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:
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or inversely by:
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Any vector field can be written in terms of the unit vectors as:
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The spherical unit vectors are related to the cartesian unit vectors by:
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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:
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