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Physics · Glossary

What is Spherical coordinates?

Definition 11.10 University Physics — Year 1 · Chapter 11 — Kinematics of a Point

Point MM is located by its distance r=OMr = OM, the colatitude θ∈[0,π]\theta \in [0, \pi] between OM→\vect{OM} and e→z\vect e_z, and the azimuth φ\varphi of its projection on the xyxy plane: OM→=r e→r\vect{OM} = r\,\vect e_r with

e→r=sin⁡θcos⁡φ e→x+sin⁡θsin⁡φ e→y+cos⁡θ e→z;\vect e_r = \sin\theta\cos\varphi\,\vect e_x + \sin\theta\sin\varphi\,\vect e_y + \cos\theta\,\vect e_z ;

e→θ\vect e_\theta (tangent to the meridian, toward increasing θ\theta) and e→φ\vect e_\varphi (tangent to the parallel) complete the local orthonormal basis. On the Earth, θ\theta is 90∘90^\circ minus the latitude and φ\varphi the longitude.

Spherical coordinates (r, , ) and their local basis: e_r radial, e_ along the meridian (toward the south on a globe), e_ along the parallel (toward the east).
Spherical coordinates (r,θ,φ)(r, \theta, \varphi) and their local basis: e→r\vect e_r radial, e→θ\vect e_\theta along the meridian (toward the south on a globe), e→φ\vect e_\varphi along the parallel (toward the east).
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