1.4 CYLINDRICAL COORDINATE SYSTEM

It would be worth commenting here that, the new coordinate systems that we are going to learn are ones derived from the Cartesian coordinate system. In a three-dimensional Cartesian coordinate system, we represent a point, say, A( x ₁, y ₁, z ₁) as shown in Fig. 1 10. To locate this point, one has to travel a distance x ₁ along the x-axis, then y ₁ parallel to the y-axis and z ₁ parallel to the z-axis. The other way round, if we consider planes x= x ₁, y= y ₁, and z= z ₁, then the intersection of these three planes is the point A( x ₁, y ₁, z ₁) in space. The cylindrical coordinate system is a derived coordinate system in three dimensions, where the required point is at a distance ? ₁ from the z-axis in the angular direction with the x-axis and at z ₁ distance from the x- y plane. In other words, the required point is the intersection of three surfaces: Surface 1 which is the locus of all points at a constant distance ?= ? ₁ from the z-axis i.e., a right circular cylinder.

FIG. 1 10: (a): Geometrical representation of a point in cylindrical coordinates. (b): Intersection of a cylinder ?= ? ₁, plane