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 [0,𝜋].

See Spherical coördinates in differential geometry for a formal treatment.

Mathematical convention

Where 𝜌 is the radial distance, 𝜃 is the azimuthal angle and 𝜙 is the polar angle

with the signature ⃗𝐫 :[0,∞) ×[0,2𝜋) ×[0,𝜋] →ℝ3.

Physics convention

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

with the signature ⃗𝐫 :[0,∞) ×[0,𝜋] ×[0,2𝜋) →ℝ3.

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

Calculus

The following differential quantities may be useful²

1. 𝑑⃗ℓ=𝑑𝑟ˆ𝐫+𝑟𝑑𝜃ˆ𝜃+𝑟sin⁡𝜃𝑑𝜙ˆ𝜙

\begin{align*}

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

3. grad⁡𝐹=𝜕𝑇𝜕𝑟ˆ𝐫+1𝑟𝜕𝑇𝜕𝜃ˆ𝜃+1𝑟sin⁡𝜃𝜕𝑇𝜕𝜙ˆ𝜙 4. ∇2𝐹=1𝑟2𝜕𝜕𝑟(𝑟2𝜕𝐹𝜕𝑟)+1𝑟2sin⁡𝜃𝜕𝜕𝜃(sin⁡𝜃𝜕𝐹𝜕𝜃)+1𝑟2sin2⁡𝜃𝜕2𝐹𝜕𝜙2 5. div⁡⃗𝐅=1𝑟2𝜕𝜕𝑟(𝑟2𝐹𝑟)+1𝑟sin⁡𝜃𝜕𝜕𝜃(sin⁡(𝜃)𝐹𝜃)+1𝑟sin⁡𝜃𝜕𝐹𝜙𝜕𝜙 6. curl⁡⃗𝐅=1𝑟sin⁡𝜃(𝜕𝜕𝜃(sin⁡(𝜃)𝐹𝜃)−𝜕𝐹𝜃𝜕𝜙)ˆ𝐫+1𝑟(1sin⁡𝜃𝜕𝐹𝑟𝜕𝜙−𝜕𝜕𝑟(𝑟𝐹𝑟))ˆ𝜃+1𝑟(𝜕𝜕𝑟(𝑟𝐹𝜃)−𝜕𝐹𝑟𝜕𝜃)ˆ𝜙

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