 Hello namaste to everyone. Myself Rohini Mergu talking about Cartesian coordinate system. Learning outcomes. At the end of this video you will be able to define the volume element in Cartesian coordinate system. You will be able to recognize the surfaces and lanes in Cartesian coordinate system. You will be able to write the equations for differential length, surface, area and volume in Cartesian coordinate system. You will be able to figure out the minimum and maximum ranges for different coordinate axes. Now the question is why do we need the coordinate system at all? Can you think? Can you tell me why do we need the coordinate system at all? So the answer is in order to measure the direction properly the knowledge of coordinate system is needed or coordinate geometry is required. A coordinate system defines a set of three different directions at each and every point in space. The origin of the coordinate system is the reference point, relative to which we locate every point or every other point in space. A position vector, a position vector defines the position of the point in space relative to origin. So these three reference directions are referred as coordinate directions. The reference directions are called coordinate directions and usually taken to be mutually perpendicular also called as aruthogonal axis. Coordinate system is quite clear in GPS global positioning of system or global positioning of satellites and when the satellite is positioned. In order to understand the position of the satellite we need the coordinate geometry, we need the coordinate system. So this is why we need the coordinate system. Now can we think on what are the different types of coordinate systems? So the different types of coordinate systems are Cartesian which is also called as rectangular coordinate system which has coordinates x, y and z, cylindrical coordinate system which has coordinates rho, phi, z, spherical coordinate system which has coordinates r, theta and phi. Spherical coordinate system having coordinates r, theta, phi. What is actually the coordinate system which is Cartesian coordinate system that we are going to see in this video. Cartesian coordinate system also called as rectangular coordinate system. It consists of three planes. It consists of three planes. One is x constant plane, this is x constant plane. The other is y constant plane and the third one is z constant plane. When x is constant the varying two are yz. So this is also called as yz plane. When x is constant other two are varying it is called yz plane. As I have drawn this plane at origin so I can say that this is x equal to 0. Even I can draw this plane forward, even I can move this plane backward. Accordingly the value of x will be changing but it will be constant for this particular plane. So x constant plane also called as yz plane then y constant plane as y is constant other two are varying it is xz plane. You can see here this is x is changing z is changing. If I look at the point here the value of xz is different. If I look at the point here the value of xz is different. If I look at the point here the value of x and z is different but on all the points here in this plane the value of y is 0. So it is y constant plane or xz plane. Same is the case in z equal to 0 plane. Here as z is constant the other two are varying which is x and y. So it is also called as xy plane. Moving further point in Cartesian or rectangular coordinate system. The point can be located in Cartesian or rectangular coordinate system like this. The point pxyz is located as intersection of three mutually perpendicular planes x, y and z. What I mean to say this when I want to locate the point I can locate this point by the intersection of the three planes x, y and z and the one particular point which I get that is point pxyz. So in Cartesian coordinate system in the right-handed coordinate system we can show the axis like this x on the side y and then set. Can we show the different axis like here can I show x here can I show y yes that is quite possible but in the right-handed system the system is like this. When I move from x to y my thumb is indicating upward direction that is z. So this is called as right-handed coordinate system even I can show it like this is x this is y and this is z. So that way it should keep on rotating that is right-handed coordinate system. In this right-handed coordinate system if I want to locate a point at all how I can locate if I want to locate the point pxyz. So I have to move from origin some particular value x some particular value along y and then I locate first the point pxy in the xy plane then I'll project this point by z units. When I project this point by z unit I get the point pxyz like this. So this way I can locate the point in three-dimensional xyz system in Cartesian coordinate system. Now differential lens the three lens along the three axis are dx, dy, dz. So general terms I can consider that these these lengths are dx, dy and dz. As length is a vector quantity I should show it with dl bar. This bar indicates it's a vector. So dl bar is nothing but some variation along x axis and the unit vector ax bar that is dx ax bar. The variation along y axis which is dy and the unit vector ay bar plus the variation along the z axis and the unit vector az bar. So this is how the dl bar is. On this side you can see there are different planes which I have shown the one which is shown in pink. Here x is constant it is x constant plane. If I consider the one which is shown in yellow here as this is the y axis you can say it is shown at particular point of y. So I can say that this plane which is shown in yellow is y constant plane. The plane which is shown here in violet color which is at some particular value of z. So this is called z constant plane. So these three planes are shown and intersection of these three planes we get some point here right and that point is p xyz and the variation along dx along dy along dz and if I want to write the length vector I can write like this. This is called as differential length. Can I write differential surface or area? Yes of course we can write the differential surface and area. Surface or area is a multiplication of two lengths. In Cartesian coordinate system the volume element is the cube and it consists of six surfaces. So here the volume element in Cartesian coordinate system is shown which is a cube and the surface area is or surface or area is a vector quantity. So how I can write? So here you can see when I want to write for front surface this is my front surface. For front surface the length is dy, the vertical length is dz. So it's multiplication of two lengths dy and dz. Fine I have written here ds1 front dy dz but when I'm saying it's a vector quantity it is ds bar actually. So it is ds1 bar can be given as dy dz ax bar. The unit vector outward from front is ax bar. If I write for back surface the unit vector is outward that is in the opposite direction of x. So it is minus ax bar with the same length dy dz minus x. See when we write the unit vectors for the volume element all the unit vectors are directed outwards that should be considered. All the unit vectors I should consider directed outwards. So if I write for right surface for right surface the length is this is dx this is dz what about direction? The direction is outward like this. So this direction is same as y so it is ay for right but if I write for left surface the outward direction is opposite to this y direction. So it is minus ay. Lengths will remain same as dx dz. If I write for top surface this length is dy and this length is dx so it is dx dy. What about direction? This direction is az vertically upward z direction. If I write for bottom then it is minus az. So these are the surfaces in differential surfaces in the Cartesian coordinate system. Differential volume. Differential volume in Cartesian coordinate system is a cube as I already said and volume is nothing but multiplication of three lengths which three lengths dx dy and dz. So this volume I can give as dv as multiplication of dx dy dz and it's a scalar quantity. So no question of direction or the unit vector coming here. So dv is dx dy dz. This is differential volume in Cartesian coordinate system. Let us think about the ranges. Range of the coordinates x y and z. So the range for different coordinate axis we are talking about. If it is x axis as we know the value of x can be minus 1 minus 2 and so on. The value of x can be 0. The value of x can be plus 1 plus 2 and plus 10 whatever the maximum value. So we know that the range of x is from minus infinity to plus infinity. Similarly the range for y is from minus infinity to plus infinity. The range for z is minus infinity to plus infinity. So these are the vectors or rather coordinates in the Cartesian coordinate system x, y, z and its ranges are from minus infinity to plus infinity. These are the references used for preparing this video. Thank you.