Editing Spherical coordinate system (section)

== Generalization ==
{{see also|Ellipsoidal coordinates}}
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

<math display="block">ax^2 + by^2 + cz^2 = d.</math>

The modified spherical coordinates of a point in P in the ISO convention (i.e. for physics: ''radius'' {{mvar|r}}, ''inclination'' {{mvar|θ}}, ''azimuth'' {{mvar|φ}}) can be obtained from its [[Cartesian coordinate system|Cartesian coordinates]] {{math|(''x'', ''y'', ''z'')}} by the formulae

<math display="block">\begin{align}
 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{align}</math>

An infinitesimal volume element is given by

<math display="block">
\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.
</math>

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

<math display="block">
\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}.
</math>