How do you derive the differential volume element in spherical coordinates d(tau) from Cartesian?

Questions
1

Dear Community:

How does one derive the d(tau), which is r^2*[Sin(theta)]d(r)d(theta)d(phi) from d(x)d(y)d(z)?

Thank you,
Patrick Merrill

2

Here is a way:

D[FromSphericalCoordinates[{r, \[Theta], \[CurlyPhi]}],
   {{r, \[Theta], \[CurlyPhi]}}] // Det // Simplify
3

Faster:

CoordinateChartData["Spherical",
  "VolumeFactor",
 {r, \[Theta], \[CurlyPhi]}]
4

Dear Gianluca,

Perfect!

Thank you!!

Patrick Merrill