 Namaste to everyone myself Rohini Mirgu talking about relation between the coordinate systems. Learning outcomes at the end of this video you will be able to determine the relationship between cylindrical and cartesian coordinates and vice versa. You will be able to determine the relationship between spherical and cartesian coordinates and vice versa. Before we move further as we are finding out the relation between the coordinate system, let us list the types of coordinate system and its coordinates. Types of coordinate systems are cartesian or rectangular coordinate system xyz, cylindrical coordinate system rho phi z, spherical coordinate system r theta phi. So just revise the cylindrical coordinate system which is rho phi z. It consists of two planes phi constant plane and z constant plane. The phi constant plane is shown in yellow z constant plane the one which is shown in purple color violet color and one cylinder which is rho constant cylinder is shown here in pink one the cylinder. So this is cylindrical coordinate system and if I want to find the relation between the cylindrical and cartesian coordinate system. Let me just repeat cylindrical coordinates are rho phi z cartesian coordinates are x, y and z. So I want to find the cylindrical to cartesian conversion that means I know rho phi z I want x, y, z. So here a piece of a cylinder is shown this is a small piece of a cylinder having radius rho angle phi angle is phi and the length is z and also I can see this this rho now I want to find x, y, z that means if this point p is here if I draw perpendicular on x axis this is x axis isn't it. So if I draw perpendicular on x axis what is this distance so the same is projected upward if I draw perpendicular from this point p onto the y axis what is this distance so that we are trying to find out. So can you see here this is phi so this angle also goes to be phi if this angle is phi and this radius is rho this front side this front side will be rho sin phi right as this is rho sin phi this is parallel line a parallel line on the opposite side is rho sin phi and this rho sin phi when projected downwards is nothing but the y axis. So I can say that y is nothing but rho sin phi similarly I can see here the adjacent side this is rho the angle is phi the adjacent side is rho cos phi the same length if projected downward it is nothing but x axis so I can say that x is rho cos phi and z is as it is z in cylindrical and cartesian will remain same so x is rho cos phi y is rho sin phi and z is z if I move want to move reverse what I mean is if I want that cartesian if I want the cylindrical coordinates from the cartesian cartesian is known to me and I want to find cylindrical so for that just recall whatever relation that just now we have written x is rho cos phi y is rho sin phi z is z now if I want rho rho is available in these two equations only so what I will do is I will square and add these two so that sin phi and cos phi will go away because sin square phi plus cos square phi is one so when I square and add these two x square plus y square is equal to rho cos phi square plus rho sin phi square so I get it as a rho square inside the bracket cos square phi plus sin square phi this is one cos square plus sin square is one so I get as x square plus y square is equal to rho square or rho is equal to under root of x square plus y square one relation we got for the coordinate that is rho second coordinate let us do for phi again phi is available only in these two if I divide y by x like y by x I get rho sin phi is y I get rho cos phi as x so rho gets cancelled out and sin upon cos is nothing but tan phi so I get y by x is x is equal to tan phi and phi is tan inverse y by x so I got rho I got phi z is remaining as it is so the relation Cartesian to cylindrical so I know Cartesian I am trying to find out cylindrical so it is how like this rho is it is like this rho is equal to under root of x square plus y square phi is equal to tan inverse of y by x z is equal to z now we will see the relation between Cartesian and spherical so just revise the spherical coordinate system in which the coordinates are r theta phi spherical coordinate system consists of one circle which is r constant circle as shown here one plane which is phi constant plane as shown in yellow color one cone theta constant cylinder sorry one cone which is theta constant cone which is shown here right then the relation between spherical and Cartesian coordinate system can be given like this spherical to Cartesian means I know r theta phi and I am trying to find out x y and z so for that let us refer this diagram here this is x axis this is y this is z the angle with respect to x axis is phi the angle made with z axis is theta this is radius r which is in spherical coordinate system now I want to find out first x so if this is the point p point p projected on the x axis this distance is x point p projected on y axis this is y so if I want to find x I can find x like whatever this distance here it is named as rho so this distance into the french side sorry into the adjacent side because x is the adjacent so this distance into the adjacent side that means cos phi so I can say rho cos phi will be my x but what is this rho this rho if I project upward here you can see this one this can be given as the front side of the theta and the radius is r so it is r sin theta so front side is r sin theta so this is also r sin theta so if this is r sin theta the adjacent side x is r sin theta cos phi so this is written here r sin theta cos phi then if I want to find y y is same as like this it is front side as it is front side I can say that this length into sin phi what is this length this length is r sin theta so I can say that y which is the see this length which is r sin theta into sin phi so y is equal to r sin theta sin phi and if I want to find z that is this distance isn't it so if I want to find z z is the adjacent side for this angle theta so I can say that it is r hypotenuse is r so it is r cos theta so r cos theta is z so spherical to Cartesian relations are x is equal to r sin theta cos phi y is equal to r sin theta sin phi z is equal to r cos theta now I want to do reverse that means Cartesian is known to me xyz and I am trying to find out spherical which is r theta phi so let us do that for that just recall these relations x is r sin theta cos phi y is r sin theta sin phi z is r cos theta now I try to find out r so when I try to find out r r is available in all three so let us square and add all these three x square plus y square plus z square so if I square this you can see here in these two terms r square sin theta comes out as a constant inside the bracket cos square phi plus sin square phi plus r cos theta again in these two this cos square plus sin square going to be one and then left over term is r square sin theta plus r square cos square theta in these two again r square is a constant inside the bracket sin square theta plus cos square theta this term again going to be one so I get only this much as my equation and then I can say r as equal to under root of x square plus y square plus z square. So, here the difference to be remembered in cylindrical coordinate system rho is under root of x square plus y square only whereas in spherical coordinate system the radius r is equal to under root of x square plus y square plus z square. So, z square term is added in the radius. One term we have calculated which is r now let us see how to find theta. If I want to find theta you can see that I can use only this equation is enough to find theta last equation that is z r cos theta. Theta is cos inverse of z by r. Now z I keep as it is r just now we have calculated. So, I can say that theta is equal to cos inverse of z upon under root of x square plus y square plus z square and if I want to find phi phi is available in these two equations. So, I will take these two equations ratio y by x which is r sin theta sin phi upon r sin theta cos phi r sin theta cancels out sin upon cos goes to be tan. So, that I get phi as tan inverse y by x. So, second coordinate also we have calculated third coordinate also we have calculated. So, Cartesian to spherical conversion is r is equal to under root of x square plus y square plus z square theta is equal to cos inverse of z upon under root of x square plus y square plus z square phi is equal to tan inverse y upon x. These are the references used for preparing this video. Thank you everyone. Thank you.