. This red plane is nothing but phi constant plane when it stops at the particular point that time the phi is constant. Now we can see this another blue one plane which is changing with respect to height. It is moving along the z axis, this is z constant plane. At this point z has some fixed value when it stops at the particular point that time the z has a fixed value. So, it is a z constant plane. A point rho, phi, z in cylindrical coordinate system can be located by intersection of all these three planes that is the cylinder which is rho constant cylinder, then z constant plane this is z constant plane and phi constant plane this is phi constant plane. So, intersection of these three will give us the point P which is rho, phi and z. So, I get this point P. Simple way I can show it like this also. I can show a cylinder here. I can consider some point at having the radius rho as of the cylinder z at the particular height z and making an angle phi with respect to x axis. So, this to be remembered phi is the angle made with respect to x axis. Now, can I write differential lens in the cylindrical coordinate system? Yes, we will be able to write the differential lens as we have written in Cartesian we can write in the cylindrical as well. As there are three lens along the three axis which axis the axis is rho, phi and z for these three axis we can write the lens as d rho, rho, d phi, d z. Now, you may say why it is rho d phi, why not just d phi? As we have written like in Cartesian dx, dy, dz here also expected like that, but let me tell you phi is an angle, phi is not a length. So, to convert it in the length we need to multiply with rho, but why only rho? Let me explain you here. So, this length which is vertical is dz, fine. The length in the radial direction this way is d rho. Now, the question comes of phi. So, when I go like this is phi and if I move in the same direction in by d phi, this part is d phi, when this is d phi I need this length actually, I want this length actually. So, when I want this length I should multiply that this is a small angle d phi. So, I should multiply it with this length and then I can get this length which is d phi angle. So, this length will be rho this length into the angle d phi. So, it is rho d phi and not just d phi. We cannot just take d phi because d phi is an angle it is not a length. So, to take it as a length it should be multiplied with rho, rho d phi is a length. So, dl bar is d rho a rho bar plus rho d phi a phi bar plus dz a z bar. So, we can write the differential length accordingly. Differential surface or area yes we are able to write that also we can take a small section of a cylinder, a small volume element of the cylinder and then we are able to write the surfaces. The surface is a vector quantity it is a d s bar so it should be along with some unit vectors. So, when I am writing the different surfaces let us say this surface if I am writing I should multiply the two lengths which are two lengths here here the circular part as well as vertical part. So, I can say it is rho d phi into dz. So, when I have written rho d phi into dz the direction should be surely the third coordinate which is a rho bar. This is if for the front part the back part back of that the same volume element will be shown with minus a phi. Do not correlate this 1, 2, 3 with this do not correlate this 1, 2, 3 with this one. So, these are the surface names which I have given ok. So, ds 3, ds 2 I have explained as it is on the opposite side or opposite side the same surface with the opposite direction of the unit vector is ds 2, ds 3 if I want to write which is you can see here vertical length dz and the inward and outward length which is d rho, d rho, d z and the direction should be surely the third vector that is a phi bar. So, this is if it is the right side the left surface will be with the minus a phi. Similarly, I can write ds 5, ds 5 if I want to write I can multiply it as the two lengths as the inward outward direct change is d rho then this length is rho d phi. So, multiplication of these is rho d rho d phi and direction is upward a z if I want to write for bottom one it will be with minus a z. So, this way I can write all the six surfaces for the small volume element in the cylindrical coordinate system. Can I write differential volume? Yes, it is very simple to write a differential volume. Volume is nothing but multiplication of three lengths. So, which are the three differential lengths d rho it is not just d phi it is rho d phi and d z is not it. So, I have shown here in the diagram as well d rho, rho d phi and d z then I can write dv. dv is a scalar quantity which is multiplication of these three it is rho d rho d phi d z. Yes, we can also write the ranges for the different axis like rho is ranging from 0 to infinity. Why 0? Why not minus infinity? Because rho is a radius we always measure radius from some center or the reference point and then we measure it. So, rho is as it is a radius it is starting point is 0 and it can go maximum to infinity as you can see this cylinder can go up to infinity but it is starting is 0 radius is 0. Phi as you can see here phi with respect to x axis it can it is from x equal to sorry phi equal to 0 to phi is equal to 2 pi and z as it is vertical variation it is from again minus infinity to plus infinity there is same as in in Cartesian coordinate system. Unit vectors if I want to take the self dot product it is always easy a rho by a rho dot a rho is 1 a phi dot a phi is equal to 1 a z dot a z is 1. But when I want to take the cross product I should follow this triangle a rho to a phi bar a phi bar to a z bar and a z bar to a rho bar a rho bar cross a phi bar a rho bar cross a phi bar results in a z bar. But if I go reverse like a phi cross a rho that means I am going in the opposite direction of this arrow. So, in that case my result is third vector third unit vector only but with the minus sign minus a z bar similarly we can see here a phi bar cross a z bar is a rho bar but a z bar cross a phi bar a z bar cross a phi bar is minus a rho bar and we can also find out like a z bar cross a rho bar is equal to a phi bar and but a rho a rho bar cross a z bar I am going in the opposite direction of the arrow so it is minus of a phi bar. So, this way we can find the cross products of the unit vectors in a cylindrical coordinate system. These are the references used for preparing this video. Thank you.