1.5 SPHERICAL COORDINATE SYSTEM

The way we have developed a cylindrical coordinate system, we now intend to develop a spherical coordinate system with reference to the Cartesian coordinate system. In a three-dimensional Cartesian coordinate system, we represent a point, say, P( x ₁, y ₁, z ₁) as shown in Fig. 1 15. To locate this point, one has to travel a distance x ₁ along the x-axis, y ₁ parallel to the y-axis and z ₁ parallel to the z-axis. The other way around, if we consider a plane x= x ₁, y=y ₁, and z=z ₁ , then the intersection of these three planes is the point P( x ₁, y _1, z ₁) in space. The spherical coordinate system is a coordinate system derived in three dimensions where the required point is at a distance r ₁ from the origin. The position vector of this space point makes an angle ? ₁ with the z-axis and the position vector of the projected point (in the XY-plane) makes an angle with the x-axis. The meaning of is invariant here with reference the one in cylindrical coordinates. 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 from the origin i.e., a sphere of radius r ₁ with the center...