Dynamics/Kinematics/Coordinate Systems/Spherical

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Introduction[edit | edit source]

A spherical coordinate system is also useful for describing motion.

Material taken from Vector fields in cylindrical and spherical coordinates

Position[edit | edit source]

Vectors are defined in spherical coordinates by (r, θ, φ), where

  • r is the length of the vector,
  • θ is the angle between the positive Z-axis and the vector in question (0 ≤ θ ≤ π), and
  • φ is the angle between the projection of the vector onto the X-Y-plane and the positive X-axis (0 ≤ φ < 2π).

(r, θ, φ) is given in Cartesian coordinates by:

or inversely by:

Any vector field can be written in terms of the unit vectors as:

The spherical unit vectors are related to the cartesian unit vectors by:

Note: the matrix is an orthogonal matrix, that is, its inverse is simply its transpose.

So the cartesian unit vectors are related to the spherical unit vectors by: