Spherical Polar Coordinates

The position of particle P relative to origin O, in spherical polar co-ordinate system, is described by radial distance r, polar angle θ, and azimuth Ø, as shown in fig. Geometry of the figure shows the relations between Cartesian and spherical polar co-ordinates:

Transformation equations between (x, y, z) and (r, θ, Ø)



The unit vector er points towards r-axis, i.e. in the direction of vector r along which only co-ordinate r changes, θ and Ø remain fixed. Hence, we can write r = r er. Unit vector e_θ is tangent at P to circle SPT; displacement along circle SPT changes only co-ordinate θ, distance r and angle Ø remain fixed. Unit vector e_Ø is tangent at P to circle PP’ produced by the rotation of OP about z-axis: displacement along this circle changes only co-ordinate Ø, distance OP = r and angle θ remain fixed.

The general differential displacement of particle P in spherical polar coordinates is given by,

d r = dr e_r + rd θ e_θ + r sin θ d Ø e_Ø                                (iv)

One can satisfy himself that the above relation is true by considering special cases: (i) if d θ = d Ø = 0, d r = dr e_r gives displacement vector when distance r changes by dr along line OP; similarly, (ii) if dr = d Ø = 0, and angle θ changes by d θ, displacement vector d r = ( r d θ) e θ, and (iii) if dr = d θ = 0, change of angle Ø by d Ø produces a displacement d r = (ρ d Ø) eØ where ρ = r sin is the radius of the circle as we rotate P about Z-axis.

The unit vectors e_r, e_θ and e_Ø can be expressed in terms of i, j, k as follows

e_r = sin θ cos Ø i + sin θ sin Ø j + cos θ k                         (v)

e_θ = cos θ cos Ø i + cos θ sin Ø j – sin θ k                       (vi)

e_Ø = –sin Ø i + cos Ø j                                                          (vii)

the unit vectors (e_r, e_θ, e_Ø), unlike (i, j, k), are not constant vectors but change in direction as co-ordinates θ and Ø change. However, at each point, they constitute an orthogonal right-handed co-ordinate system, that is, we have

e_r . e_θ = e_r . e_Ø = e_θ . e_Ø = 0                                                (viii)

e_r × e_θ = e_Ø . e_θ × e_Ø = e_r, e_Ø × e_r = e_θ                              (ix)

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