Coördinate system

Spherical coördinates

Spherical coördinates have two variants which essentially vary only in argument order. One variant is traditionally used in maths while another is used in physics. In general, the argument that occurs in all three cartesian axes and two sine functions will have the domain .

See Spherical coördinates in differential geometry for a formal treatment.

Mathematical convention

Where is the radial distance, is the azimuthal angle and is the polar angle

with the signature .

Physics convention

The physics convention (ISO 80000-2:2019) is swapped: Where is the radial distance, is the polar angle and is the azimuthal angle¹

with the signature .

This has the benefit of generating a conventionally-orientated parameterisation of a sphere's surface

Calculus

The following differential quantities may be useful²

\begin{align*}

d \tau = r^2 \sin\theta\,dr\,d\theta\,d\phi = r^2 \,dr\, d\Omega
\end{align*}
$$

3. 4. 5. 6.

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