structural:reformatting, information:extraction
Generalization

It is also possible to deal with ellipsoids in Cartesian coordinates by using a modified version of the spherical coordinates.

Let P be an ellipsoid specified by the level set

a x 2 + b y 2 + c z 2 = d. {\displaystyle ax^{2}+by^{2}+cz^{2}=d.}

The modified spherical coordinates of a point in P in the ISO convention (i.e. for physics: radius r, inclination θ, azimuth φ) can be obtained from its Cartesian coordinates (x, y, z) by the formulae

x = 1 a r sin θ cos φ, y = 1 b r sin θ sin φ, z = 1 c r cos θ, r 2 = a x 2 + b y 2 + c z 2. {\displaystyle {\begin{aligned}x&={\frac {1}{\sqrt {a}}}r\sin \theta \,\cos \varphi,\\y&={\frac {1}{\sqrt {b}}}r\sin \theta \,\sin \varphi,\\z&={\frac {1}{\sqrt {c}}}r\cos \theta,\\r^{2}&=ax^{2}+by^{2}+cz^{2}.\end{aligned}}}

An infinitesimal volume element is given by

d V = | ∂ (x, y, z) ∂ (r, θ, φ) | d r d θ d φ = 1 a b c r 2 sin θ d r d θ d φ = 1 a b c r 2 d r d Ω. {\displaystyle \mathrm {d} V=\left|{\frac {\partial (x,y,z)}{\partial (r,\theta,\varphi)}}\right|\,dr\,d\theta \,d\varphi ={\frac {1}{\sqrt {abc}}}r^{2}\sin \theta \,\mathrm {d} r\,\mathrm {d} \theta \,\mathrm {d} \varphi ={\frac {1}{\sqrt {abc}}}r^{2}\,\mathrm {d} r\,\mathrm {d} \Omega.}

The square-root factor comes from the property of the determinant that allows a constant to be pulled out from a column:

| k a b c k d e f k g h i | = k | a b c d e f g h i |. {\displaystyle {\begin{vmatrix}ka&b&c\\kd&e&f\\kg&h&i\end{vmatrix}}=k{\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}}.}