🧄Volume Elements in Non-Cartesian Coordinates (Jacobian Method)

The volume element calculation relies universally on the Jacobian determinant $|\operatorname{det}( J )|$, which is the scaling factor that relates the infinitesimal volume in Cartesian space to the infinitesimal product of the new coordinates. The process requires calculating the partial derivatives of the Cartesian coordinates with respect to the new ones, finding the determinant of the resulting matrix, and taking its absolute value. Specifically, for hyperbolic coordinates, the calculation simplifies to $2|v|$, making the volume element $d V=2|v| d u d v$. For parabolic coordinates, the determinant is $-\left(s^2+t^2\right)$, resulting in a positive volume element of $d V=\left(s^2+t^2\right) d t d s$. These results show how the physical area stretching (or shrinking) is dependent on the location specified by the new coordinate values.

Volume Elements in Non-Cartesian Coordinates (Jacobian Method) | Proof and Derivationviadean on Notion