 Namaste! Hello everyone! Myself Rony Mergo talking about spherical coordinate system, learning outcomes. At the end of this video you will be able to define the volume element in spherical coordinate system. You will be able to write the equations for differential length, surface area and volume in spherical coordinate system. You will be able to figure out the minimum and maximum ranges for different coordinate axes. Learning outcomes we have seen. Now let us go for the next part, list the coordinates in Cartesian and cylindrical coordinate system. I have already uploaded the videos on Cartesian and cylindrical coordinate system. Just revise the and list the coordinates in Cartesian and cylindrical coordinate system. So the coordinates in Cartesian coordinate system are x, y, z and the coordinates in cylindrical coordinate system are rho, phi, z. So today in this in this video we are going to see the spherical coordinate system which has coordinates r, theta, phi. So this spherical coordinate system consists of one circle, one plane and one cone. One circle is r constant circle that is shown here, the circle, one plane which is phi constant plane, the one which is shown in a low color that is phi constant plane, a cone, one cone which is theta constant cone shown in violet color we can see here. So this is one circle, one plane and a cone together forms the spherical coordinate system having the coordinates r, theta, phi. You can see here the center one the circle size is moving from 0 and it is increasing it can go to maximum value. This plane which is shown in red color is moving from 0 and moving in 360 degrees. We can see this cone, cone is starting from 0 moving to 180 degrees and also the same cone is repeating in downward direction the same values are repeating of the theta in downward direction. Theta is always measured with respect to z axis. Let me show you in the further diagrams. How I can locate a point r, theta, phi in spherical coordinate system? The point p is the intersection of all these three coordinates like one circle which is r constant circle. So we take r constant or a constant here it is shown. Then we have to take the theta constant cone this is theta constant cone plane I can see here then phi constant plane this is a phi constant plane I can see here. So intersection of these three will give me the point p which is r theta phi. If I want to draw it simply like this I can draw a small section of a sphere and I can take the angle theta phi with respect to x axis if this is angle phi I can consider the angle theta which is with respect to z axis this is angle theta this is a point p and this is the radius r. So this way I can show point p. The differential lens the differential lens r if this is dr the small length is dr if this is r the small variation along that length is dr if this is phi the small variation along this is d phi and if this is theta the small variation in this direction is d theta. But whether can I say d theta and d phi as lens yes dr is a length fine but can I say d theta and d phi as a length surely we cannot. Theta and d phi are not the these are the angles and not the lens then what are the lens let us see this. So if I want to show you what is the length if I want to find this length this length the angle is d theta to find this length I should multiply it with this distance. So this distance is r and the angle is d theta so it is coming to be r d theta as you can see here r d theta. If at all I want to find this distance this one so I can project it downward here this one so I need to find out this distance first what is this. So this one is nothing but r sin theta it is the front side for theta so it is r sin theta if the same is same is projected downward it is r sin theta and this much angle d phi so r sin theta d phi is nothing but this length this is r sin theta d phi is this length. So the lens in the spherical coordinate system r dr in the radial direction dr in this direction it is r sin theta d phi and you can see this one is along the cone variation is r d theta these are the three lens in differential lens in spherical coordinate system three lens along the three axis r dr r d theta r sin theta d phi the length is a vector quantity remember so we can write it as a d l bar bar indicates a vector which is equal to dr ar bar plus r d theta a theta bar plus r sin theta d phi a phi bar so this way I can write d l bar so these are the lens dr r d theta and r sin theta d phi we can also write the surfaces differential surface or area which is multiplication of two lens so surface or area is nothing but a vector quantity which two lens I should multiply so if I write the surface ds 1 the multi if I multiply dr if I multiply dr and r d theta the left over is a phi so I can write unit vector as a phi and the opposite direct side will be with minus a phi when I multiply the two lens as r d theta and r sin theta d phi I will get r square sin theta d theta d phi as ar dr is not there the unit vector comes as ar the opposite side of the same surface is minus ar again here we can see if I multiply this dr and r sin theta d phi I get r sin theta dr d phi and a theta bar and the opposite side of the opposite surface will be with minus a theta this way I can write differential surfaces in spherical coordinate system differential volume volume is nothing but multiplication of three lens dr r d theta r sin theta d phi volume is nothing but a scalar quantity so dv can be written as dr r d theta r sin theta d phi it's multiplication so I get dv as r square sin theta dr d theta d phi this is the differential volume now I can write the range also you can see the sphere which is more changing its radius from 0 and it can go to the maximum value that is infinity so r is ranging from 0 to infinity phi is same like in cylindrical coordinates the steam it is ranging from 0 and it is moving the complete circle that is 360 degrees so its range of phi is 0 to 2 pi but remember the middle coordinate in spherical coordinate system is theta and not phi so theta is forming a cone which is with respect to z axis if I measure its angle is 0 to 180 degrees as it is measured with respect to z axis it's you can see here it's 0 to 180 degrees so likewise theta is 0 to pi so r is ranging from 0 to infinity theta is 0 to pi and phi is 0 to 2 pi now can we think about what angles I have shown here which is showing me 5 constant plane and which is showing me theta constant plane so to understand it I have shown the watermelon slices and layman slices the way we cut the watermelon and put the slices these slices are showing me 5 constant plane for this piece there is some constant value of 5 for this piece there is another constant value of 5 that means when I'm moving the knife my knife is nothing but some 5 constant plane which is moving along so this particular slice has having 5 constant now when I cut the layman and when I'm cutting the layman with the knife my knife is showing me the theta constant plane for this layman when I connect it to the origin or to the center point it will be forming giving me some angle theta this lemon slice will give different value of theta this lemon slice will give me different value of theta so these are lemon slices are theta constant planes unit vectors of course dot product is easy to find a r dot a r bar is 1 if a theta dot a theta bar is 1 a phi bar dot a phi a phi bar is equal to 1 but to find the cross product I should follow the strangle the arrows like this like a r cross a theta is a phi the third unit vector I'm moving in the same direction of the arrow if I'm moving in the opposite direction of the arrow like a theta cross a r is minus a phi similarly a theta cross a phi is a r and a phi cross a theta is minus a r and a phi cross a r same direction of the arrow is a theta and a r cross a phi is minus of a theta this way I can find the cross product of the unit vectors these are the references used for preparing this video thank you